LAGRANGIAN INTERSECTION FLOER THEORY - ANOMALY AND OBSTRUCTION -

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono

Typeset by AMS-TEX

CONTENTS

Chapter 1. Introduction. §1 Introduction §2 Review of the Floer homology and outline of the main construction.

Chapter 2. Geometric and Algebraic Constructions. §3 Moduli space of stable maps of open Riemann surfaces. §4 Obstruction classes. §5 Boundary operators. §6 Calculation of the Floer homology; spectral sequence. 6.1 Behaviour of G_{ℓ_{0}}(L_{0}, L_{1}) under the Hamiltonian isotopy. 6.2 Calculation of HF (L; Φ): the Bott-Morse case. §7 Obstruction class of the top dimension.

Chapter 3. Applications. §8 Estimate of the number of intersection points. §9 Automorphism groups of marked discs. §10 Application to the Maslov class. §11 Fixed points of anti-holomorphic involutions. 11.1 Orientation. 11.2 Arnold-Givental conjecture.

Chapter 4. Algebraic Deformation of Lagrangian Submanifolds and Floer Cohomology. §12 Outline of Chapter 4. §13 Filtered A_{∞} algebra. 13.1 Algebraic framework. 13.2 Filtered A_{∞} algebra associated to Lagrangian submanifold. §14 Deformed Floer cohomology. 14.1 Algebraic framework. 14.2 Filtered A_{∞} bimodule structure on C(L, L′; Λ_{nov}). §15 Homotopy equivalence and deformations. 15.1 Algebraic framework. 15.2 Homotopy equivalence and deformations of C(L; Λ_{0,nov}). 15.3 Deformation of the Floer cohomology §16 Clean intersections and Bott-Morse theory.

Chapter 5. Transversality and Gluing. §17 Construction of the Kuranishi structure I. §18 Construction of the Kuranishi structure II. §19 Construction of the fundamental chain. §20 Homotopy unit.

Chapter 6. Orientation. §21 Orientation on the moduli space of unmarked pseudoholomorphic discs. 21.1 The case of holomorphic discs. 21.2 Examples of non-orientable family index. 21.3 The case of connecting orbits in the Floer theory. §22 Convention and preliminaries. §23 Orientation on the singular strata of the moduli space. §24 The obstruction chain is a cycle– Proof of Theorem 4.9. §25 Orientation on the moduli space defining boundary operators. 25.1 Definition of the orientation on M_{1}(β; P_{1}, …, P_{ℓ}). 25.2 Orientation on the moduli space of connecting orbits. 25.3 The Bott-Morse case I. 25.4 A_{∞} algebra case. 25.5 Orientation on the moduli space of constant maps. 25.6 The Bott-Morse case II. 25.7 Orientation on M_{k+1}^{main}(L, j : β; para).

Chapter 7. Lagrangian Surgery. §26 Construction of examples by Lagrangian surgery. 26.1 The case of 4 dimensional flat tori. 26.2 The case of 6 dimensional flat tori. §27 Local model of holomorphic discs in ℂ^{n}. §28 Wall crossing and monodromy §29 Gluing.

Appendices. §A1 Smooth correspondence. §A2 Rerview of Kuranishi structure. §A3 Normally polynomial perturbation and triangulation of the moduli space -the semi positive case - §A4 Construction of the spectral sequence from a continuous family of filtrations. §A5 Independece of the choice of countably generated subcomplex. §A6 Construction of the Kuranishi map.

Chapter 1. Introduction.

§1. Introduction.

Floer homology in symplectic geometry was invented by Floer [Fl1] for an attempt to prove Arnold’s conjecture [A1] for the fixed points of Hamiltonian diffeomorphisms (or exact symplectic diffeomorphisms), i.e., the symplectic diffeomorphisms obtained by (time-dependent) Hamiltonian flow: for any non-degenerate Hamiltonian diffeomorphism ϕ : (M, ω) → (M, ω),

(1.1)  # Fix ϕ ≥ ∑_{k} rank H_{k}(M).

By considering the diagonal Δ ⊂ (M × M, ω ⊕ −ω), which is a Lagrangian submanifold in the product, one can instead ask its intersection theoretic version, which is indeed the original approach taken by Floer in [Fl1].

Let L_{0}, L_{1} ⊂ (M, ω) be two compact Lagrangian submanifolds. What the general Floer theory asks for is to assign a (graded) abelian group, denoted by HF (L_{0}, L_{1}), to each pair (L_{0}, L_{1}) of Lagrangian submanifolds that one would “wish” satisfies the following properties:

(1.2.1) The group is invariant under the Hamiltonian (or exact) diffeomorphisms. Namely if ψ_{i}, i = 0, 1 are two Hamiltonian diffeomorphisms, then

HF_{∗}(L_{0}, L_{1}) ≃ HF_{∗}(ψ_{1}L_{0}, ψ_{2}L_{1}).

(1.2.2) If L_{0} = L_{1} then the Floer homology group coincides with the standard homology group of Lagrangian submanifold L_{0}. Namely HF_{∗}(L, L) ≃ H_{∗}(L; ℤ_{2}).

When L_{0} is transverse to L_{1}, Floer defined HF_{∗}(L_{0}, L_{1}) as the homology of the chain complex

(1.3)  CF_{∗}(L_{0}, L_{1}) = ∑_{p∈L_{0}∩L_{1}} ℤ_{2}[p]

where the matrix element ⟨∂p, q⟩ of the boundary operator ∂ is defined by counting the number of pseudo-holomorphic (cornered) discs with each half of its boundary lying on L_{0} and L_{1} respectively, and with its asymptotic limits given by p, q ∈ L_{0}∩L_{1} (see §2 for more details). The main task to be carried out is to justify this definition and to prove the boundary property ∂∂ = 0 of the operator ∂.

In [Fl1], Floer carried out this analysis, proved the required properties of the boundary map and defined the group HF (L_{0}, L_{1}) when L_{0} = ϕ(L), L_{1} = L with π_{2}(M, L) = 0 for any Hamiltonian diffeomorphism ϕ such that ϕ(L) intersects L transversely. He also proved both (1.2.1) and (1.2.2) for this case. After that, the second named author generalized the construction to the arbitrary pairs (L_{0}, L_{1}) of monotone Lagrangian submanifolds with minimal Maslov number ≥ 3 in [Oh2,3] (See §2 for its definition). (He also proved (1.2.1), but observed that (1.2.2) does not hold in general.) Both authors use ℤ_{2}-coefficients. At first sight, these conditions look rather restrictive and technical especially when compared to the development of the other version of Floer homology, Floer homology for Hamiltonian diffeomorphisms, and of Arnold’s conjecture (1.1).

Namely, for diffeomorphisms, the following result has been completely established by now: For any Hamiltonian diffeomorphism ϕ : M → M, one can associate a graded abelian group, Floer homology HF (M; ϕ), such that

\begin{aligned}HF (M; ϕ) & ≅ H(M) \\ ∑_{k} rank HF_{k}(M; ϕ) & ≤ ♯ Fix(ϕ) \end{aligned}

Here Fix(ϕ) is the fixed point set of ϕ.

In [Fl2], Floer defined HF (M; ϕ) for Hamiltonian diffeomorphisms and proved that HF (M; ϕ) ≅ H(M; ℤ), when (M, ω) is monotone. As a consequence, he proved Arnold’s conjecture for the monotone case. (For this matter, we should mention the earlier result by Conley-Zehnder [CZ] for the case of symplectic tori and by Floer himself for the case where π_{2}(M) = 0 as a consequence of [Fl1]). This was generalized by Hofer-Salamon [HS] and the fourth named author [On] to semipositive symplectic manifolds. The construction of the Floer homology and the proof of Arnold’s conjecture for the general cases have been carried out in [FOn2], [LuT], [Ru] over ℚ, and by the first and fourth named authors in [FOn3] over ℤ. Since HF (M; ϕ) is isomorphic to the homology group H(M) of M, the group itself does not give rise to a new invariant of the symplectic manifold (M, ω). However defining Floer homology itself has application to Arnold’s conjecture in various settings.

On the other hand, we can not expect the theory of Lagrangian intersection Floer homology would completely resemble that of diffeomorphisms. Firstly, Lagrangian intersection Floer homology concerns pairs (L_{0}, L_{1}) of Lagrangian submanifolds, but cannot be defined for completely arbitrary pairs. Secondly, unlike the case of diffeomorphisms, the relevant moduli spaces are neither (stably almost) complex spaces nor deformable to them and so do not carry canonical orientations. Providing an orientation on the moduli space of (pseudo-)holomorphic discs (or open Riemann surfaces in general) attaching to a Lagrangian submanifold is not always possible, and is a rather nontrivial problem even if so. This leads to the condition on the spin structure on the Lagrangian submanifold (see Definition below). This orientation problem in the Lagrangian intersection Floer homology was previously studied by the second named author in [Oh7] where he improved Floer’s result [Fl1] to define the ℤ-valued Floer homology for the case of cotangent bundles. Thirdly, even when HF (L_{0}, L_{1}) is defined, the (symplectic) geometric meaning of the invariants HF (L_{0}, L_{1}) becomes less clear. For example let us consider a compact Lagrangian submanifold L_{i} of ℂ^{n} (ℂ^{n} is noncompact but is tame in the sense of Gromov [Gr2] which still allows one to do the Floer theory for compact Lagrangian submanifolds in the same way). Then obviously there exists a Hamiltonian isotopy ϕ_{t} of compact support such that ϕ_{1}(L_{1}) ∩ L_{2} = ∅. This would mean that if one could define a Floer homology of the pair (L_{1}, L_{2}) of Lagrangian submanifolds that is invariant under the Hamiltonian isotopy, then it should become necessarily trivial. Hence it appears from a simple minded thinking that the Floer theory would give nothing interesting in the important case of Lagrangian submanifolds of ℂ^{n}. However there are various results on Lagrangian submanifold of ℂ^{n} which one can prove using the Floer theory, just knowing that it is defined. For example, using the Floer theory, one can prove Gromov’s theorem of nonexistence of exact Lagrangian submanifold [Gr2] and some topological obstruction to the Maslov class of compact Lagrangian embedding in ℂ^{n} as illustrated in [Oh4,5].

So it is an important task to understand when the Floer homology can be defined and when not. As pointed out in [Oh2], if we define the operator ∂ by counting the number of pseudo-holomorphic discs as Floer did in [Fl1], then ∂ will not satisfy the boundary property, i.e., ∂∂ ≠ 0 in general. In physical terms, ∂∂ ≠ 0 might be regarded as the breaking of BRST symmetry by soliton effects (i.e., by the presence of pseudo-holomorphic discs). The word “anomaly” in the sub-title of the present paper indicates this (BRST) symmetry breaking. This means that the framework under which one will construct the Floer homology of Lagrangian submanifolds is more delicate than that of Hamiltonian diffeomorphisms.

The main result that we establish in this book is an obstruction theory to defining the Floer homology of Lagrangian intersections. This obstruction theory we develop has important implications for the application of Floer homology both to the geometry of Lagrangian submanifolds and to the (homological) mirror symmetry [K3]. We now summarize the main result of this obstruction theory in the following Theorem A.

Throughout this paper, unless otherwise stated, we assume that Lagrangian submanifolds L_{i}, i = 0, 1 are relatively spin:

Definition. We call a submanifold L ⊂ M relatively spin if it is orientable and there exists a class st ∈ H^{2}(M; ℤ_{2}) that restricts to the second Stiefel-Whitney class w_{2}(L) of L.

We call a pair of submanifolds (L_{0}, L_{1}) of M relatively spin if they are orientable and there exists a class st ∈ H^{2}(M; ℤ_{2}) that restricts to the second Stiefel-Whitney classes w_{2}(L_{i}) of both of L_{i}, i = 0, 1.

Theorem A. Suppose that a pair of Lagrangian submanifolds L_{i} ⊂ M, i = 0, 1 is relatively spin. Then there exists a sequence of chains o_{2m_{k}}^{k}(L_{i}) with

[o_{2m_{k}}^{k}(L_{i})] ∈ Ker (H_{n−2m_{k}}(L_{i}; ℚ) → H_{n−2m_{k}}(M; ℚ)) ,

for k = 1, 2, ⋯ that have the following properties:

(1.4.1) o_{2m_{k}}^{k}(L_{i}) is defined if [o_{2m_{j}}^{j}(L_{i})] = 0 for every j = 0, 1, …, k − 1 with m_{j} ≠ 0. (Here m_{k} are non-negative integers, with 2m_{k} < n.) (1.4.2) o_{2m_{k}}^{k}(L_{i}) depends on the choice of almost complex structure J and of the chains B_{j}(L_{i}), bounding o_{2m_{j}}^{j}(L_{i}) for j = 0, 1, …, k − 1. (1.4.3) If o_{2m_{k}}^{k}(L_{i}) = 0 for every k, then the Floer homology HF_{∗}((L_{0}, b_{0}), (L_{1}, b_{1})) is defined whose definition depends on J, B_{j}(L_{i}). (Here b_{i} denotes the totality of B_{j}(L_{i}).) (1.4.4) In case L_{0} = L_{1}, the vanishing of o_{2m_{k}}^{k}(L_{i}) for m_{k} ≠ 0 is enough to define the Floer homology. (1.4.5) If L_{0} is transverse to L_{1}, then

♯(L_{0} ∩ L_{1}) ≥ ∑_{k} rank HF_{k}(L_{0}, L_{1}).

The Floer homology in Theorem A is defined over an appropriate Novikov ring which we will discuss later. The relatively spin condition on L_{i} is needed to be able to define an orientation of the moduli space of pseudo-holomorphic discs, which is necessary for our construction of rational fundamental chain of the moduli space. Under certain circumstances as in Chapter 3, we can define Floer homology over ℤ_{2} coefficients without assuming L_{i} to be relatively spin.

In Theorem A, we did not mention how the Floer homology depends on the choice of almost complex structure J and the chains B_{j}’s and other choices involved in the construction. We will describe this dependence in Theorem C below. Before proceeding further, we would like to state the following consequence of Theorems A,C and E combined.

Theorem B. Assume that L is relatively spin and that the natural map H_{∗}(L; ℚ) → H_{∗}(M; ℚ) is injective. Then for any Hamiltonian diffeomorphism ψ : M → M such that L ⋔ ψ(L), we have

♯(L ∩ ψL) ≥ ∑_{k} rank H_{k}(L; ℚ).

We remark that the diagonal Δ ⊂ M × M satisfies the assumption in Theorem B and so the version Arnold’s conjecture established in [FOn2], [LuT], [Ru] for Hamiltonian diffeomorphisms (over ℚ-coefficients) is a special case of Theorem B.

We next describe the dependence of our Floer homology on the various choices involved and on the Hamiltonian isotopy. To describe this dependence precisely in a systematic manner, we need to use the notion of A_{∞} algebra which we will develop in Chapter 4. Here we just state its consequence referring to Chapter 4 for more precise statements.

Theorem C. Let L ⊂ M be relatively spin and a pair (L_{0}, L_{1}) be relatively spin. Then there exists a set M(L, J, Ξ) which remains isomorphic to one another under the change of almost complex structures J, and of other choices involved in the definition, which we write Ξ. We denote the common set by M(L). M(L) enjoys the following properties:

(1.5.1) M(L) is nonempty if and only if the obstruction classes [o_{2m_{j}}^{j}(L)] in Theorem A all vanishes. (1.5.2) Each choice of J, B_{j}(L) j = 1, 2, ⋯ and other data related to transversality, determines an element [b] in M(L). (1.5.3) Floer cohomology HF^{∗}((L_{0}, b_{0}), (L_{1}, b_{1})) may vary depending on the choice of [b_{0}] ∈ M(L_{0}), [b_{1}] ∈ M(L_{1}), but is independent of J and other choices involved. (1.5.4) Let ϕ be a Hamiltonian diffeomorphism of M. Then there exists a canonical isomorphism ϕ_{∗} : M(L) → M(ϕ(L)) which depends only on the embedding L ↦ ϕ(L). (1.5.5) Let Φ_{i} = {ϕ_{i}^{s}}_{0≤s≤1} for i = 0, 1 be two Hamiltonian isotopies of M. Then there exists an isomorphism

(Φ_{0}, Φ_{1})_{∗} : HF^{∗}((L_{0}, b_{0}), (L_{1}, b_{1})) ≃ HF^{∗}((ϕ_{0}^{1}(L_{0}), ϕ_{0∗}^{τ}(b_{0})), (ϕ_{1}^{1}(L_{1}), ϕ_{1∗}^{τ}(b_{1}))).

depending only on the homotopy classes of Hamiltonian isotopies Φ_{i} with the endpoints fixed.

The space M(L) can be described as the zero set of some formal map.

Theorem D. There exists a formal map of power series

s = ∑_{i=1}^{∞} s_{i}T^{λ_{i}}

with the following properties. (Here T is a formal parameter.)

(1.6.1) s_{i} : H^{odd}(L; ℚ) → H^{even}(L; ℚ) is a formal map for each i, in the following sense:

Let m = rank H^{odd}(L; ℚ). We fix a basis of H^{odd}(L; ℚ) and denote by (a_{1}, ⋯, a_{m}) the coordinates of an element of H^{odd}(L; ℚ) with respect to the basis, where a_{i} ∈ ℚ. We put m′ = rank H^{even}(L; ℚ) and denote elements of H^{even}(L; ℚ) in a similar way.

Then we have s_{i} = (s_{i,1}, ⋯, s_{i,m′}) where

s_{i,j} = ∑_{k_{1}=0}^{∞} ⋯ ∑_{k_{m}=0}^{∞} s_{i,j;k_{1},··· ,k_{m}}a_{1}^{k_{1}} ⋯ a_{m}^{k_{m}}

is a formal power series with coefficients s_{i,j;k_{1},··· ,k_{m}} in ℚ. (1.6.2) λ_{i} ∈ ℝ, λ_{i} > 0, lim_{i} λ_{i} = ∞.

(1.6.3) Let M̃(L) be the set of all formal sums

∑_{i=1}^{∞} b_{i}T^{λ′_{i}}

such that (1.6.3.1) b_{i} ∈ H^{odd}(L; ℚ). (1.6.3.2) λ′_{i} ∈ ℝ, λ′_{i} > 0, lim_{i→∞} λ′_{i} = ∞. (1.6.3.3)

∑_{i} s_{i} (∑_{j} b_{j}T^{λ′_{j}} ) T^{λ_{i}} = 0.

Then there exists a natural surjective map : M̃(L) → M(L).

Let us explain Formula (1.6.3.3). Using the basis in (1.6.1) we write b_{j} = (b_{j,1}, ⋯, b_{j,m}). Then using the coefficient s_{i,j;k_{1},··· ,k_{m}} we define the ℓ-th component of left hand side of (1.6.3.3) by

∑_{k_{1}=0}^{∞} ⋯ ∑_{k_{m}=0}^{∞} s_{i,ℓ;k_{1},··· ,k_{m}} (∑_{j} b_{j,1}T^{λ′_{j}} )^{k_{1}} ⋯ (∑_{j} b_{j,m}T^{λ′_{j}} )^{k_{m}} ,

which is well defined as a “formal power series” since λ′_{j} > 0. (Note the “formal power series” here means the infinite sum ∑ c_{i}T^{λ′_{i}} such that c_{i} ∈ ℚ and λ′_{i} satisfies (1.6.3.2).)

Now we describe a means of possible computation of the HF ((L, b), (L, b)). Recall that when π_{2}(M, L) = 0, Floer proved HF (L, L) ≃ H(L) (This is analogous to the statement that the Floer homology of Hamiltonian diffeomorphism of M is isomorphic to the homology of M itself). As was pointed out in [Oh4], HF (L, L) ≃ H(L) is no longer true even in the case of monotone Lagrangian submanifolds (Consider any monotone Lagrangian embedding in ℂ^{n}, for example). In the monotone case [Oh4,5], a spectral sequence relating the cohomology group H(L; ℚ) of L and the Floer cohomology HF (L, L) was constructed. We now generalize this spectral sequence to our more general situations. We use the Novikov ring [N] as a coefficient ring of Floer homology. More precisely our Floer homology is defined over the universal Novikov rings Λ_{nov} or Λ_{0,nov}. They are filtered rings and the filtration is parameterized by the real number. Namely we have filtrations F^{λ}Λ_{nov} parameterized by λ ∈ ℝ such that

F^{λ}Λ_{nov} ⊆ F^{λ′}Λ_{nov}

and similarly for F^{λ}Λ_{0,nov} but this time λ parameterized by λ ≥ 0. See §6 for a precise definition of the universal Novikov ring. To construct a spectral sequence we need to use a filtration parameterized by the integer rather than the real number. So we take λ_{0} > 0 and define 𝔉^{n}Λ_{nov} = F^{nλ_{0}}Λ_{nov}, 𝔉^{n}Λ_{0,nov} = F^{nλ_{0}}Λ_{0,nov}. (See §A4.) We then take its associated graded ring gr_{∗}(𝔉^{∗}Λ_{nov}), gr_{∗}(𝔉^{∗}Λ_{0,nov}). Namely gr_{q}(𝔉^{∗}Λ_{nov}) ≃ 𝔉^{q}Λ_{nov}/𝔉^{q+1}Λ_{nov} etc.

Theorem E. Let L ⊂ M be relatively spin. Then for each b ∈ M(L), there exists a spectral sequence E_{∗} such that

(1.7.1) E_{2}^{p,q} ≃ (H^{∗}(L; ℚ) ⊗ gr_{q}(𝔉^{∗}Λ_{nov}))^{p}. Here ()^{p} means the total degree p part. (1.7.2) There exists a filtration 𝔉^{∗}HF ((L, b), (L, b)) of HF ((L, b), (L, b)) such that

E_{∞}^{p,q} ≃ (𝔉^{q}HF^{∗}((L, b), (L, b))/𝔉^{q+1}HF^{∗}((L, b), (L, b)))^{p}.

(1.7.3) The image of δ_{2} : E_{2} → E_{2} is contained in

P D(Ker : H_{∗}(L; ℚ) → H_{∗}(M; ℚ)) ⊗ gr_{q}(𝔉^{∗}Λ_{nov})

(Here P D denotes the Poincaré duality.) The image of δ_{k} : E_{k} → E_{k}, k ≥ 2 also is contained is in the subgroup induced by P D(Ker : H_{∗}(L; ℚ) → H_{∗}(M; ℚ)) ⊗ gr_{q}(𝔉^{∗}Λ_{nov}). (1.7.4) Suppose that the Maslov index of (M, L) vanishes. Then P D([L]) ∈ H^{0}(L; ℚ) ≃ E_{2}, and P D([pt]) ∈ H^{n}(L; ℚ) ≃ E_{2} survives and is linearly independent in E_{∞}. In particular HF ((L, b), (L, b)) ≠ 0.

We remark that we use the universal Novikov ring in place of usual Novikov ring (which have been used in the literature). Although this is not essential, we introduce the universal Novikov ring because we want to use the same coefficient ring throughout our discussion. We can work with the usual Novikov ring for our applications in this paper.

We would also like to remark that the Maslov index vanishes for any special Lagrangian submanifolds in the Calabi-Yau space and so (1.7.4) applies to those special Lagrangian submanifolds. We will now state some applications of these results. The following is a more precise version of Theorem B.

Theorem F. Let L be relatively spin and assume that M(L) is non-empty. Denote A = ∑ rank H(L; ℚ) and B = ∑ rank ker(H(L; ℚ) → H(M; ℚ)). Then we have

♯(L ∩ ψ(L)) ≥ A − 2B

for any Hamiltonian diffeomorphism ψ : M → M such that L ⋔ ψ(L).

Next application concerns the Maslov class μ_{L} ∈ H^{1}(L; ℤ) of Lagrangian embedding L ⊂ ℂ^{n} (see [A2] for its definition). The general folklore conjecture says that this Maslov class is non-trivial for any compact Lagrangian embedding in ℂ^{n}. There have been several partial results towards this conjecture (see [Vi2], [Po2], [Oh5]). Here we add another:

Theorem G. Let L be a compact embedded Lagrangian submanifold of ℂ^{n} that satisfies H^{2}(L; ℤ_{2}) = 0. Then its Maslov class μ_{L} ∈ H^{1}(L; ℤ) is nonzero.

As mentioned before, Floer homology of Hamiltonian diffeomorphisms was successfully applied to prove the Arnold conjecture (1.1). Similarly one can try to apply the Floer homology of Lagrangian intersections to the counting problem of the number of intersections of two Lagrangian submanifolds. However the most naive statement such as

(1.8)  ♯(L ∩ ϕ(L)) ≥ ∑ rank H_{∗}(L; ℤ_{2})

is not true for general L and general Hamiltonian diffeomorphism ϕ. In this respect, Givental made a conjecture [Gi] that (1.8) is true at least if L is the fixed point set of an anti-symplectic involution. (For example if L is the set of real points of a projective variety M defined over ℝ.) Some partial result on compact Hermitian symmetric spaces was proven before by the second named author in [Oh3] using the idea that (pseudo)holomorphic discs come in pair in this case and so the “quantum effect” would be cancelled away. However a careful analysis on the orientation of the moduli space shows that this cancellation does not happen over ℚ (or over ℤ) but works only over ℤ_{2}-coefficient in general. (See §11, 11.1). Now we prove the following :

Theorem H. Let L = Fix τ be the fixed point set of an anti-symplectic involution τ : (M, ω) → (M, ω) and L be semi-positive. Then the inequality (1.8) holds.

We refer to §10 for the definition of semi-positivity. The reason why we need to assume the semi-positivity is to handle the negative multiple cover problem. We recall that we should use ℤ_{2}-coefficient to have the cancellation of quantum effects in general which forces us to use integral cycles rather than rational cycles. We would like to emphasize that since we use ℤ_{2}-coefficients, we do not have to assume our Lagrangian submanifold is relatively spin. Using the method of [FOn3], we are planning to remove the semi-positivity assumption from Theorem H.

We refer to §26 for some other applications.

Another direction of research to which the result of this paper can be applied is that of the homological mirror symmetry proposal made by Kontsevich [K2,3]. (See [Va] for a translation of this proposal into the physics language which came later.) There are many more things left to be carried out towards this direction. We would like to explain the relation of our work to the homological mirror symmetry very briefly here (See also [Fu7,8] for more detailed explanation).

Let (M, ω) be a symplectic manifold. Its mirror will be a complex manifold (M^{†}, J_{M^{†}}). (Not all symplectic manifold has its mirror, though.) The part of mirror symmetry closely related to this paper can be summarized into “Conjecture I” below. To state it, we first remark that we can attach a flat line bundle 𝔏 to the Lagrangian submanifold L in the story of Floer homology and define the Floer homology HF ((L_{1}, 𝔏_{1}), (L_{2}, 𝔏_{2})) with local coefficients (See [Fu2,8]). (More precisely, we need to include b_{i} ∈ M(L_{i}, 𝔏_{i}) in the definitions of M(L, 𝔏) and HF ((L_{1}, 𝔏_{1}, b_{1}), (L_{2}, 𝔏_{2}, b_{2})).)

“Conjecture I”. For each pair (L, 𝔏) of relatively spin Lagrangian submanifold L ⊂ M and a flat line bundle 𝔏 on L, there exists an object E(L, 𝔏) of the derived category of the coherent sheaves on M^{†} such that

HF ((L_{1}, 𝔏_{1}), (L_{2}, 𝔏_{2})) ≃ Ext(E(L_{1}, 𝔏_{1}), E(L_{2}, 𝔏_{2}))

for the pair of such (L_{i}, 𝔏_{i}).

We put the conjecture in the quote since not all pairs (L, 𝔏) would have its mirror E(L, 𝔏). This point is where the homological mirror symmetry is related to the obstruction theory that we develop in this paper. As we mentioned before, we have the moduli space M(L, 𝔏) of the data b which parameterizes the Floer homology or deformations of the A_{∞} algebra associated to each (unobstructed) Lagrangian submanifold. Theorem D implies that M(L, 𝔏) is a quotient space of the zero point set of the (formal) map s : H^{odd}(L; ℂ) → H^{even}(L; ℂ). Here we mean by a formal map a formal power series. Among the pieces of H^{odd}(L; ℂ), the first cohomology group H^{1}(L; ℝ) is the parameter space describing deformations of the Lagrangian submanifold up to Hamiltonian isotopy. H^{1}(L; \sqrt{−1}ℝ) is regarded as the parameter space describing deformations of 𝔏, the flat line bundle on L. The other pieces of H^{odd}(L; ℂ) do not have such geometric interpretation. We regard them as the extended directions on the extended moduli space of Lagrangian submanifolds.

Let [b_{0}] ∈ M(L, 𝔏). We assume that the Floer homology HF ((L, 𝔏, b_{0}), (L, 𝔏, b_{0})) is torsion free (over the universal Novikov ring Λ_{0,nov} we introduce later). It follows from the spectral in Theorem E that

HF ((L, 𝔏, b_{0}), (L, 𝔏, b_{0})) ≅ H(L; ℚ) ⊗ Λ_{0,nov}.

In particular Floer homology is independent of b_{0} such that HF ((L, 𝔏, b_{0}), (L, 𝔏, b_{0})) is torsion free. By Theorem D we have a formal map

(1.9)  s : HF^{odd}((L, 𝔏, b_{0}), (L, 𝔏, b_{0})) → HF^{even}((L, 𝔏, b_{0}), (L, 𝔏, b_{0}))

such that its zero set is M̃(L, 𝔏). (M̃(L, 𝔏) is independent of b_{0}.)

One the other hand, let E be a coherent sheaf on M^{†}. Then the Kuranishi theory of deformations of the complex structure on the vector bundle provides the Kuranishi map

s : Ext^{1}(E; E) → Ext^{2}(E; E).

Such that the neighborhood of E in the moduli stack of the isomorphism classes of coherent sheaves is isomorphic to s^{−1}(0)/ Aut(E).

One can also extend this map to other extended directions to define

(1.10)  s : Ext^{odd}(E; E) → Ext^{even}(E; E).

The zero set of this extended Kuranishi map describes the neighborhood of E in the (extended) moduli space of the objects of the derived category of coherent sheaves, that is the homotopy equivalence classes of ℤ_{2}-graded complexes.

Here is a more precise form of “Conjecture I”.

Conjecture J. Let (L, 𝔏) be as above. Then (1.11.1) The formal map (1.9) is homotopic to a map converging in a neighborhood of the zero. We denote by M̃_{conv}(L, 𝔏) the zero set of this map. (1.11.2) The convergent power series version of the Floer cohomology HF_{conv}^{∗}((L, 𝔏, b), (L, 𝔏, b)) is defined for each b ∈ M̃_{conv}(L, 𝔏) sufficiently close to zero, and has properties similar to the usual Floer homology. (1.11.3) The convergent power series version of the moduli space M_{conv}(L, 𝔏) is also defined. There exists an automorphism group Aut(L, 𝔏) acting on M̃_{conv}(L, 𝔏) such that M_{conv}(L, 𝔏) = M̃_{conv}(L, 𝔏)/ Aut(L, 𝔏). (1.11.4) If b ∈ M_{conv}(L, 𝔏), then there exists E = E(L, 𝔏, b), an object of derived category of M^{†}, such that

(1.12)  Ext^{∗}(E; E) ≃ HF_{conv}^{∗}((L, 𝔏, b), (L, 𝔏, b)).

(1.11.5) The convergent power series version of the obstruction map (1.9) induces a map HF_{conv}^{odd}(L, 𝔏, b) → HF_{conv}^{even}(L, 𝔏, b) for each b ∈ M̃_{conv}(L, 𝔏) sufficently close to 0. It coincides with the Kuranishi map (1.10) through the isomorphism (1.12). (1.11.6) Aut(E(L, 𝔏, b)) is isomorphic to the isotropy group

I_{b} = {g ∈ Aut(L, 𝔏) | g · b = b}.

The isomorphism of zero in (1.11.5) is equivariant. (1.11.7) The q-deformed cup product on HF_{conv}^{∗}((L, 𝔏, b), (L, 𝔏, b)), which is a convergent power series version of one defined in Chapter 4, is well-defined and coincides with the Yoneda product on Ext^{∗}(E; E) through the isomorphism (1.12). The q-deformed (higher) Massey product on HF_{conv}^{∗}((L, 𝔏, b), (L, 𝔏, b)) coincides with the (higher) Massey-Yoneda product on Ext^{∗}(E; E).

(Actually (1.11.7) will follow from (1.11.5),(1.11.6).) Conjecture J predicts coincidence of the two moduli spaces of the pairs (L, 𝔏) on M and of the objects E in the derived category on M^{†}. Using the terminology of string theory, it would mean coincidence of the moduli spaces of two different kinds of branes. In this way, Conjecture J is closely related to the geometric mirror symmetry conjecture by Strominger-Yau-Zaslow [SYZ].

In Conjecture J, we consider the case with one Lagrangian submanifold. One can generalize this to the cases with several Lagrangian submanifolds using the notion of A_{∞} category introduced by the first named author in [Fu1], instead of A_{∞} algebra. (In fact to correct some flaw in [Fu1] (which was pointed out by Kontsevich [K3], noting the result from the second named author’s papers [Oh2]), one needs to use the idea from this paper. We plan to discuss it elsewhere.)

Construction of the Floer homology and the product structure on HF^{∗}((L, 𝔏, b), (L, 𝔏, b)) mentioned above uses the moduli space of pseudo-holomorphic maps from the disc D^{2}. We may also consider pseudo-holomorphic maps from the open Riemann surface of higher genus, and consider more general flat bundle than a flat line bundle on L. If we perform these constructions for the zero section of the cotangent bundle of a 3-manifold, this is expected to coincide with the Chern-Simons perturbation theory, [AS1,2], [Bar], [GMM], [K1]. This fact was pointed out by Witten in [W]. See also [Fu2], [FOh]. Its mirror would be a version of the quantum Kodaira-Spencer theory by Bershadsky, Ceccoti, Ooguri, and Vafa [BCOV], more precisely, its version for the moduli space of coherent sheaves. In [BCOV], the quantum Kodaira-Spencer theory of the moduli space of complex structure on a manifold was described. Unfortunately, stating a conjecture of this coincidence in the mathematical level of rigor like in Conjecture J is not possible for us yet.

One might, however, be able to say something rigorous in the genus one case. Namely in the symplectic side, the genus one part would correspond to Hutchings’s invariant [Hu], [HL] together with quantum effects (related to the number of pseudoholomorphic annuli). It is a kind of Reidemeister torsion. (See also [Fu4].) In the complex side, it would correspond to the analytic torsion of \overline{∂} operator with Hom(E, E) as its coefficient. (The relation of the analytic torsion of \overline{∂} operator to the Quillen metric on the moduli space of coherent sheaves etc., has been discussed by various people, for example by Bismut [Bi].) A first step towards the higher genus analogue to Conjecture J would be to prove their coincidence for the genus one case. We hope to discuss this elsewhere.

Now a brief description of each chapter and section comes in order. In §2, we first review the Floer theory of Lagrangian submanifolds due to Floer [Fl1] and second named author [Oh2,5], and point out where their arguments fail in the general situation. We then describe the first obstruction and the first step towards the idea of modifying Floer’s boundary operator by adding correction terms.

In Chapter 2, we give a basic part of the proof of Theorems A and E. In §3, we define the moduli space of pseudo-holomorphic discs bounding the given Lagrangian submanifold L. For its compactification, we generalize the notion of stable maps introduced in [K4], [KM] to the case of open Riemann surfaces. This moduli space plays a central role in this book. Using this, we define the obstruction cycle o_{2m_{k}}^{k}(L). In §4 we explain how to use the chains B_{k}(L) bounding o_{2m_{k}}^{k}(L) to modify Floer’s boundary operator so that ∂∂ = 0 will become satisfied. §§6,7 are devoted to the proof of Theorem E. In this chapter, we postpone two technical but essential parts of the proof, the problems of orientation and transversality, to later chapters.

In Chapter 3, we prove Theorems F, G and H. Besides the results from Chapter 2, we need some additional arguments to work over ℤ_{2}-coefficients. For the applications in this chapter, we want go as far as possible with ℤ_{2}-coefficients because it provides the strongest results. We need to work over ℤ_{2}-coefficients partly because the moduli space of pseudo-holomorphic discs is not always oriented as in the case of Theorem F where we do not assume L to be oriented and to be spin, and partly because, as we already mentioned before, vanishing of the obstruction class under the assumption of Theorem G works only over ℤ_{2}-coefficients in general.

In Chapter 4, we prove Theorem C. This chapter is the central chapter of this book. We refer to §12 for the summary of this chapter. In §13 we introduce the notion of A_{∞} algebra and associate an A_{∞} algebra (C(L), 𝔪_{∗}) to each Lagrangian submanifold L (which is relatively spin). We next give a (algebraic) construction of the moduli space M̂(C) associated to each A_{∞} algebra C. In the case of A_{∞} algebra (C(L), 𝔪_{∗}) that we define for the Lagrangian submanifold L, the moduli space M(C(L), 𝔪_{∗}) is precisely M(L) in Theorem C. In §14, we construct the Floer homology HF_{∗}((L_{0}, b_{0}), (L_{1}, b_{1})) that appeared in Theorem C. In §15, we define the notion of homotopy equivalence between two A_{∞} algebras and show that the homotopy type of A_{∞} algebra (C(L), 𝔪_{∗}) depends only on the Hamiltonian isotopy class of L’s but not on J or Hamiltonian perturbations of L etc. In §16, we complete the proof of Theorem C.

In Chapter 5, we provide analytic details of our construction of the moduli space of pseudo-holomorphic discs and of its fundamental chain. (We use these throughout this book.) The method of proof is a combination of the one from [FOn2] (where similar analysis was carried out in the context of pseudo-holomorphic spheres) and the one from [FOh] (where the case of cotangent bundle is studied.)

To study the transversality problem, we apply the notion of Kuranishi structure. A space with Kuranishi structure is the C^{∞} analog of schemes or stacks in algebraic geometry. It was developed in [FOn2] to handle the transversality problem in general on the moduli space of pseudo-holomorphic curves. Each stratum of the compactified moduli space of pseudo-holomorphic discs, corresponds to the moduli space of pseudo-holomorphic maps from a singular open Riemann surface with a fixed combinatorial type of singularity. Such a moduli space is obtained as a fiber product of moduli spaces of pseudo-holomorphic discs (without singularities). Also we use the fiber product of various moduli spaces to prove many identities (especially the A_{∞} formulae) in Chapters 2 and 4.

So in §A2, we review the notion of Kuranishi structure and define the fiber product of spaces with Kuranishi structure. §§17,18 are devoted to the analytic details necessary for the construction of Kuranishi structures on the moduli space of pseudo-holomorphic discs. (This corresponds to Chapter 3 of [FOn2].) For this analysis, we use a natural parameterization and standard singular metrics of the genus zero open Riemann surface with marked points on the boundary, which was introduced in [FOh]. Another point to be worked out is the gluing construction of pseudo-holomorphic maps from the open Riemann surface in the case where the asymptotic boundary condition at the end is degenerate. This belongs to the realm of Bott-Morse theory. We follow [Fu3] on this point.

All necessary analytic details are completed in §§19, 20.

There we need to discuss one more important point which did not appear in [FOn2]. Namely we need to discuss the transversality problem of the moduli space of trivial (constant) maps from the disc. This moduli space will give the classical part (the zeroth-order term) of the operator 𝔪 of the A_{∞} algebra. The classical part of 𝔪_{2} corresponds to the cup product or the intersection pairing. We remark that the cup product is not well-defined in the chain level (because of the transversality problem), although it is well-defined in the homology level. This point is closely related to several basic points of the rational homotopy theory. In §19, we construct an A_{∞} deformation of the cup product, which is defined in the chain level, and discuss its relation to the deformation of the moduli space of pseudo-holomorphic discs.

In §20, we discuss the problem of constructing a unit in our A_{∞} algebra. We start with constructing the homotopy unit in the sense defined there and show that any A_{∞} algebra with a homotopy unit is homotopy equivalent to one with a unit.

Chapter 6 is devoted to another important issue postponed in the earlier chapters, the sign or the orientation of the moduli space of pseudo-holomorphic discs. In §21, we study the moduli space of pseudo-holomorphic discs without marked points. This problem is naturally related to the index theory of family of elliptic operators. In our case of holomorphic discs (or the case of open Riemann surfaces with boundary), we need an index theory of family of elliptic operators with boundary conditions. We will show how the second Stiefel-Whitney class comes into this index theory, and prove that the moduli space of pseudo-holomorphic discs without marked points is orientable if the Lagrangian submanifold is relatively spin. More precisely, we prove the following theorem

Theorem K. The moduli space of pseudo-holomorphic discs is orientable, if L ⊂ (M, ω) is relatively spin Lagrangian submanifold. Furthermore the choice of relative spin structure on L determines an orientation on M(L; β) canonically for all β ∈ π_{2}(P, L) in a coherent way.

We refer to Chapter 6 for the definition of relative spin structure.

To illustrate that this index bundle of the Cauchy-Riemann operator with boundary conditions is not orientable in general, we give an explicit example for which the index bundle (of the Cauchy-Riemann operator on the disc with totally real boundary condition) is not orientable. We would like to remark that the result of this section has been obtained independently by Silva [Sil].

Other sections of Chapter 6 are devoted to defining coherent orientations on various moduli spaces of pseudo-holomorphic discs with respect to the fiber product.

This problem turns out to be an essential point in our construction of the A_{∞}- algebra. In fact, the problem of giving the coherent orientations is closely tied to that of finding a good sign convention for the A_{∞} formulae, and also very much related to the super symmetry. According to Getzler-Jones [GJ] and Deligne [De], there are basically two different kinds of the sign convention for the A_{∞} formulae, which we now briefly recall the differences between them here:

Consider the (singular) chain complex C(L) of L. Let x_{i} ∈ C and its degree be d_{i}.

In the first convention, we shift its degree by one and regard it as an element of degree d_{i} + 1 in the shifted complex. In this convention, the sign change we get when commuting two elements in the product is (d_{1} + 1)(d_{2} + 1).

In the second convention, each element will have two different degrees, one the usual degree in the graded complex and the other the super degree. In this convention, an element x_{i} ∈ C(L) has the usual degree d_{i} coming from the grading and has super degree 1. The sign change we get when we commute two elements in the product is d_{1}d_{2} + 1.

For this matter, Getzler-Jones used the notation 𝔪̃ for the operator with the first sign convention and 𝔪 for the second sign convention. They are related by the map b in pp 259-260 [GJ]. Deligne [De] referred the second convention to Case II, (But his case I in [De] is different from our first convention). Our convention in this book seems to coincides with the first sign convention. (Note that the second sign convention was used in [Fu7].)

From the geometric point of view, these two conventions correspond to two different ways of equipping with orientation on the moduli space of pseudo-holomorphic discs with marked points on the boundary. Thus, §22 - §25 are devoted to a detailed description of the orientation of the various moduli spaces, and to the verification of the A_{∞} formulae with sign in the first convention. We include a thorough account on this orientation problem because it is a complicated and confusing matter. Without thorough analysis, there would be no way to see the correct formulae with sign throughout the book.

The main purpose of Chapter 7 is to illustrate, by some explicit examples, various constructions carried out in this book. We will give an example of the Lagrangian submanifold for which our obstruction is nonzero (Theorem 26.17) and an example (Theorem 26.18) for which there does exist a pseudo-holomorphic disc bounding it but the obstruction class still vanishes in the cohomology level. We also carry out some calculations of the boundary operator.

The examples we discuss in Chapter 7 are obtained by the Lagrangian surgery of affine Lagrangian submanifolds in the flat symplectic torus. The main results that we are based on to study those examples is Theorem 26.2 and the results from [Fu7]: Theorem 26.2 studies the singular perturbation problem of pseudoholomorphic discs, after Lagrangian surgery, out of the moduli space of pseudoholomorphic polygons bounding flat Lagrangian tori which has been described in [Fu7]. The singular perturbation problem needed in the proof of Theorem 26.2 is based on Taubes’ type of gluing techniques. This gluing involves an explicit local model of holomorphic discs in ℂ^{n} as in [FOh]. The detail of this gluing construction is given in §27 and §29. §28 provides an explicit example of one parameter family of Lagrangian submanifolds arising from the Lagrangian surgery that illustrates the wall crossing phenomenon of the moduli space of pseudo-holomorphic discs and the Floer homology.

Remarks on how this book is presented are now on order. In the beginning we provide less rigorous but more intuitive description avoiding highly technical details, but by the end of the book we will achieve complete rigorous statements and proofs of main theorems. So we would like to point out how much rigor will be achieved at different chapters. Section 2 is written only to motivate our construction and is not intended to give precise nor rigorous statements, especially in those put in the quote. In Chapter 2, all the statements and proofs are rigorous except two points: one concerning transversality and the other concerning the signs in various formulae. Completely precise statements of theorems on the moduli spaces that we use require the notion of Kuranishi structure introduced in [FOn2]. However we avoid mentioning this in Chapter 2. After Chapter 2, we use the Kuranishi structure which enables us to make all the statements precise and rigorous. Although some of the proofs are postponed till later chapters, we will explicitly mention it whenever the proof is postponed. Details on the transversality are carried out in Chapter 5 and those on the orientation in Chapter 6. We use these to carry out main construction of the A_{∞} algebra and deformed Floer cohomology of Lagrangian submanifolds in Chapter 4, and provide some applications of our deformed Floer cohomology in Chapter 3. Several of the most delicate points of various proofs are postponed to the appendices, especially to Appendix 3, 4 and 5. Statements and proofs in Chapter 5, 6 and 7 are all precise and rigorous (again except those few points we mention explicitly to postpone to the appendices).

We owe the idea of constructing a series of obstruction classes to M. Kontsevich. We have also learned many things from his papers which are used in an essential way, especially in Chapter 4.

The first named author (K.F.) would like to thank IHES, where he had fruitful discussion with M. Kontsevich related to this project during his visit in 1997. First, third and fourth named authors (K.F., H.O., K.O.) would also like to thank Y.- G. Oh and KIAS for hospitality during their stays there in 1999 and 2000. They are also grateful to ETH, University of Warwick and Johns Hopkins University for hospitality during their visit. During their visit, they benefited much from these institutions providing good atmosphere in which they could focus their attention on this work. The second named author (Y.O.) thanks K. Fukaya and RIMS for the financial support and hospitality during his stay in RIMS. The fourth named author would like to thank IHES for its hospitality during his stay in the summer of 1998.

The first named author (K.F.) is partially supported by Japan Grant-in-Aid for Scientific Research # 09304008 & 12874009, the third named author (H.O.) by # 12640066 & 10640070, and the fourth named author (K.O.) by # 09640095 & 11440015. The second named author (Y.O.) is partially supported by USA NSF grant # DMS-9971446.

Notations.

(1) (L, x): Lagrangian submanifold with the base point x or the connected component of L containing x

(2) Ω(L, L′): the space of parameterized paths between L and L′

(3) Ω(L, L′ : ℓ_{0}) or Ω_{ℓ_{0}}(L, L′): the space of paths with a based path ℓ_{0} or the component of Ω(L, L′) containing ℓ_{0}

(4) Ω̃_{ℓ_{0}}(L, L′): Covering space of Ω_{ℓ_{0}}(L, L′)

(5) A = A_{ℓ_{0}} : Ω̃_{ℓ_{0}}(L, L′) → ℝ: the action functional defined by A([ℓ, w]) = ∫ w^{∗}ω

(6) G(L, x) or G(L): the covering group π_{2}(M, L; x)/ ∼

(7) G(L, L′ : ℓ_{0}) or G_{ℓ_{0}}(L, L′) or G(L, L′): the covering group of Ω̃_{ℓ_{0}}(L, L′) → Ω_{ℓ_{0}}(L, L′)

(8) Cr_{ℓ_{0}}(L, L′) = Crit A_{ℓ_{0}} ⊂ Ω̃_{ℓ_{0}}(L, L′)

(9) CF (L, L′) or CF (L, L′; ℓ_{0}): the free module over Cr_{ℓ_{0}}(L, L′) with the universal Novikov ring as its coefficients

(10) HF (L, L′) or HF (L, L′; ℓ_{0}): the (Floer) homology or cohomology over CF (L, L′; ℓ_{0}) Conventions

(1) The Hamiltonian vector field X_{H} is defined by dH = ω(X_{H}, ·).

(2) An almost complex structure is called tame to ω if the bilinear from ω(·, J·) is positive and compatible to ω if the bilinear form defines a Riemannian metric (which is also bounded at infinity if M is non-compact).

§2. Review of the Floer homology and outline of the main construction.

Our construction of the obstruction classes o_{k}(L) and modification of the standard Floer’s boundary operator ∂ will be based on a careful study of the moduli spaces of pseudo-holomorphic discs which are attached to the given Lagrangian submanifold.

Although our construction and proof of the basic properties of the various moduli spaces that we consider require delicate analysis, they are essentially a combination and modification of those which have already appeared in the literature and so our emphasis will not lie on these technical analytic points. We postpone these analytic details to Chapter 5. In this section and in Chapter 2, we describe the essential geometric and algebraic constructions that we need, pretending that these technical analytic points are already taken care of.

We first recall the construction of Floer homology of monotone Lagrangian pair (L_{0}, L_{1}) and explain a trouble we meet when attempting to generalize it to general Lagrangian submanifolds. We refer to the second named author’s papers [Oh2,4,5] for more details. However we will make clearer about the proper geometric setup needed for our construction of Lagrangian intersection Floer homology over the Novikov rings. This point was not addressed in [Oh2,4,5] or in other previous literature.

After then, we explain the very first step to overcoming the above mentioned trouble.

Let L_{i}, i = 0, 1 be two Lagrangian submanifolds of (M, ω), which are transverse to each other. We do not impose any conditions on the Lagrangian submanifolds L_{i} in the beginning but mention them when needed.

2.1. Covering of the path space Ω(L_{0}, L_{1}; ℓ_{0}) and the Maslov index.

We first choose a connected component from each of L_{0} and L_{1}, which is done by specifying marked points

x_{0} ∈ L_{0}, x_{1} ∈ L_{1}.

We denote the corresponding component by (L_{0}, x_{0}) and (L_{1}, x_{1}).

Then we choose and fix a path ℓ_{0} between x_{0} ∈ L_{0} and x_{1} ∈ L_{1} and consider the component of the path space

Ω(L_{0}, L_{1}) = {ℓ : [0, 1] → M | ℓ(0) ∈ L_{0}, ℓ(1) ∈ L_{1}}

that contains ℓ_{0}. We denote the component either by Ω(L_{0}, L_{1}; ℓ_{0}) or Ω_{ℓ_{0}}(L_{0}, L_{1}). We now describe a certain covering space of the component Ω_{ℓ_{0}}(L_{0}, L_{1}) of Ω(L_{0}, L_{1}). Consider the set of all pairs (ℓ, w) satisfying:

(2.1.1) w(0, ·) = ℓ_{0}, (2.1.2) w(τ, 0) ∈ L_{0}, w(τ, 1) ∈ L_{1} for all 0 ≤ τ ≤ 1, (2.1.3) w(1, ·) = ℓ.

The universal covering space of Ω_{ℓ_{0}}(L_{0}, L_{1}) can be represented by the set of homotopy classes of the pair (ℓ, w). As in the case of Hamiltonian diffeomorphism [HS], we will consider a different kind of regular covering space.

First, we consider any closed loop

c : S^{1} → Ω_{ℓ_{0}}(L_{0}, L_{1})

which will also define a pair of closed loops in L_{0} and L_{1} for t = 0, 1 respectively. Noting that every symplectic vector bundle over S^{1} is trivial, the bundle c^{∗}T M over S^{1} × [0, 1] is symplectically trivial. Therefore any such trivialization defines two closed loops of Lagrangian subspaces

α_{0}, α_{1} : S^{1} → Λ(ℂ^{n})

by

α_{0}(τ) = T_{c(τ,0)}L_{0}, α_{1}(τ) = T_{c(τ,1)}L_{1},

in the trivialization. We fix any such trivialization

Ψ : c^{∗}T M → S^{1} × [0, 1] × ℂ^{n}

and denote by μ_{Ψ}(α_{i}) the Maslov index of the loop α_{i} in the trivialization Ψ. (See [A2], [AG], [Vi1]). One can easily check (See [Lemma 2.8, Oh2-I]) the following

Lemma 2.2. The difference

μ_{Ψ}(α_{1}) − μ_{Ψ}(α_{0})

is independent of the choice of trivialization Ψ but depends only on the loop c.

We denote this common number by μ(c) and call it the Maslov index of the loop c in Ω(L_{0}, L_{1}). It defines an integer valued homomorphism

μ : π_{1}(Ω_{ℓ_{0}}(L_{0}, L_{1}), ℓ_{0}) → ℤ.

Using μ and the symplectic form ω, we define an equivalence relation ∼ on the set of all pairs (ℓ, w) satisfying (2.1). We denote by w#w′ the concatenation of w and w′ along ℓ, which will define a loop in Ω_{ℓ_{0}}(L_{0}, L_{1}) based at ℓ_{0}.

Definition 2.3. We say that (ℓ, w) is equivalent to (ℓ, w′) and write (ℓ, w) ∼ (ℓ, w′) if the following conditions are satisfied

∫_{w} ω = ∫_{w′} ω  i.e.  ∫_{\overline{w}#w′} ω = 0 (2.3.1)

μ(\overline{w}#w′) = 0 (2.3.2)

where \overline{w} is the disc w with the opposite orientation.

We define a covering space of Ω_{ℓ_{0}}(L_{0}, L_{1}) by

Ω̃_{ℓ_{0}}(L_{0}, L_{1}) = {(ℓ, w) | satisfying (2.1)}/ ∼ .

We denote by [ℓ, w] the equivalence class of (ℓ, w).

We would like to note that there is a canonical lifting of ℓ_{0} ∈ Ω_{ℓ_{0}}(L_{0}, L_{1}), [ℓ_{0}, ℓ̃_{0}] ∈ Ω̃_{ℓ_{0}}(L_{0}, L_{1}) where ℓ̃_{0} is the map ℓ̃_{0} : [0, 1]^{2} → M with ℓ̃_{0}(s, t) = ℓ_{0}(t).

2.2. Floer’s complex.

We define a functional A = A_{ℓ_{0}} : Ω̃_{ℓ_{0}}(L_{0}, L_{1}) → ℝ by

(2.4)  A([ℓ, w]) = ∫ w^{∗}ω.

A simple standard calculation shows that the set of critical points of A on Ω̃_{ℓ_{0}}(L_{0}, L_{1}) are those [ℓ_{p}, w] where ℓ_{p} : [0, 1] → M is the constant path corresponding to an intersection point p ∈ L_{0} ∩L_{1}. We denote by Cr_{ℓ_{0}}(L_{0}, L_{1}) the set of all critical points of

A : Ω̃_{ℓ_{0}}(L_{0}, L_{1}) → ℝ.

We next study the gradient lines of A with respect to some L^{2}-type metric. We fix a t-dependent family J = {J_{t}}_{0≤t≤1} compatible almost complex structures on M and consider the induced family of Riemannian metrics g_{J} := ω(·, J·) on M. This will in turn induce an L^{2}-metric on Ω̃_{ℓ_{0}}(L_{0}, L_{1}) by

≪ ξ, η ≫= ∫_{0}^{1} g_{J_{t}}(ξ(t), η(t)) dt.

We now define the moduli space M̃_{J}([ℓ_{p}, w], [ℓ_{q}, w′]) as follows: M̃_{J}([ℓ_{p}, w], [ℓ_{q}, w′]) is the set of maps

u : ℝ × [0, 1] → M

with (2.5.1) u(ℝ × {0}) ⊂ L_{0}, u(ℝ × {1}) ⊂ L_{1}, (2.5.2) u satisfies

\{ \begin{matrix}\frac{∂u}{∂τ} + J(t, u)\frac{∂u}{∂t} = 0 \\ lim_{τ→−∞} u(τ, t) = p, lim_{τ→+∞} u(τ, t) = q \end{matrix}

(2.5.3) w#u ∼ w′.

Here w#u is the obvious concatenation of w and u along the constant path ℓ_{p}.

¿From now on, we will suppress J from various notations whenever possible. We will make the J-dependence explicit only when needed.

We next assign a consistent grading to Cr_{ℓ_{0}}(L_{0}, L_{1}). For this, we also need to fix a section λ^{0} of ℓ_{0}^{∗}ΛM where ΛM is the bundle of Lagrangian Grasmmanians of T M such that

(2.6)  λ^{0}(0) = T_{ℓ_{0}(0)}L_{0}, λ^{0}(1) = T_{ℓ_{0}(1)}L_{1}.

Let (ℓ_{p}, w) ∈ Cr_{ℓ_{0}}(L_{0}, L_{1}) ⊂ Ω̃_{ℓ_{0}}(L_{0}, L_{1}) be an element whose projection corresponds to the intersection point p ∈ L_{0} ∩ L_{1}. Choose any trivialization

Φ : w^{∗}T M → [0, 1]^{2} × ℂ^{n} ≃ ℝ^{2n}

such that (2.7.1) Φ(0, t)(λ^{0}(t)) ≡ ℝ^{n} (2.7.2) Φ(1, 0)(T_{p}L_{0}) = ℝ^{n}, Φ(1, 1)(T_{p}L_{1}) = \sqrt{−1}ℝ^{n}

This Φ will induce a loop λ_{Φ} : ∂[0, 1]^{2} → Λ(ℂ^{n}) defined by

(2.8)  \begin{aligned}λ_{Φ}(s, 0) & = T_{w(s,0)}L_{0}, λ_{Φ}(1, t) = e^{\frac{πit}{2}} ℝ^{n}, \\ λ_{Φ}(s, 1) & = T_{w(s,1)}L_{1} λ_{Φ}(0, t) = ℝ^{n} \end{aligned}

We define the degree of [ℓ_{p}, w] by

(2.9)  μ_{λ_{0}}([ℓ_{p}, w]) = μ(λ_{Φ}).

If follows from Lemma 2.2 that (2.9) does not depend on the trivialization Φ satisfying (2.8). This provides a consistent grading to Cr_{ℓ_{0}}(L_{0}, L_{1}), provided the based path ℓ_{0} and the section λ^{0} of Λ(M) over ℓ_{0} are chosen.

In this section, we will presume the following

“Theorem 2.10” Let μ : Cr_{ℓ_{0}}(L_{0}, L_{1}) → ℤ be the grading defined as above. Then the space M̃([ℓ_{p}, w], [ℓ_{q}, w′]) is a smooth manifold of dimension μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]). We also assume that the pair (L_{0}, L_{1}) is relatively spin. Then the space will carry an orientation.

The statement on the dimension is a consequence of a well-established argument using the gluing formula for indices (if we replace the dimension by the virtual dimension). See Chapter 6 for more precise argument. The statement on the orientation will be proved in Chapter 6. (In Chapter 6, we will specify the precise data on which the orientation depends.) The statement on the transversality implicit in “Theorem 2.10” is in fact false in the standard sense even for generic almost complex structures because of the multiple covering problem. In this section, we presume all of these to ease the exposition of essential geometric constructions leading to our obstruction classes o_{k}(L).

We recall that our J = {J_{t}} does not depend on τ and so there is a natural ℝ-action on M̃([ℓ_{p}, w], [ℓ_{q}, w′]) defined by the translation along the τ-direction. We put

M([ℓ_{p}, w], [ℓ_{q}, w′]) = M̃([ℓ_{p}, w], [ℓ_{q}, w′])/ℝ.

The standard Floer’s cochain complex (CF (L_{0}, L_{1}; ℓ_{0}), δ_{0}) is defined as Definition 2.11

CF^{k}(L_{0}, L_{1}) = CF^{k}(L_{0}, L_{1}; ℓ_{0}) = ˆ⨁_{{[ℓp,w]∈Cr(L_{0},L_{1}) \atop μ([ℓp,w])=k}}ℚ[ℓ_{p}, w]

δ_{0}[ℓ_{p}, w] = ∑_{μ(ℓ_{q},w)=μ(ℓ_{p},w)+1} #(M([ℓ_{p}, w], [ℓ_{q}, w′])[ℓ_{q}, w′].

Here ˆ⊕ means an appropriate completion which will be explained in §5. Refining Floer’s proof in [Fl1], the second named author [Oh2] proved that δ_{0} ∘ δ_{0} = 0 in the case of monotone Lagrangian submanifolds with the minimal Maslov index > 2 with some topological restriction on the pair (L_{0}, L_{1}). As in other Floer theory, it is based on the compactification of the moduli space M([ℓ_{p}, w], [ℓ_{q}, w′]) of dimension 1.

For the case of Hamiltonian diffeomorphism, similar constructions have been used. However, there is a crucial difference for the case of Lagrangian intersections from the case of Hamiltonian diffeomorphisms: The boundary ∂\overline{M}([ℓ_{p}, w], [ℓ_{q}, w′])) consists of more than

⋃_{μ([ℓ_{r},w′′])=μ([ℓ_{p},w])+1} M([ℓ_{p}, w], [ℓ_{r}, w′′]) × M([ℓ_{r}, w′′], [ℓ_{q}, w′]).

More precisely, the compactification of M([ℓ_{p}, w], [ℓ_{q}, w′]) has extra codimension one components other than those of “split connecting orbits”. The extra components come from bubbling-off discs. From the index formula, we know that bubbling-off spheres are phenomena of codimension at least two, while bubbling-off discs is of codimension one in general.

2.3. First Obstruction.

We now need to carefully study those extra codimension one components mentioned above. In this section, we discuss only the simplest case leaving the general case to Chapter 2.

For each given β ∈ π_{2}(M, L; x), we define the moduli space of pseudo-holomorphic discs attached to L.

Definition 2.12. We define

M̃(L, J : β) = M̃((L, x), J : β) = \{w : D^{2} → M | \overline{∂}_{J}w = 0, w(∂D^{2}) ⊂ L, [w] = β\}

and denote by μ_{L} : π_{2}(M, L) → ℤ the standard Maslov index homomorphism (See [Vi1], [Oh2]). We write M̃(β) for M̃(L, J : β) in case no confusion can occur. However we would like to emphasize that the almost complex structure J may vary for different Lagrangian submanifolds L.

“Theorem 2.13” The moduli space M̃(β) is a manifold of dimension n + μ_{L}(β).

Again the dimensional statement is standard [Gr2] and only the transversality is of issue, whose study will be postponed until Chapter 5.

Definition 2.14. We define an equivalence relation by putting β ∼ β′ in π_{2}(M, L) if and only if

[ω](β) = [ω](β′) and μ_{L}(β) = μ_{L}(β′),

and denote G(L) = G(L, x) = π_{2}(M, L; x)/ ∼. For each β ∈ G(L), we put

M̃(β) = ⋃_{β̃:[β̃]=β} M̃(L, J : β̃).

Hereafter, we will use the symbol β also for the elements of G(L).

We remark that the group P SL(2; ℝ) = Aut(D, j_{D}) acts on M̃(β) by φ · w = w ∘ φ^{−1}. We put

M(β) = M̃(β)/P SL(2; ℝ).

Now we denote by G_{ℓ_{0}}(L_{0}, L_{1}) the group of deck transformations of the covering

Ω̃_{ℓ_{0}}(L_{0}, L_{1}) → Ω_{ℓ_{0}}(L_{0}, L_{1}).

It is easy to check that this is a regular covering i.e., Ω̃_{ℓ_{0}}(L_{0}, L_{1})/G_{ℓ_{0}}(L_{0}, L_{1}) ≃ Ω_{ℓ_{0}}(L_{0}, L_{1}) for any fixed ℓ_{0} ∈ Ω_{ℓ_{0}}(L_{0}, L_{1}). From now on, we will omit the subscript ℓ_{0} from Ω̃_{ℓ_{0}}(L_{0}, L_{1}) and G_{ℓ_{0}}(L_{0}, L_{1}) for the rest of this section when there is no danger of confusion.

Note that there exist natural homomorphisms

G(L_{i}) → G(L_{0}, L_{1}), i = 0, 1

which we describe now. We will just describe the case for i = 0. The case for i = 1 is entirely similar. It suffices to find an action of G(L_{0}) on Ω̃(L_{0}, L_{1}) that is compatible with the covering Ω̃(L_{0}, L_{1}) → Ω(L_{0}, L_{1}). For this, let [ℓ, u] ∈ Ω̃(L_{0}, L_{1}) and w_{β} : (D^{2}, ∂D^{2}) → (M, L) with [w_{β}] = β ∈ G(L_{0}). Take any path joining ℓ_{0}(0) and w_{β}(1) and use this path to define a boundary connected sum u#w_{β}. It is easy to see that the ∼ equivalence class of (ℓ, u#w_{β}) is independent of the choice of the path and the representatives w_{β} and u.

Noting that

Cr_{ℓ_{0}}(L_{0}, L_{1})/G_{ℓ_{0}}(L_{0}, L_{1}) ≅ L_{0} ∩ L_{1},

we define homomorphisms

E = E_{ℓ_{0}} : G(L_{0}, L_{1}) → ℝ, μ : G(L_{0}, L_{1}) → ℤ

by

(2.15.1)  E(g) = A(g · [ℓ, w]) − A([ℓ, w])
(2.15.2)  μ(g) = μ(g · [ℓ, w]) − μ([ℓ, w]).

It is easy to see that the right hand sides of (2.15) are independent of [ℓ, w] ∈ Ω̃(L_{0}, L_{1}) and depends only on g ∈ G(L_{0}, L_{1}).

We note that the following diagrams commute

Figure 2.1

respectively. Since the map E × μ : G(L_{0}, L_{1}) → ℝ × ℤ is injective, we conclude that G(L_{0}, L_{1}) is abelian. Similarly G(L_{i}), i = 0, 1 are also abelian.

As before, we fix a “generic” choice J = {J_{t}}_{0≤t≤1} of t-dependent family of almost complex structures connecting two almost complex structures J_{0} and J_{1} which are ‘generic’ for L_{0} and L_{1} respectively.

“Definition 2.16”. Let [ℓ_{p}, w], [ℓ_{q}, w′] ∈ Cr(L_{0}, L_{1}) and β_{(0)} ∈ G(L_{0}) and β_{(1)} ∈ G(L_{1}). For each i = 0, 1, we define the following moduli space

N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{i}, β_{(i)}) = {(u, w_{β_{(i)}}) | u ∈ M([ℓ_{p}, w], [ℓ_{q}, w′′])
w_{β_{(i)}} ∈ M(L_{i}, J_{i}; β_{(i)})
u(ℝ × {i}) ∩ w_{β_{(i)}}(∂D^{2}) ≠ ∅
β_{(i)} · [ℓ_{q}, w′′] = [ℓ_{q}, w′]}.

(This definition is only a provisional one. We will give the precise definition in §5).

Figure 2.2

“Theorem 2.17” N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{i}, β_{(i)}) for both i = 0, 1 are smooth manifolds of dimension μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]) − 2.

“Proof” We mention here only the counting of virtual dimension. We consider only for i = 0. By “Theorem 2.10”, the moduli space M([ℓ_{p}, w], [ℓ_{q}, w′′]) is of dimension μ([ℓ_{q}, w′′]) − μ([ℓ_{p}, w]) − 1. On the other hand, since M(β_{(0)}) is of dimension n + μ(β_{(0)}) − 3, the condition

u(ℝ × {0}) ∩ w_{β_{(0)}}(∂D^{2}) ≠ ∅

changes the dimension of the moduli space by n + μ(β_{(0)}) − 3 + 2 − n = μ(β_{(0)}) − 1. Therefore N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}) is of dimension μ([ℓ_{q}, w′′]) − μ([ℓ_{p}, w]) − 1 + μ(β_{(0)}) − 1. Since we assume β_{(0)} · [ℓ_{q}, w′′] = [ℓ_{q}, w′], we have from (2.15.2),

μ([ℓ_{q}, w′′]) − μ([ℓ_{p}, w]) − 1 + μ(β_{(0)}) − 1 = μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]) − 2

which proves the statement on the dimension. □

Now we have

“Theorem 2.18” We have the identity

∂\overline{M}([ℓ_{p}, w], [ℓ_{q}, w′])
= ⋃_{μ([ℓ_{r},w′′])=μ([ℓ_{q},w′])−1} M([ℓ_{p}, w], [ℓ_{r}, w′′]) × M([ℓ_{r}, w′′], [ℓ_{q}, w′])

+ ⋃_{β_{(0)}} N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}) + ⋃_{β_{(1)}} N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}).

“Proof.” Let u_{i} be a divergent sequence of elements of M([ℓ_{p}, w], [ℓ_{q}, w′]). By Gromov’s compactness theorem, one of the followings must occur in the limit (after taking a subsequence if necessary):

(2.19.1) u_{i} splits into a sum of various connecting orbits. (2.19.2) Bubbling-off spheres occurs at some point of ℝ × [0, 1]. (2.19.3) Bubbling-off discs occurs at some boundary point of ℝ × [0, 1], i.e., at some point in ℝ × {0, 1}.

(2.19.1) corresponds to the first term of “Theorem 2.18” and (2.19.3) corresponds to the second term there. (2.19.2) is a phenomenon of codimension 2 and so does not occur in the boundary of codimension 1. The precise details will be given in Chapter 5. □

By the usual cobordism argument, “Theorem 2.18” implies the identity

(2.20)  \begin{aligned}0 = ⟨δ_{0}δ_{0}[ℓ_{p}, w], [ℓ_{q}, w′]⟩ & + ∑_{β_{(0)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}) \\ & + ∑_{β_{(1)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}) \end{aligned}

where ⟨δ_{0}δ_{0}[ℓ_{p}, w], [ℓ_{q}, w′]⟩ denotes the coefficient of [ℓ_{q}, w′] in δ_{0}δ_{0}[ℓ_{p}, w].

In the case of monotone Lagrangian submanifold with minimal Maslov number > 2 as in [Oh2,4], one can prove that the extra terms in (2.20) become zero by a dimension counting argument. The proof of this is based on the observation [Oh2] that if β_{(i)} ∈ π_{2}(M, L_{i}), μ(β_{(i)}) > 2, then N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{i}, β_{(i)}) = ∅ for a generic choice of J = {J_{t}}_{0≤t≤1} when μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]) = 2. From the point of view of [Oh2], we can re-interpret the argument in [Oh2] in the following way: We consider the following moduli space

M^{∂}(L_{i}; β_{(i)}) = (M̃(L_{i}; β_{(i)}) × S^{1})/P SL(2; ℝ)

and the evaluation map

ev^{∂} : M^{∂}(L_{i}; β_{(i)}) → L_{i}; ev^{∂}([w, t]) = w(t).

Note that the (virtual) dimension of M^{∂}(L_{i}; β_{(i)}) is μ(β_{(i)})+n−2 and so μ(β_{(i)}) > 2 implies that

dim M^{∂}(L_{i}; β_{(i)}) > dim L_{i}.

We remark that the condition that “there exists w_{β_{(i)}} ∈ M̃(L_{i}; β_{(i)}) with u(ℝ × {i}) ∩ w_{β_{(i)}}(∂D^{2}) ≠ ∅” is equivalent to the condition that

u(ℝ × {i}) ∩ ev(M^{∂}(L_{i}; β_{(i)})) ≠ ∅.

This condition (which we impose in Definition 2.17) is degenerate in case when dim M^{∂}(L_{i}; β_{(i)}) > dim L_{i} in that we have

dim(u(ℝ × {i})) ∩ ev(M^{∂}(L_{i}; β_{(i)})) > 0

provided it is non-empty. This is the reason why M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{0}) is of smaller dimension than expected. Therefore N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{0}) becomes empty, whose virtual dimension is 0 for the monotone Lagrangian submanifold with minimal Maslov number > 2.

For the general Lagrangian submanifold, the above dimension counting argument does not apply because the condition u(ℝ × {i}) ∩ ev(M^{∂}(L_{i} : β_{(i)})) ≠ ∅ is not degenerate.

One important idea introduced in this paper is that we impose the homological condition on L_{i} that the chain ev(M^{∂}(L_{i} : β_{(i)})) (which becomes a cycle when β_{(i)} is of the minimal possible symplectic area) is homologous to zero, and then use a chain bounding this cycle to modify the standard boundary map δ_{0} to δ in order to achieve δ ∘ δ = 0.

Suppose that we have a chain B(L_{i}; β_{(i)}) such that

(2.21)  ∂B(L_{i}; β_{(i)})“ = ”(−1)^{n+1}ev(M^{∂}(L_{i}; β_{(i)}))

(In fact it is not the case in general that ev(M^{∂}(L_{0}; β_{0})) defines a cycle, and so we need to modify the chain itself before considering the equation (2.21). See §4.)

Now using this chain, we define

M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)})
“ = ”{u | u ∈ M([ℓ_{p}, w], [ℓ_{q}, w′′]) for some [ℓ_{q}, w′′] ∈ Cr(L_{0}, L_{1}),
u(ℝ × {0}) ∩ B(L_{0}; β_{(0)}) ≠ ∅, β_{(0)} · [ℓ_{q}, w′′] = [ℓ_{q}, w′]}

N ([ℓ_{p}, w], [ℓ_{q}, w′] :L_{0}, β_{(0)} : L_{0}, β′_{(0)})
“ = ”{(u, w_{β′_{(0)}}) | u ∈ M([ℓ_{p}, w], [ℓ_{q}, w′′]) for some [ℓ_{q}, w′′] ∈ Cr(L_{0}, L_{1}),
u(ℝ × {0}) ∩ B(L_{0}; β_{(0)}) ≠ ∅,
u(ℝ × {0}) ∩ w_{β′_{(0)}}(∂D^{2}) ≠ ∅,
(β_{(0)} + β′_{(0)}) · [ℓ_{q}, w′′] = [ℓ_{q}, w′]}

We also define M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}), N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}; L_{1}, β_{(1)}), N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}; L_{0}, β_{(0)}) and N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}; L_{1}, β_{(1)}) in a similar way.

Figure 2.3 (M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}))

Figure 2.4 (N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}; L_{1}, β′_{(1)}))

By the same way as in “Theorem 2.17, 2.18”, we prove

“Theorem 2.22” The moduli spaces M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{i}, β_{(i)}) for i = 0, 1 are smooth manifolds of dimension μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]) − 1. The moduli spaces

N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{i}, β_{(i)}; L_{j}, β_{(j)}) for i, j = 0, 1

are smooth manifolds of dimension μ([ℓ_{q}, w′])−μ([ℓ_{p}, w])−2. Furthermore, we have a compactification \overline{M} of M such that the identity

∂M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)})

= ∑_{[ℓ_{r},w′′]} M([ℓ_{p}, w], [ℓ_{r}, w′′]; L_{0}, β_{(0)}) × M([ℓ_{r}, w′′], [ℓ_{q}, w′])

+ ∑_{[ℓ_{r},w′′]} M([ℓ_{p}, w], [ℓ_{r}, w′′]) × M([ℓ_{p}, w′′], [ℓ_{q}, w′] : L_{0}, β_{(0)})

+ ∑_{β′_{0}} N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}; L_{0}, β′_{(0)})

+ ∑_{β_{1}} N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}; L_{1}, β_{(1)})

is satisfied and similarly so is for ∂M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}).

The “proof” of “Theorem 2.22” is a straightforward generalization of the “proofs” of “Theorems 2.17, 2.18” using the identity (2.21). We omit the proof of this since we will prove more general results in a rigorous way in Chapters 2 and 5.

We now “define” the first correction term

(2.23) \begin{aligned}δ_{1}[ℓ_{p}, w] & = ∑_{μ([ℓ_{q},w′])=μ([ℓ_{p},w])+1} ∑_{β_{(0)}} #M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)})[ℓ_{q}, w′] \\ & + ∑ ∑_{β_{(1)}} #M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)})[ℓ_{q}, w′]. \end{aligned}

Then “Theorems 2.18 and 2.22” imply

(2.24)  \begin{aligned}0 & = ⟨(δ_{0} + δ_{1})(δ_{0} + δ_{1})[ℓ_{p}, w], [ℓ_{q}, w′]⟩ \\ & + ∑_{β_{(0)},β′_{(0)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}; L_{0}, β′_{(0)}) \\ & + ∑_{β_{(0)},β_{(1)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}; L_{1}, β_{(1)}) \\ & + ∑_{β_{(1)},β_{(0)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}; L_{0}, β_{(0)}) \\ & + ∑_{β_{(1)},β′_{(1)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)}; L_{1}, β′_{(1)}). \end{aligned}

Roughly speaking, the error terms after the first line in (2.24) are of the “second order”, while the terms in

∑_{β_{(0)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}) + ∑_{β_{(1)}} #N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{(1)})

in (2.20) is of “first order”.

We can repeat the similar constructions to define higher order correction terms δ_{i}, under the assumption that the earlier obstruction cycles vanish. We will prove in Chapter 2 that

δ = ∑_{i} δ_{i}

satisfies the property δ ∘ δ = 0.

Remark 2.25. Note that the based path ℓ_{0} plays an important role in our construction. The point of view that our Lagrangian intersection theory should be about the based space of paths, rather than the space of paths will be also important in defining the chain homomorphism under the Hamiltonian isotopy of Lagrangian submanifolds (See §15). The choice of section λ^{0} of ℓ_{0}^{∗}ΛM is also needed to provide a grading of the Floer complex and to define a degree preserving chain homomorphism between the Floer homology over the Novikov rings under the Hamiltonian isotopy. We refer to §15 for more explanations on this point.

On the other hand, for two given pairs (ℓ_{0}, λ^{0}) (ℓ′_{0}, λ^{′0}) such that ℓ_{0} is homotopic to ℓ′_{0}, a homotopy Λ = {ℓ_{s}}_{0≤s≤1} between them induces a natural chain map

\begin{matrix}h_{Λ} : CF (L_{0}, L_{1}; ℓ_{0}) → CF (L_{0}, L_{1}; ℓ′_{0}); \\ h_{Λ}([ℓ_{p}, w]) = T^{A_{Λ}}e^{n_{Λ}}[ℓ_{p}, w′]; w ≃ ℓ_{Λ}#w′ \end{matrix}

where

A_{Λ} = ∫ Λ^{∗}ω, n_{Λ} = μ(λ_{Λ}).

Here μ(λ_{Λ}) is the Maslov index of the obvious boundary Lagrangian loop λ_{Λ} over the bundle Λ^{∗}T M → [0, 1]^{2}, which can be seen to be well-defined. This chain map is an isomorphism which preserves both filtration and grading of the complex and can be shown to be independent of the choice of homotopy Λ. In this sense our construction depends only on the homotopy class of the based path ℓ_{0}.

Chapter 2. Algebraic and Geometric Constructions.

§3. Moduli space of stable maps of open Riemann surfaces.

In this section, we further generalize the notion of stable maps of (open) Riemann surfaces that is introduced in [K4], [KM].

We first recall that a marked semi-stable curve (S, z⃗) is a pair of (complex) one dimensional variety S and z⃗ = (z_{1}, ⋯, z_{m}), z_{i} ∈ S and z_{i} ≠ z_{j} for i ≠ j where S has at worst ordinary double points as singular points and all z_{i}’s are away from double points. A marked semi-stable curve (S, z⃗) is stable if and only if the automorphism group of (S, z⃗) is finite (See [KM] for example).

Definition 3.1. A real marked semi-stable curve is a system

(S, τ, z⃗, z⃗^{±}) = (S, τ, (z_{1}, …, z_{m}), ({z_{1}^{+}, z_{1}^{−}}, …, {z_{m′}^{+}, z_{m′}^{−}}))

where S is a semi-stable curve (without boundary) and τ : S → S is an antiholomorphic involution such that

\begin{matrix}τ(z_{i}) = z_{i} for i = 1, …, m \\ τ(z_{i}^{+}) = z_{i}^{−} for i = 1, …, m′ \end{matrix}

and that z_{1}, …, z_{m}, z_{1}^{+}, …, z_{m′}^{+}, z_{1}^{−}, ⋯, z_{m′}^{−} are all distinct and are nonsingular.

Note that (S, τ, z⃗, z⃗^{±}) is unchanged when we interchange z_{i}^{+} and z_{i}^{−} since the pairs {z_{i}^{+}, z_{i}^{−}} are assumed to be unordered.

We define the genus of a real marked semi-stable curve (S, τ, z⃗, z⃗^{±}) to be the genus of S. We denote by M_{g,m,m′}^{ℝ} the set of all isomorphism classes of real marked stable curve of genus g and with (m, m′) marked points. For each (S, τ, z⃗, z⃗^{±}) ∈ M_{g,m,m′}^{ℝ}, we consider the set of real points of τ,

S^{τ} = {z ∈ S | τz = z}.

We note that if S^{τ} is non empty, it is a union of finitely many circles glued at finitely many points.

Definition 3.2. A marked semi-stable curve with boundary of genus g and with (m, m′) marked points is a real marked semi-stable curve of genus g and with (m, m′) marked points together with an orientation ori of S^{τ}, assuming that S^{τ} is non empty.

In case g ≥ 2 (which we do not use in this paper), we need two additional assumptions on ori which we will describe below.

We call a marked open semi-stable curve (Σ, τ, z⃗, z⃗^{±}, ori) stable if the corresponding semi-stable curve (Σ, (z_{1}, …, z_{m}, z_{1}^{+}, z_{1}^{−}, …, z_{m′}^{+}, z_{m′}^{−})) is stable.

For each marked semi-stable open curve (S, τ, z⃗, z⃗^{±} , ori), we select a closed subset Σ ⊂ S with ∂Σ = S^{τ} in the following way: Let S_{i} ⊂ S be an irreducible component with S_{i} ∩ S^{τ} ≠ ∅. The set S_{i}\S^{τ} will have two connected components. We select one of the components, denoted by Σ_{i}, so that the induced orientation on ∂Σ_{i} = S^{τ} ∩ S_{i} coincides with the given orientation, ori. In case when S^{τ} ∩ S_{i} is connected (which is always the case when g = 0), the existence of Σ_{i} will be automatic. Otherwise, we impose the existence of Σ_{i} as a part of the definition of semi-stable curves with boundary.

We take Σ_{0}, a union of components of S\ ⋃_{i:S_{i}∩S^{τ} =∅} \overline{Σ_{i}}, so that the set Σ_{0} ∪ ⋃_{i:S_{i}∩S^{τ} =∅} Σ_{i} is connected, Σ_{0} ∩ τ(Σ_{0}) = ∅ and that the set

(Σ_{0} ∪ ⋃_{i:S_{i}∩S^{τ} =∅} Σ_{i}) ∪ (τ(Σ_{0}) ∪ ⋃_{i:S_{i}∩S^{τ} =∅} τ(Σ_{i}))

is dense in S. In case g = 0, such Σ_{0} always exists. Otherwise, Σ_{0} may or may not exist and so we impose that Σ_{0} exists as a part of the definition of semi-stable curves with boundary. If Σ_{0} exists, then Σ_{0} is always unique. We then put

Σ = \overline{Σ_{0} ∪ ⋃_{i:S_{i}∩S^{τ} =∅} Σ_{i}}

(See Figure 3.1.)

Figure 3.1.

Once we have selected Σ, we can and will choose the marked points z_{i}^{+}, z_{i}^{−} so that

z_{i}^{+} ∈ Σ.

From now on, we will denote the semi-stable curve with boundary (S, τ, z⃗, z⃗^{±}, ori) by (Σ, z⃗, z⃗^{+}).

We note that (Σ, z⃗, z⃗^{+}) determines (S, τ, z⃗, z⃗^{±}, ori) in an obvious way.

Definition 3.3. We denote by M_{g,m,m′}^{b} the set of all isomorphism classes of marked stable curve with boundary of genus g and with (m, m′) marked points. We can provide a topology on M_{g,m,m′}^{b} using the topology on the set M_{g,m+2m′} of stable curves. (See [FOn2] §10 for details).

From now on, we will consider only the case of g = 0 and so denote M_{m,m′}^{b} in place of M_{0,m,m′}^{b}. We also write M_{m}^{b} for M_{m,0}^{b}. Some discussion on M_{m}^{b} was given in [FOh]. Our discussion here basically follows it, although we need a slight modification.

Lemma 3.4. M_{m,m′}^{b} has exactly (m − 1)! connected components.

Proof. Let (Σ, z⃗, z⃗^{+}) = (S, τ, z⃗, z⃗^{±}, ori) ∈ M_{m}^{b}. The orientation of S^{ℝ} = S^{τ} determines a cyclic order of {z_{1}, …, z_{m}}. There are (m − 1)! ways to cyclically order a set of order m. It is not difficult to see that this choice of cyclic order determines a connected component of M_{m,m′}^{b}. □

Now, we study orientation of the moduli space M_{m,m′}^{b}. We remark that there exists an obvious fibration M_{m,m′}^{b} → M_{m}^{b} whose fiber is a complex variety and so that it suffices to define an orientation of M_{m}^{b}.

We denote the upper half plane by ℍ.

Definition 3.5 We consider (ℍ, (z_{1}, …, z_{m})) = (ℂP^{1}, ℝP^{1}, (z_{1}, …, z_{m})) where (z_{1}, …, z_{m}), z_{i} ∈ ℝP^{1} respects the cyclic order of ℝP^{1} with respect to the usual orientation on ℝP^{1} = ℝ ∪ {∞}. There is a component of M_{m}^{b} that contains this element. We call this the main component and denote it by M_{m}^{b,main}. (In [FOh], we used only this main component.) Let us denote by M_{m}^{b,main, reg} the set of all elements (S, τ, z⃗, ori) of M_{m}^{b,main} such that S is nonsingular.

In [FOh], we proved that the main component M_{m}^{b,main,reg} is diffeomorphic to ℝ^{m−3} as follows: We define a map M_{m}^{b,main,reg} → M_{m−1}^{b,main,reg} by forgetting the last marked point z_{m}, (ℂP^{1}, ℝP^{1}, (z_{1}, …, z_{m})) → (ℂP^{1}, ℝP^{1}, (z_{1}, …, z_{m−1})). This is a fiber bundle whose fiber is diffeomorphic to ℝ. Note that M_{3}^{b,main,reg} consists of a single point. Therefore, we conclude M_{m}^{b,main,reg} ≅ ℝ^{m−3} by induction. Using this diffeomorphism, we now define an orientation of M_{m}^{b,main,reg}: A fiber of M_{m}^{b,main,reg} → M_{m−1}^{b,main,reg} at (ℂP^{1}, ℝP^{1}, (z_{1}, … z_{m−1})) ∈ M_{m−1}^{b,main,reg} for m ≥ 4 can be identified with an open subset of ℝP^{1} which is the interval between z_{m−1} and z_{1}. Hence each fiber is naturally oriented, which together with the orientation of M_{m−1}^{b,main} induces that of M_{m}^{b,main}. Thus by induction, we obtain an orientation of the main component.

Now, let us look at other components of M_{m}^{b}. Each permutation σ : {1, …, m} → {1, …, m} induces a homeomorphism σ_{∗} : M_{m}^{b} → M_{m}^{b} defined by

σ_{∗} : (S, τ, (z_{1}, …, z_{m})) ↦ (S, τ, (z_{σ(1)}, …, z_{σ(m)})).

Denote by σ_{0} : {1, …, m} → {1, …, m} the shift

σ_{0}(k) = k + 1 mod m + 1.

This (σ_{0})_{∗} : M_{m}^{b,reg} → M_{m}^{b,reg} preserves each component of M_{m}^{b,reg} and in particular the main component.

Lemma 3.6. (σ_{0})_{∗} : M_{m}^{b,main,reg} → M_{m}^{b,main,reg} is orientation preserving if and only if m is odd.

Proof. Let M̂_{m}^{b,main,reg} be the set of all (z_{1}, …, z_{m}) ∈ (ℝP^{1})^{m} such that [ℂP^{1}, ℝP^{1}, (z_{1}, …, z_{m})] ∈ M_{m}^{b,main,reg}. Note that M̂_{m}^{b,main,reg}/P SL(2, ℝ) ≅ M_{m}^{b,main,reg}. The map (σ_{0})_{∗} : M_{m}^{b,main,reg} → M_{m}^{b,main,reg} lifts to M̂_{m}^{b,main,reg} as a P SL(2, ℝ)-invariant map, which is nothing but the shift map on (ℝP^{1})^{m}. This shift map σ_{0} has the parity m − 1 (mod 2). This finishes proof of Lemma 3.6. □

Proposition 3.7. There exists a unique orientation on M_{m}^{b,reg} such that :

(3.7.1) It coincides with the orientation of M_{m}^{b,main,reg} described as above. (3.7.2) σ_{∗} : (S, τ, (z_{1}, …, z_{m})) ↦ (S, τ, (z_{σ(1)}, …, z_{σ(m)})) is orientation preserving if and only if σ is even.

Proof. We note that σ_{∗}(M_{m}^{b,main,reg}) ∩ M_{m}^{b,main,reg} ≠ ∅ if and only if σ = σ_{0}^{i} for some i, and that σ_{0} is even is and only if m is odd. Proposition 3.7 then follows from Lemma 3.6 by transferring the orientation on M_{m}^{b,main,reg} to other components through appropriate σ_{∗}. It is obvious that the corresponding orientation satisfies (3.7.1) and (3.7.2). □

In Chapter 6, we will use Proposition 3.7 to provide an orientation of the moduli space of stable maps from genus 0 open Riemann surface, which we now define below. (At this moment, we do not know how to construct an orientation on M_{g,m,m′} when g ≥ 1.)

Let (M, ω, J) be a symplectic manifold (M, ω) together with a compatible almost complex structure J and let L ⊂ (M, ω) be a Lagrangian submanifold.

Definition 3.8. A genus 0 stable map from an open curve with (m, m′) marked points is a pair ((Σ, z⃗, z⃗^{+}), w) such that (Σ, z⃗, z⃗^{+}) is a genus 0 semi-stable curve with (m, m′) marked points, and w : (Σ, ∂Σ) → (M, L) is a pseudo-holomorphic map such that the set of all φ : Σ → Σ satisfying the following properties is finite:

(3.8.1) φ is biholomorphic. (3.8.2) φ(z_{i}) = z_{i}, φ(z_{i}^{+}) = z_{i}^{+}. (3.8.3) w ∘ φ = ω.

Each genus 0 stable map from open curve ((Σ, z⃗, z⃗^{+}), w) determines a homology class w_{∗}([Σ]) ∈ H_{2}(M, L; ℤ) in an obvious way. (In fact, even a homotopy class w_{∗}([Σ]) ∈ π_{2}(M, L)).

We say that ((Σ, z⃗, z⃗^{+}), w) is isomorphic to ((Σ′, z′, z⃗^{′+}), w′) if there exists a biholomorphic map φ : Σ → Σ′ such that

(3.9.1) w′ ∘ φ = w (3.9.2) φ(z_{i}) = z′_{i}, φ(z_{i}^{+}) = z′_{i}^{+}.

Definition 3.10. (3.10.1) Let β ∈ H_{2}(M, L; ℤ) and denote by M_{m,m′}(β) the set of all isomorphism classes of genus 0 stable maps from open curve with (m, m′) marked points, ((Σ, z⃗, z⃗^{+}), w) such that w_{∗}([Σ]) = β. We denote by Int M_{m,m′}(β) the set of all elements ((Σ, z⃗, z⃗^{+}), w) ∈ M_{m,m′}(β) such that Σ is nonsingular, and put

∂M_{m,m′}(β) = M_{m,m′}(β) − Int M_{m,m′}(β).

(3.10.2) We denote

M_{m}^{main}(β) = {((Σ, z⃗), w) | (Σ, z⃗) ∈ M_{m}^{b,main}, [w] = β}.

We will define a topology of the moduli space M_{m,m′}(β) in a way similar to §10 [FOn2]. We would like to note that for β ≠ 0

Int M_{0,0}(β) = M(β), Int M_{1,0}(β) = M^{∂}(β)

where the right hand sides of the above are the ones defined in §2.

The following are the essential analytic results that we need for the construction of our obstruction class.

“Theorem 3.11” There exists a homomorphism μ : H_{2}(M, L; ℤ) → ℤ such that M_{m,m′}(β) is a smooth manifold of dimension n + μ(β) + m + 2m′ − 3 if the almost complex structure J on M is generic.

(Again “Theorem 3.11” as stated does not hold in general because of the transversality problem.)

We next state the following Theorem 3.12 whose proof will be given in Chapter 6. Let L ⊂ M be a relatively spin Lagrangian submanifold: There exists a class st ∈ H^{2}(M; ℤ_{2}) which restricts to the second Stiefel-Whitney class of L. We will prove in §21 in Chapter 6 that M_{m,m′}(β) is orientable under this condition.

In fact we will prove the following stronger theorem there. We refer to §21 for the definition of relative spin structure of the embedding L ⊂ M.

Theorem 3.12. Let L be relatively spin. Then a choice of relative spin structure of L ⊂ M canonically induces an orientation of M_{m,m′}(β).

We next consider the evaluation maps

Definition 3.13. We define ev : \overline{M}_{m,m′}(β) → L^{m} × M^{m′} by

ev((Σ, z⃗, z⃗^{+}), w) = (w(z_{1}), …, w(z_{m}), w(z_{1}^{+}), …, w(z_{m′}^{+})).

We regard M_{m,m′}(β) as an (integral) current on L^{m}×M^{m′}. Namely for a smooth form u of degree dim(L^{m} × M^{m′}) − dim(M_{m,m′}(β)) on L^{m} × M^{m′}, we define the integral current T_{M_{m,m′} (β)} by

∫_{L^{m}×M^{m′}} T_{M_{m,m′} (β)} ∧ u = ∫_{M_{m,m′} (β)} ev^{∗}u,

and identify M_{m,m′}(β) with T_{M_{m,m′} (β)}. (See §A1 for more detail.) We often write (M_{m,m′}(β), ev) for this current by this identification. Its boundary (or dM_{m,m′}(β) in case we regard it as a current) is described by the following :

“Theorem 3.14” The boundary of the chain M_{m,m′}(β) consists of all equivalent classes of ((Σ, z⃗, z^{+}), w) ∈ M_{m,m′}(β)) with Σ having exactly one singular point.

The current (\overline{M}_{m,m′}(β), ev) modified by some correction terms which we define in the next section, will be our obstruction cycle.

§4. Obstruction classes

In this section, we define the obstruction classes mentioned in Theorem A. As stated there, we will construct them inductively. We first recall the following finiteness theorem whose proof is a consequence of Gromov’s compactness theorem [Gr2] and will be omitted since it is standard by now. We will always implicitly assume that a generic (time-independent) almost complex structure is fixed.

Theorem 4.1. M_{m,m′}(β) is compact. Moreover for each positive constant C > 0, the set

{β ∈ G(L) | M_{m,m′}(β) ≠ ∅, E(β) ≤ C}

is finite.

(G(L) is defined in Definition 2.11.) We define

(4.2)  G_{++}(L) = {β ∈ G(L) | M_{m,m′}(β) ≠ ϕ}

and G_{+}(L) to be the monoid generated by G_{+}(L). In other words, each element in G_{+}(L) is written as ∏ β_{i}^{n_{i}} with n_{i} ≥ 0 and β_{i} ∈ G_{++}(L). By Theorem 4.1, we can enumerate G_{+}(L) so that the followings are satisfied :

(4.3.1) G_{+}(L) = {β_{0} = 0, β_{1}, β_{2}, ⋯ }. (4.3.2) 0 < E(β_{1}) ≤ E(β_{2}) ≤ · · ·. (4.3.3) β_{i} ≠ β_{j} if i ≠ j. (4.3.4) lim E(β_{i}) = ∞.

In case E(β) = E(β′) for some β, β′, there is more than one way of enumerating G_{++}(L). However, the theory we develop will be independent of the enumeration.

Now, we are ready to start with our inductive construction of the obstruction cycles o_{k}(L) = o(L : β_{k}). o_{k}(L) will be defined when o_{i}(L) is homologous to zero for i < k : when o_{i}(L) ∼ 0, we take a chain B(β_{i}) of L such that

(4.4)  ∂B(β_{i}) = (−1)^{n+1}o_{i}(L)

and will define o_{k}(L) using B(β_{i}) for i < k.

From now on, we use the moduli space M_{m+1}(β) with m ≥ 0 and enumerate the marked points starting from 0. So the evaluation maps are enumerated as ev = (ev_{0}, ev_{1}, ⋯).

We regard (M_{1}(β_{1}), ev_{0}) as a current on L as is explained at the end of the last section and in §A1. To start with the induction, we need :

Lemma 4.5. Let β_{1} ∈ G_{++}(L) be the class with minimal possible non-zero area (with respect to the given almost complex structure). Then ∂(M_{1}(β_{1}), ev_{0}) = 0 as a current.

Proof. We consider the moduli space M_{1}(β_{1}) and its boundary ∂M_{1}(β_{1}). By “Theorem 3.14”, ∂M_{1}(β_{1}) is contained in the set of ((Σ, z_{1}), w)) such that Σ has exactly one singular point. Since the energy of β_{1} is the minimum among the nontrivial pseudo-holomorphic discs, pseudo-holomorphic maps with the energy β_{1} cannot bubble off, which proves that ∂M_{1}(β_{1}) = ∅ and so ∂M_{1}(β_{1}) = 0 as a current. □

Definition 4.6. We define

o_{1}(L) = [M_{1}(β_{1}), ev_{0}] ∈ H_{n−2+μ(β_{1})}(L; ℚ).

Next, we assume that o_{i}(L) and B(β_{i}) have been defined so that (4.4) is satisfied for all i < k. We use the evaluation map

ev = (ev_{0}, ev_{1}, …, ev_{m}) : M_{m+1}(β_{k}) → L^{m+1}

to define the following moduli space.

Definition 4.7. Let i_{1}, …, i_{m} < k. We define

\begin{matrix}M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{m}}) = & ± M_{m+1}(β_{k} − β_{i_{1}} − ⋯ − β_{i_{m}}) \\ _{(ev_{1},... ,ev_{m})} × (B(β_{i_{1}}) × ⋯ × B(β_{i_{m}})). \end{matrix}

The precise sign on the right hand side will be given in §24 (Definition 24.1) in Chapter 6. Using this moduli space, we define our obstruction cycle.

Definition 4.8. We define a current

\begin{aligned}o_{k}(L) & := o(L; β_{k}) \\ & = ∑_{m=0,1,2,... ; i_{1},... ,im<k ; β_{k}−∑{m \atop j=1} β_{i_{j}} ∈G_{++}(L)} \frac{1}{m!}(M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{m}}), ev_{0}) \end{aligned}

where ev_{0} is the evaluation map at the marked point corresponding to the cycle β_{k}.

The main result of this section is the following.

Theorem 4.9. We define the currents o_{j}(L) = o(L; β_{j}) as above for j < k. Suppose that (4.4) is satisfied for all j < k, and define o_{k}(L) as in Definition 4.8. Then we have

∂o_{k}(L) = 0.

To motivate our definition of o_{k}(L), for each permutation σ ∈ S_{m}, we study the induced map

σ_{∗} : M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{m}}) → M_{1}(β_{k}; β_{i_{σ(1)}} ⋯ β_{i_{σ(m)}})

defined by

(4.10)  σ_{∗}((Σ, z⃗), w), (x_{1}, …, x_{m})) = ((σ_{∗}(Σ, z⃗), w), (x_{σ(1)}, ⋯, x_{σ(m)}))

where σ_{∗} : \overline{M^{b}}_{m+1} → \overline{M^{b}}_{m+1} is defined by

σ_{∗}(Σ, τ, (z_{0}, z_{1}, …, z_{m})) = (Σ, τ, (z_{0}, z_{σ(1)}, …, z_{σ(m)}).

Lemma 4.11. The map (4.10) is orientation preserving for all σ ∈ S_{m}.

We refer §25.1 Lemma 25.3 (and also Remark 24.2) for the proof.

Lemma 4.11 justifies the coefficient m! we put in Definition 4.8. In other words, if we use only (D, z⃗) in the main component, we can drop it. (This remark will be important later in the applications in Chapter 3 where we have to work over ℤ_{2} coefficients for which we are not allowed to divide by m!.)

The following theorem, which is a consequence of “Theorem 3.14”, will play an essential role in the proof of “Theorem 4.9”.

Theorem 4.12. We have

(4.13) \begin{matrix}∂M_{m+1}(β_{k}) = \\ ∑_{β′+β′′=β_{k}} ∑_{0<m_{1}≤m_{2}≤m} ±M_{m+1−m_{2}+m_{1}+1}(β′)_{ev_{m_{1}}^{β′}} ×_{ev_{0}^{β′′}} M_{m_{2}−m_{1}+1}(β′′). \end{matrix}

Here the symbol _{ev_{m_{1}}^{β′}} ×_{ev_{0}^{β′′}} means the use of the m_{1}-th evaluation map

ev_{m_{1}}^{β′} : M_{m+1−m_{2}+m_{1}+1}(β′) → L

and the 0-th evaluation map

ev_{0}^{β′′} : M_{m_{2}−m_{1}+1}(β′′) → L

to take the fiber product over L. The choice of sign ± is a little complicated to describe, which will be carried out in Chapter 6. (See Proposition 23.2.)

Now we can describe the various evaluation maps

ev_{i}^{β_{k}} : ∂M_{m+1}(β_{k}) → L

used in the identification (4.13). For simplicity, we consider the case m_{1} = 1, namely, the case that the second marked point z_{1,β′} of the β′-disc is glued with the first marked point z_{0,β′′} of the β′′-disc and the two marked points disappear after gluing. (See also Remark 23.3).

Addendum 4.14. Let i ∈ {1, …, m}. Then ev_{i}^{β_{k}} : ∂M_{m+1}(β_{k}) → L through the identification (4.13) is described as follows:

(4.15.1) If i = 0, then

ev_{0}^{β_{k}} = ev_{0}^{β′}.

(4.15.2) If 1 ≤ i ≤ m_{2} − m_{1}, then

ev_{i}^{β_{k}} = ev_{i}^{β′′}.

(4.15.3) If m_{2} − m_{1} < i ≤ m, then

ev_{i}^{β_{k}} = ev_{i+1−m_{2}+m_{1}}^{β′}.

Proof. Theorem 4.12 and Addendum 4.14 rather immediately follow from “Theorem 3.14” (except the orientation, which we will discuss in Chapter 6). The proof can be conveniently done using figures. The following figure represents an element of M_{m+1}(β).

Figure 4.1.

“Theorem 3.14” implies that codimension one components of M_{m+1}(β) lying in ∂M_{m+1}(β) can be described by similar figures. □

Figure 4.2

Now, we are ready to prove Theorem 4.9. We introduce the following set

(4.16)  \begin{aligned}N_{1}(β_{k}; β_{i_{1}}, & …, β̂_{i_{j}}, …, β_{i_{m}}; β_{i_{j}}) \\ & = M_{m+1}(β_{k} − β_{i_{1}} − ⋯ − β_{i_{m}}) _{(ev_{1},... ,ev_{m})}× \\ & (B(β_{i_{1}}) × ⋯ × o(β_{i_{j}}) × ⋯ × B(β_{i_{m}})). \end{aligned}

We again describe M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{m}}) and N_{1}(β_{k}; β_{i_{1}}, …, β̂_{i_{j}}, …, β_{i_{m}}; β_{i_{j}}) in the following figure.

Figure 4.3 (M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{m}}))

Here the symbol B(β_{i}) in the figure means that the corresponding marked point should be sent to B(β_{i}). We also recall that we use the evaluation map ev_{0} at z_{0} to define the corresponding geometric chain.

Next, we decompose N_{1}(β_{k}; β_{i_{1}}, …, β̂_{i_{j}}, ⋯, β_{i_{m}}; β_{i_{j}}) into a union of the components corresponding to the terms of o(β_{i_{j}}) in Definition 4.8. Then one of them is represented by the following figure

Figure 4.4 (N_{1}(β_{k}; β_{i_{1}}, …, β̂_{i_{j}}, …, β_{i_{m}}; β_{i_{j}}))

Figure 4.4 represents the component corresponding to M_{1}(β_{i_{j}}; β_{ℓ_{1}}, …, β_{ℓ_{n}}) in o_{i_{j}}(L). Again we use the evaluation map at z_{0}.

Now we consider the boundary of the component of M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{m}}) that is described by Figure 4.2. “Theorem 4.12” implies that when this element approaches the boundary (of codimension 1), one of the following should occur.

(4.17.1) The disc splits into two. (4.17.2) The value at some marked point (other than z_{0}) approaches ∂B(β_{i_{j}}).

(4.17.2) corresponds to an element in

⋃_{j=1}^{m} N_{1}(β_{k}; β_{i}, …, β̂_{i_{j}}, …, β_{i_{m}}; β_{i_{j}})

which is described by Figure 4.4. (4.17.1) in fact is described by the same figure as Figure 4.4. Hence they cancel each other. (See §24 in Chapter 6 for the discussion on the orientation we need to show the cancellation.) The proof of Theorem 4.9 is now complete. □

So far, we have defined the obstruction classes o_{k}(L) by induction. We next prove the following :

Theorem 4.18. For the natural map i_{∗} : H_{∗}(L; ℚ) → H_{∗}(M; ℚ) we have

i_{∗}(o_{k}(L)) = 0 in H_{∗}(M; ℚ).

For the proof of this theorem, we introduce another moduli space which this time defines a chain on M.

Definition 4.19. Let i_{1}, …, i_{m} < k. We define

\begin{matrix}M_{0,1}^{Σ}(β_{k}; β_{i_{1}}, … β_{i_{m}}) \\ = ±M_{m,1}(β_{k} − β_{i_{1}} − ⋯ − β_{i_{m}})_{(ev_{1},···ev_{m})} × (B(β_{i_{1}}) × ⋯ × B(β_{i_{m}})). \end{matrix}

Here the sign on the right hand side is exactly the same as in Definition 4.7.

Definition 4.20. We define a current on M by

b_{k}(L) = ∑_{{m=0,1,2,... \atop i_{1},... ,im<k} ; β_{k}−β_{i_{1}} −···−β_{im} ∈G_{++}(L)} \frac{1}{m!}(M_{0,1}^{Σ}(β_{k}; β_{i_{1}}, …, β_{i_{m}}), ev^{+})

where ev^{+} : M_{m,1}(β_{k} − β_{i_{1}} − ⋯ − β_{i_{m}}) → M is the evaluation map at the unique interior marked point in the domain.

Theorem 4.18 will immediately follow from the following proposition.

Proposition 4.21.

∂((−1)^{n+1}b_{k}(L)) = i_{∗}(o_{k}(L)).

Proof. The boundary of the current (M_{0,1}^{Σ}(β_{k}; β_{i_{1}}, …, β_{i_{m}}), ev^{+}) consists of the following three terms:

(4.22.1) The disc splits into two. (4.22.2) The value at a marked point on ∂Σ approaches ∂B(β_{i_{j}}). (4.22.3) The interior marked point z^{+} approaches ∂Σ.

By the same way as in Theorem 4.12, the terms (4.22.1) and (4.22.2) cancel each other when we take the boundary of b_{k}(L). On the other hand, the term (4.22.3) gives rise to i_{∗}(o_{k}(L)). The argument on the orientation problem is now the same as in the proof of Theorem 4.9. Hence we have finished the proof. □

§5. Boundary operators.

In this section, we use the chain B(β_{k}) that bounds our obstruction cycle o_{k}(L) and modify Floer’s original boundary operator. For given almost complex structures J_{0} and J_{1} generic for L_{0} and L_{1} respectively, we fix a generic choice of t-dependent family J = {J_{t}}_{0≤t≤1} of almost complex structures. The meaning of the genericity will be clear from our discussions and so not be specified at this point.

In this section, we explicitly specify the Lagrangian submanifold and write B(L_{0} : β_{k}) for B(β_{k}) etc.

Suppose that the pair of Lagrangian submanifolds (L_{0}, L_{1}) of (M, w) is relatively spin. We assume that their obstruction classes o_{k}(L_{i}), i = 0, 1 all vanish and the based path ℓ_{0} is given. We now gives the description of the (compactified) moduli space

M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}}).

In our discussion, the bubbling-off spheres do not play any essential role because it is a phenomenon of codimension 2 and the way how to handle with it has been already well-studied in [FOn2] as far as we work over the rational number. Thus we first single out the sphere bubbles in the compactification and consider the map u : Σ → M where Σ is [0, 1] × ℝ plus a finite number of sphere components and m + n marked points on the boundary. Of course such Σ is conformal to Σ′ minus two marked points. Here (Σ′, z⃗) is an element of M_{m+n+2}. We assume the marked points z_{0} and z_{1} will be identified to +∞ and −∞, respectively. We identify z_{2}, ⋯, z_{m+n+2} as m + n points on ∂Σ = {0, 1} × ℝ. (We use the same symbol z_{i} for them.)

Definition 5.1. Let [ℓ_{p}, w], [ℓ_{q}, w′] ∈ Cr(L_{0}, L_{1}). We first define the moduli space M_{m,n}([ℓ_{p}, w], [ℓ_{q}, w′]). Let (Σ′, z⃗) ∈ M_{m+n+2}^{b}. Let (Σ, z_{2}, ⋯, z_{m+n}) be [0, 1] × ℝ plus bubbles and m + n marked points, which corresponds to (Σ′, z⃗) in the sense explained above. We denote by ∂_{0}Σ the part of ∂Σ between z_{0} and z_{1} where z_{2}, … z_{m+1} are laid counterclockwise from z_{0} and ∂_{1}Σ = \overline{∂Σ − ∂_{0}Σ}. We consider the maps u : Σ → M and z⃗ satisfying the following properties: (5.2.1) On ℝ × [0, 1], u satisfies

\frac{∂u}{∂τ} + J(t, u)\frac{∂u}{∂t} = 0.

When the root of a sphere bubble tree is attached to a point (τ, t) ∈ ℝ × [0, 1], then all the sphere bubbles in the tree are J_{t}-holomorphic. (5.2.2) z_{i} ∈ ∂_{0}Σ for i = 0, …, m + 1 and z_{i} ∈ ∂_{1}Σ for i = m + 2, …, m + n + 1. (5.2.3) lim_{τ→−∞} u(t, τ) = p. (5.2.4) lim_{τ→+∞} u(t, τ) = q. (5.2.5) u(∂_{0}Σ) ⊂ L_{0}. (5.2.6) u(∂_{1}Σ) ⊂ L_{1}. (5.2.7) (w#u) ∼ w′. (5.2.8) ((Σ, z⃗), u) is stable, i.e. the set of automorphisms φ : Σ → Σ with φ(z_{i}) = z_{i}, u ∘ φ = u is finite. We define an equivalence relation ∼ by saying that ((Σ, z⃗), u) ∼ ((Σ′, z⃗′), u′) if and only if there exists a biholomorphic map φ : Σ → Σ′ such that

φ(z_{i}) = z′_{i} and u′ ∘ φ = u.

Then we define M_{m,n}([ℓ_{p}, w], [ℓ_{q}, w′]) to be the set of all ∼ equivalence classes of ((Σ, z⃗), u) satisfying (5.2.1), …, (5.2.8).

Note that m, n here are number of marked points which are assumed to be mapped to L_{0}, L_{1} respectively. (In §3 we denoted by n the number of marked point in the interior. The notation here is different from there.)

We define the evaluation map

ev = (ev_{2}, …, ev_{m+1}, ev_{m+2}, …, ev_{n+m+1}) : M_{m,n}([ℓ_{p}, w], [ℓ_{q}, w′]) → L_{0}^{m} × L_{1}^{n}. by

ev(((Σ, z⃗), u)) = (u(z_{2}), ⋯, u(z_{m+1}), u(z_{m+2}), ⋯, u(z_{n+m+1})).

Let

β_{0,i_{1}}, …, β_{0,i_{m}} ∈ G(L_{0}), β_{1,j_{1}}, …, β_{1,j_{n}} ∈ G(L_{1})

and define the space

M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}})

to be the fiber product

M_{m,n}([ℓ_{p}, w], [ℓ_{q}, w′′])_{ev} × (∏_{k=1}^{m} B(L_{0} : β_{0,i_{k}}) × ∏_{k=1}^{n} B(L_{1} : β_{1,j_{k}})) ,

where

(Σβ_{0,i_{k}} + Σβ_{1,j_{ℓ}}) · w′′ ∼ w′.

In other words, M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}}) is the set of all ∼ equivalence classes of ((Σ, z⃗), u) satisfying (5.2.1), …,(5.2.6),(5.2.8) and : (5.2.7) (Σβ_{0,i_{k}} + Σβ_{1,j_{ℓ}}) · (w#u) ∼ w′. (5.2.8) u(z_{k}) ∈ B(L_{0} : β_{0,i_{k}}), k = 2, …, m + 1. (5.2.9) u(z_{k}) ∈ B(L_{1} : β_{1,j_{m+n+2−k}}), k = m + 2, …, n + m + 1.

We denote by

Int M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}})

the set of all elements ((Σ, z⃗), u) such that Σ is smooth.

Similarly as in §2 and §4, we use figures to describe elements of Int M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{1,j_{m}}).

(Here the order of markings B(L_{1}; β_{j_{k}}) or B(L_{0}; β_{i_{ℓ}}) may not be preserved, respectively).

Under these notations, we remark the following identities:

\begin{aligned}M([ℓ_{p}, w], [ℓ_{q}, w′] : ∅; ∅) & = M([ℓ_{p}, w], [ℓ_{q}, w′]) \\ M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}; ∅) “ & = ”M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{0,i_{1}}) \\ M([ℓ_{p}, w], [ℓ_{q}, w′] : ∅; β_{1,j_{1}}) “ & = ”M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{1,j_{1}}) \\ M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, β_{0,i_{2}}) “ & = ”M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{0,i_{1}}, L_{0}, β_{0,i_{2}}) \\ M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}; β_{1,j_{1}}) “ & = ”M[ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{0,i_{1}}; L_{1}, β_{1,j_{1}}) \\ M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{1,j_{1}}, β_{1,j_{2}}) “ & = ”M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{1}, β_{1,j_{1}}, L_{1}, β_{1,j_{2}}). \end{aligned}

Here the terms on the left are as in Definition 5.1 and those on the right are as in §2. We would like to remark that #(u(ℝ×{0})∩w_{β_{0,i_{1}}} (∂D^{2})) := k could be more than 1, in which case (u, w_{β_{0,i_{1}}} ) gives k elements of M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}; ϕ) while it gives just one element of M([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}; β_{0,i_{1}}). This is the reason why we only put “ = ” above.

The moduli space N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}) in “Definition 2.13” in §2 is also written as

N ([ℓ_{p}, w], [ℓ_{q}, w′] : L_{0}, β_{(0)}) = M_{1,0}([ℓ_{p}, w], [ℓ_{q}, w′′])_{ev} × o(L_{0} : β_{(0)}).

where β_{(0)} · w′′ = w′. Again, in the case when pseudo-holomorphic disc intersects with o(L_{0} : β_{(0)}) at two or more points, “Definition 2.13” is not correct and the definition given above using the fiber product is the correct one.

Now we have the following :

“Theorem 5.3.” Int M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}}) is a smooth oriented manifold of dimension

μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]) − 1.

The proof of the dimension formula is straightforward. The orientation will be defined in §25.2 in Chapter 6 (Definition 25.6) and the transversality part is in fact not true even for a generic choice of almost complex structures. The transversality will be discussed in Chapter 5.

Definition 5.4. When μ([ℓ_{q}, w′]) = μ([ℓ_{p}, w]) + 1, we define the operator

δ_{β_{0,i_{1}} ,··· ,β_{0,im};β_{1,j_{1}} ,··· ,β_{1,jn}}

by the matrix element

\begin{matrix}⟨δ_{β_{0,i_{1}} ,··· ,β_{0,im};β_{1,j_{1}} ,··· ,β_{1,jn}} [ℓ_{p}, w], [ℓ_{q}, w′]) \\ = #M([ℓ_{p}, w], [ℓ_{q}, w′]) : β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}}). \end{matrix}

We now explain the completion

CF^{k}(L_{0}, L_{1}) := ˆ⨁_{{[ℓp,w]∈Cr(L_{0},L_{1}) \atop μ([ℓp,w])=k}}ℚ[ℓ_{p}, w]

of the module ⨁ ℚ[ℓ_{p}, w], which we mentioned in Definition 2.7.

Definition 5.5. CF^{k}(L_{0}, L_{1}) is defined to be the set of (infinite) sums

∑_{μ([ℓ_{p},w])=k} a_{[ℓ_{p},w]}[ℓ_{p}, w]

such that for each C ∈ ℝ, a_{[ℓ_{p},w]} ∈ ℚ and the set

{[ℓ_{p}, w] | A(ℓ_{p}, w) ≤ C, a_{[ℓ_{p},w]} ≠ 0}

is of finite order. We then define the ℤ-graded module

CF (L_{0}, L_{1}) = ⨁_{k} CF^{k}(L_{0}, L_{1}).

We also define the Novikov ring Λ(L_{0}, L_{1}) as a completion of the group ring ℚ[G(L_{0}, L_{1})].

Definition 5.6. Λ^{k}(L_{0}, L_{1}) denotes the set of all (infinite) sums ∑_{{g∈G(L_{0},L_{1}) \atop μ(g)=k}} a_{g}[g]

such that a_{g} ∈ ℚ and that for each C, the set

{g ∈ G(L_{0}, L_{1}) | E(g) ≤ C_{g}, a_{g} ≠ 0}

is of finite order. We put

Λ(L_{0}, L_{1}) = ⨁_{k} Λ^{k}(L_{0}, L_{1}).

The ring structure on Λ(L_{0}, L_{1}) is defined by

(5.7)  ( ∑_{g∈G(L_{0},L_{1})} a_{g}[g]) · ( ∑_{g∈G(L_{0},L_{1})} b_{g}[g]) = ∑_{g_{1},g_{2}∈G(L_{0},L_{1})} a_{g_{1}}b_{g_{2}}[g_{1}g_{2}].

It is easy to see that the term in the right hand side is indeed an element in Λ(L_{0}, L_{1}), i.e., satisfies the finiteness condition in its definition. Thus Λ(L_{0}, L_{1}) = ⊕_{k}Λ^{k}(L_{0}, L_{1}) becomes a graded ring under the multiplication (5.7).

This Novikov ring Λ(L_{0}, L_{1}) acts on CF (L_{0}, L_{1}) by

(∑ b_{g}[g]) · (∑ a_{[ℓ_{p},w]}[ℓ_{p}, w]) = ∑ b_{g}a_{[ℓ_{p},w]}(g · [ℓ_{p}, w]).

Again it is not difficult to see that the term in the right hand side is well-defined as an element in CF (L_{0}, L_{1}). Therefore CF (L_{0}, L_{1}) is a graded Λ(L_{0}, L_{1})-module.

Proposition 5.8. For each [ℓ_{p}, w], the (infinite) sum

∑_{β_{0,i_{1}} ,··· ,β_{0,im};β_{1,j_{1}} ,··· ,β_{1,jn}} \frac{1}{m!n!}δ_{β_{0,i_{1}} ,··· ,β_{0,im};β_{1,j_{1}} ,··· ,β_{1,jn}} [ℓ_{p}, w]

is an element of CF (L_{0}, L_{1}) . Proof. It suffices to prove that for any given C > 0, the set of all [ℓ_{q}, w′] ∈ Cr(L_{0}, L_{1}) satisfying the following (5.9) is of finite order:

(5.9.1)  A([ℓ_{q}, w′]) ≤ C.
(5.9.2)  M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, ⋯, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}}) ≠ ∅.

Note that (5.9.2) implies that (5.10) M(L_{0}; β_{0,i_{d}}) and M(L_{1}; β_{1,j_{e}}) are non-empty.

If we define w′′ by

(5.11)  (∑_{d} β_{i_{d}} + ∑_{e} β_{j_{e}}) · [ℓ_{q}, w′′] = [ℓ_{q}, w′],

then (5.9.2) also implies that

(5.12)  M([ℓ_{p}, w], [ℓ_{q}, w′′] : ∅; ∅) ≠ ∅.

Therefore, we have

(5.13)  A([ℓ_{p}, w]) ≤ A([ℓ_{q}, w′′]).

We also have

(5.14)  ω(β_{0,i_{d}}), ω(β_{1,j_{e}}) > 0

from (5.10) because for any w ∈ M(L_{i}, J_{i}; β_{i,i_{d}} for i = 0, 1 we have

ω(β_{i,k}) = E_{J_{i}}(w) = \frac{1}{2} ∫ |dw|_{J}^{2} > 0

as long as w is non-constant. Furthermore we have

(5.15)  ∑_{d} ω(β_{0,i_{d}}) + ∑_{e} ω(β_{0,j_{e}}) + A([ℓ_{q}, w′′]) = A([ℓ_{q}, w′])

from (5.11). Then (5.9.1), (5.14) and (5.15) together imply

A([ℓ_{q}, w′′]) ≤ C.

Now this together with (5.12) implies, via Gromov’s compactness theorem, that there exist only a finite number of such (ℓ_{q}, w′′). On the other hand, (5.13), (5.15) and (5.9.1) imply that

(5.16)  0 < ω(β_{0,i_{d}}), ω(β_{1,j_{e}}) ≤ C − A([ℓ_{p}, w]).

We note that #{g ∈ G_{++}(L; J)|E_{J}(g) ≤ C} is finite and M(β) = ∅ if β ∉ G_{++}(L; J). Therefore again from Gromov’s compactness, (5.16) implies that there exist only a finite number of possibilities of β_{0,i_{1}}, ⋯, β_{0,i_{m}}; β_{1,j_{1}}, ⋯, β_{1,j_{n}} that satisfy (5.9). Hence it follows from (5.11) that there are only a finite number of [ℓ_{p}, w′] ∈ Cr(L_{0}, L_{1}) satisfying (5.9). This finishes the proof of Proposition 5.8. □

Proposition 5.8 implies that the operator δ = δ^{J} : CF^{k}(L_{0}, L_{1}) → CF^{k+1}(L_{0}, L_{1}) defined by

δ = ∑_{m,n} ∑_{β_{0,i_{1}} ,··· ,β_{0,im};β_{1,j_{1}} ,··· ,β_{1,jn}} \frac{1}{m!n!}δ_{β_{0,i_{1}} ,··· ,β_{0,im};β_{1,j_{1}} ,··· ,β_{1,jn}}

is well-defined. Now we prove the boundary property of δ.

Theorem 5.18. δ defined as above satisfies

δ ∘ δ = 0.

Proof. Let [ℓ_{p}, w], [ℓ_{q}, w′] ∈ Cr(L_{0}, L_{1}) with μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]) = 2. We will show that the matrix elements of δ satisfies ⟨δ ∘ δ[ℓ_{p}, w], [ℓ_{q}, w′]⟩ = 0 for all [ℓ_{p}, w] and [ℓ_{q}, w′]. The matrix element is by definition equal to

(5.19) \begin{aligned}∑ \frac{1}{m!n!m′!n′!} & #(M([ℓ_{p}, w], [ℓ_{r}, w′′] : β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}})× \\ & M([ℓ_{r}, w′′], [ℓ_{q}, w′] : β_{0,i′_{1}}, …, β_{0,i′_{m′}}; β_{1,j′_{1}}, …, β_{1,j′_{n′}})) \end{aligned}

where the sum is taken over all [ℓ_{r}, w′′] with μ([ℓ_{r}, w′′]) − μ([ℓ_{p}, w]) = 1 and over

β_{0,i_{1}}, …, β_{0,i_{m}}; β_{1,j_{1}}, …, β_{1,j_{n}}.

See the following figure.

Figure 5.2

To prove that (5.19) is zero, we need to study the following moduli space

(5.20)  \begin{matrix}M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, & …, β_{0,i_{m}}, β_{0,i′_{1}}, …, β_{0,i′_{m′}}; \\ β_{1,j_{1}}, …, β_{1,j_{n}}, β_{1,j_{1}}, …, β_{1,j′_{n′}}) \end{matrix}

which is one dimensional by “Theorem 5.3” when μ([ℓ_{q}, w′]) − μ([ℓ_{p}, w]) = 2. The moduli space of pairs of trajectories appearing in (5.19), which is figured in Figure 5.2, form a part of the boundary of (5.20). However there are two other types of boundary components which are described as follows: Let u_{d} : Σ → M be a sequence of elements of (5.20) which is divergent. (Σ is ℝ × [0, 1] plus possibly sphere bubbles.) Then besides the above “splitting” of trajectories, we have the possibilities

(5.21.1) A pseudo-holomorphic disc bubbles-off. (5.21.2) lim_{d→∞} u_{d}(z_{k}) ∈ ∂B(L_{1}; β_{1,j_{m+n+2−k}}) = (−1)^{n+1}o(L_{1}; β_{1,j_{m+n+2−k}}) or

lim_{d→∞} u_{d}(z_{k}) ∈ ∂B(L_{0}; β_{0,i_{k}}) = (−1)^{n+1}o(L_{0}; β_{0,i_{k}}).

Again one can prove as in case of (4.17.1) and (4.17.2) that (5.21.1) and (5.21.2) cancel each other. We refer §25.2 for more detail about the orientations. This finishes the proof of Theorem 5.18. □

Now, we have finally arrived at our definition of Floer cohomology.

Definition 5.22. We define

HF (L_{0}, L_{1}) = HF (L_{0}, L_{1}; ℓ_{0}) = Ker δ/ Im δ.

Note that this definition depends on various choices of parameters, especially on the choice of bounding chains B(L_{i}, J_{i}; β) i = 0, 1. We will study this dependence carefully later in Chapter 4.

§6. Calculation of the Floer homology; spectral sequence

6.1. Behavior of G_{ℓ_{0}}(L_{0}, L_{1}) under the Hamiltonian isotopy.

First we recall that G_{ℓ_{0}}(L_{0}, L_{1}) is the group of deck transformations of the covering space Ω̃_{ℓ_{0}}(L_{0}, L_{1}) → Ω_{ℓ_{0}}(L_{0}, L_{1}) where ℓ_{0} is path and Ω_{ℓ_{0}}(L_{0}, L_{1}) is the component of Ω(L_{0}, L_{1}) containing ℓ_{0} (see Section §2). We have the following result which tells us how G_{ℓ_{0}}(L_{0}, L_{1}) behaves under Hamiltonian isotopies of L_{0} and L_{1}.

Let Φ_{0} = {ϕ_{0}^{t}}_{0≤t≤1} and Φ_{1} = {ϕ_{1}^{t}}_{0≤t≤1} be two Hamiltonian isotopies moving L_{0} and L_{1} respectively. Define ℓ_{0}^{Φ_{0}Φ_{1}} : [0, 1] → M to be the path by

(6.1)  ℓ_{0}^{Φ_{0}Φ_{1}}(t) = ϕ_{1}^{t} ∘ (ϕ_{0}^{1−t})(ℓ_{0}(t))

and consider the component Ω_{ℓ_{0}^{Φ_{0}Φ_{1}}} (ϕ_{0}^{1}(L_{0}), ϕ_{1}^{1}(L_{1})) of Ω(ϕ_{0}^{1}(L_{0}), ϕ_{1}^{1}(L_{1})). Denote by G_{ℓ_{0}^{Φ_{0}Φ_{1}}} (ϕ_{0}^{1}(L_{0}), ϕ_{1}^{1}(L_{1})) the corresponding covering group.

Proposition 6.2. The assignment

I_{Φ_{0}Φ_{1}} : ℓ ↦ ℓ^{Φ_{0}Φ_{1}}

above defines a natural isomorphism

I_{Φ_{0}Φ_{1}} : G_{ℓ_{0}}(L_{0}, L_{1}) ≅ G_{ℓ_{0}^{Φ_{0}Φ_{1}}} (ϕ_{0}^{1}(L_{0}), ϕ_{1}^{1}(L_{1}))

such that

E ∘ I_{Φ_{0}Φ_{1}} = E, μ ∘ I_{Φ_{0}Φ_{1}} = μ.

Proof. Without loss of any generality, we will assume Φ_{0} ≡ id. The following lemma is immediate from the definition of Ω_{ℓ_{0}}(L_{0}, L_{1}).

Lemma 6.4. Let γ : S^{1} → Ω_{ℓ_{0}}(L_{0}, L_{1}) be a loop and ν : S^{1} × [0, 1] → M be the corresponding map induced by γ. Then the followings are equivalent:

(6.4.1) γ is lifted to a loop γ̃ : S^{1} → Ω̃_{ℓ_{0}}(L_{0}, L_{1}). (6.4.2) ∫_{S^{1}×[0,1]} ν^{∗}ω = 0 and μ([γ]) = 0 where μ : π_{1}(Ω(L_{0}, L_{1}), ℓ_{0}) → ℤ is the Maslov index as defined in (2.2).

Now for given Hamiltonian isotopy Φ = {ϕ^{t}}_{0≤t≤1}, we define a map I_{Φ} : Ω(L_{0}, L_{1}) → Ω(L_{0}, ϕ^{1}(L_{1})) as follows: For each ℓ ∈ Ω(L_{0}, L_{1}), we define

I_{Φ}(ℓ)(t) = ϕ^{t}(ℓ(t))

To prove Proposition 6.2, it will be enough to prove Lemma 6.4. For given loop γ : S^{1} → Ω(L_{0}, L_{1}), the following two conditions are equivalent:

(6.4.1) γ can be lifted to γ̃ : S^{1} → Ω̃(L_{0}, L_{1}) (6.4.2) I_{Φ}(γ) : S^{1} → Ω(L_{0}, ϕ^{1}(L_{0})) can be lifted to a loop { \atop I_{Φ} ∘ γ} : S^{1} → Ω̃(L_{0}, φ^{1}(L_{1})).

Proof. Suppose (6.4.1) holds and let ν : S^{1} × [0, 1] → M be the natural map associated to γ which is defined by ν(s, t) = γ(s)(t). Obviously the natural map ν_{Φ} : S^{1} × [0, 1] → M associated to the loop I_{Φ}(γ) : S^{1} → Ω(L_{0}, ϕ^{1}(L_{1})) is given by

(6.5)  ν_{Φ}(s, t) = ϕ^{t}(ν(s, t)) = ϕ^{t}(γ(s)(t)).

By Lemma 6.3, it is enough to prove

(6.6.1)  ∫ ν_{Φ}^{∗}ω = ∫ ν^{∗}ω

and

(6.6.2)  μ([γ]) = μ([I_{Φ}(γ)]).

Let H : M × [0, 1] → ℝ be the Hamiltonian generating the isotopy Φ. We compute

∫ ν_{Φ}^{∗}ω = ∫_{S^{1}×[0,1]} ω (\frac{∂ν_{Φ}}{∂s} , \frac{∂ν_{Φ}}{∂t} ) dsdt.

Using (6.5), we compute

\begin{aligned}\frac{∂ν_{Φ}}{∂s} & = T ϕ^{t} (\frac{∂ν}{∂s} ) \\ \frac{∂ν_{Φ}}{∂t} & = X_{H}(ν) + T ϕ^{t} (\frac{∂ν}{∂t} ) \end{aligned}

and so

∫ ν_{Φ}^{∗}ω = ∫_{S^{1}×[0,1]} ω (T ϕ^{t} (\frac{∂ν}{∂s} ) , X_{H} (ν) + T ϕ^{t} (\frac{∂ν}{∂t} )) dsdt

= ∫_{S^{1}×[0,1]} ω (T ϕ^{t} (\frac{∂ν}{∂s} ) , T ϕ^{t} (\frac{∂ν}{∂t} )) dsdt

+ ∫_{S^{1}×[0,1]} ω (T ϕ^{t} (\frac{∂ν}{∂s} ) , X_{H}(ν)) dsdt

= ∫ ν^{∗}ω + ∫_{S^{1}×[0,1]} \frac{∂}{∂s}H (ϕ^{t} (ν(s, t))) dsdt

= ∫ ν^{∗}ω.

This proves (6.6.1). For the proof of (6.6.2), we consider the pull-back bundle ν^{∗}T M. We choose a symplectic trivialization Ψ : ν^{∗}T M → S^{1} × [0, 1] × ℂ^{n}. Using this, we can naturally trivialize ν_{Φ}^{∗}T M as follows: Let (s, t) ∈ S^{1}×[0, 1] and consider the map

dϕ^{t} : T_{ν(s,t)}M → T_{ϕ^{t}(ν(s,t))}M = T_{ν_{Φ}(s,t)}M.

We define a trivialization of ν_{Φ}^{∗}T M by composing

Ψ_{Φ} : ν_{Φ}^{∗}T M →^{(dϕ^{t})^{−1}}ν^{∗}T M→^{Ψ}S^{1} × [0, 1] × ℂ^{n}.

Indeed we can consider one parameter family of bundle ν_{Φ^{ϵ}}^{∗}T M and the trivializations

Ψ_{Φ^{ϵ}} : ν_{Φ^{ϵ}}^{∗}T M →^{(dϕ^{ϵt})^{−1}}ν^{∗}T M→^{Ψ}S^{1} × [0, 1] × ℂ^{n}

associated to the map

ν_{Φ^{ϵ}}(s, t) = ϕ^{ϵt}(ν(s, t)).

This family induces a pair (α_{1}^{ϵ}, α_{0}^{ϵ}) of continuous family of loops in Λ(ℂ^{n}) and so the standard Maslov indices of them are unchanged. This proves

μ(ν) = μ(ν_{Φ})

from the definition (2.2) of the Maslov index μ(ν) since this is independent of the trivializations we choose. This finishes the proof of Lemma 6.4. □

Now, we go back to the proof of Proposition 6.2. It immediately follows from Lemma 6.3 that the map I_{Φ} defined as above induces a natural map of the covering groups, i.e.,

I_{Φ} : G(L_{0}, L_{1}) → G(L_{0}, ϕ^{1}(L_{1})).

The compatibility

E ∘ I_{Φ} = E

also follows from (6.6.1). Hence comes the proof of Proposition 6.2. □

From now on in this section, we will consider the case when

(6.7)  L_{0} = L and L_{1} = ϕ^{1}(L)

where ϕ^{1} is the time one map of a Hamiltonian isotopy Φ = {ϕ^{t}}_{0≤t≤1}. When Φ is given, the path space Ω(L, ϕ^{1}(L)) has the distinguished component, which we denote by

\begin{aligned}Ω(L, Φ) = \{ℓ : [0, 1] → M | & ℓ(0) ∈ L, ℓ(1) ∈ ϕ^{1}(L) and \\ & [t ↦ ϕ_{t}^{−1}(t)] = 0 in π_{1}(P, L)\} \end{aligned}

where the based path ℓ_{0} is given by the path

ℓ_{0}(t) = ϕ^{t}(x)

for a fixed point x ∈ L_{0}. Note that [t ↦ ϕ_{t}^{−1}z(t)] = 0 in π_{1}(P, L) is equivalent to saying that the path [t ↦ ℓ(t)] is homotopic to [t ↦ ϕ^{t}(ℓ(0))] in Ω(L_{0}, L_{1}).

We also note that the dependence of Ω(L, Φ) on Φ is not strong but depends only on the homotopy class of the path Φ = {ϕ^{t}}_{0≤t≤1} from id to ϕ^{1}. When Φ ≡ id, we will also denote Ω(L, id) by

Ω_{0}(L, L) = {ℓ : [0, 1] → M = 0 in π_{1}(P, L)}

or by Ω(L, L) omitting the subscript 0. We now study the covering group corresponding to the component Ω(L, Φ). We denote this covering group G(L, Φ).

Proposition 6.8. There exists a natural isomorphism

I_{id} : G(L, id) → G(L)

that is compatible with E.

Proof. Let (ℓ, w) and (ℓ, w′) be two elements of Ω̃(L, id) (with respect to a fixed path ℓ_{0}). We glue w and w′ along ℓ and obtain a map

w#w′ : [0, 1] × [0, 1] → M

such that

w#w′(s, t) = \{ \begin{matrix}w(2s, t) & for 0 ≤ s ≤ \frac{1}{2} \\ w(2 − 2s, t) & for \frac{1}{2} ≤ s ≤ 1 \end{matrix}

which has the property

(1) (w#w′)(s, 0), (w#w′)(s, 1) ∈ L,

(2) (w#w′)(0, ·) = (w#w′)(1, ·) = ℓ_{0}.

Since Φ ≡ id, we may choose a constant path ℓ_{0} ≡ p ∈ L in defining Ω̃(L, id) and so we may consider w#w′ as a map w#w′ : (D^{2}, ∂D^{2}) → (M, L). We set β = [w#w′] ∈ π_{2}(M, L) and then clearly

β · [ℓ, w′] = [ℓ, w].

This implies that the map G(L) → G(L, id) = G_{ℓ_{0}}(L, L) defined in §2 is surjective. Both injectivity and compatibility with E are easy to prove and omitted. □

In Chapter 4, we will prove that for given Hamiltonian isotopies Φ_{0} = {ϕ_{0}^{t}} and Φ_{1} = {ϕ_{1}^{t}}, there exists a natural isomorphism

(6.9)  (I_{Φ_{0}Φ_{1}})_{∗} : HF (L_{0}, L_{1}; ℓ_{0}) → HF (ϕ_{0}^{1}(L_{0}), ϕ_{1}^{1}(L_{1}); ℓ_{0}^{Φ_{0}Φ_{1}}),

for an appropriate choices of B’s. We denote by HF (L, Φ) the Floer homology that we defined for the pair (L, ϕ^{1}(L)). Combining (6.9), we prove that G(L, Φ) are naturally isomorphic under the homotopy of Φ.

Remark 6.10. It is interesting to note the case in the isomorphism (6.9) when Φ_{0}, Φ_{1} are Hamiltonian isotopies such that

L_{0} = ϕ_{0}^{1}(L_{0}), L_{1} = ϕ_{0}^{1}(L_{1}).

If they define non-contractible loops in the space of Lagrangian submanifolds, (6.9) may not be the identity isomorphism in general. In other words, the “monodromy isomorphism” (6.9) could detect π_{1} of Hamiltonian diffeomorphism groups of (M, ω). (Compare [Sei].)

With this remark in mind, we will compute HF (L, Φ) when Φ is a Hamiltonian path {ϕ_{t}}_{0≤t≤1}.

6.2. Calculation of HF (L; Φ): the Bott-Morse case.

If we define our Floer homology allowing Bott-type clean intersections (e.g., Φ ≡ id), then we can prove that there exists a canonical isomorphism

HF (L, L) → HF (L; Φ)

where HF (L, L) is the Floer homology in this generalized context. In this section, we define a spectral sequence computing this HF (L, L). In fact, Floer homology depends on the choice of the chains B(β) satisfying (4.4). Therefore it would be more precise to write HF ((L, b), (L, b)), where b represents the totality of B(β). However Theorem 6.11 below holds for any choice of B(β) and so the dependence on B(β) will be suppressed below.

We work over the universal Novikov ring Λ_{nov}, which we introduce now. Let T, e be two formal variables.

Definition 6.11. The ring Λ_{nov} is the totality of all formal sums ∑ a_{i}T^{λ_{i}}e^{n_{i}} such that (6.12.1) λ_{i} ∈ ℝ and n_{i} ∈ ℤ. (6.12.2) lim_{i→∞} λ_{i} = ∞. We define its subset Λ_{0,nov} by :

{∑ a_{i}T^{λ_{i}}e^{n_{i}} | λ_{i} ≥ 0, and n_{i} = 0 if λ_{i} = 0} .

We define the product of elements of Λ_{nov} in an obvious way. Λ_{nov} is then a commutative ring with the unit 1, and Λ_{0,nov} is its subring. We put

deg T^{λ}e^{n} = 2n.

Λ_{nov}, and Λ_{0,nov} then are graded rings. Let Λ_{nov}^{(k)}, and Λ_{0,nov}^{(k)} be the grading k part of them. We define a filtration F^{λ}Λ_{nov} of Λ_{nov} by

F^{λ}Λ_{nov} = {∑ a_{i}T^{λ_{i}}e^{n_{i}} | λ_{i} ≥ λ } .

It induces a filtration on Λ_{0,nov}. Those filtrations define uniform structures on Λ_{nov}, Λ_{0,nov}. By definition they are complete.

The above filtration on Λ_{nov} induces another filtrations on Λ_{nov} and Λ_{0,nov} defined by

𝔉^{q}Λ_{nov} = F^{qλ_{0}}Λ_{nov}, 𝔉^{q}Λ_{0,nov} = F^{qλ_{0}}Λ_{0,nov}.

Here λ_{0} is a positive number to be determined later. (q ∈ ℤ_{≥0} for Λ_{0,nov} and q ∈ ℤ for Λ_{nov}.) We then have

gr_{q}(𝔉Λ_{nov}) = 𝔉^{q}Λ_{nov}/𝔉^{q+1}Λ_{nov}, gr_{q}(𝔉Λ_{0,nov}) = 𝔉^{q}Λ_{0,nov}/𝔉^{q+1}Λ_{0,nov},

The following is the main result we prove in this section.

Theorem 6.13. Let L ⊂ M be a relatively spin Lagrangian submanifold. We assume that its obstruction classes o(β) vanish for all β. Then there exists a spectral sequence E_{r}^{p,q} with the following properties:

(6.14.1) E_{2}^{p,q} = ⨁_{k} H^{k}(L; ℚ) ⊗ gr_{q}(Λ_{nov}^{(p−k)}). (6.14.2) There exists a Floer homology HF (L, L; Λ_{0,nov}) with Λ_{0,nov} coefficient and its filtration 𝔉^{∗}HF (L, L; Λ_{0,nov}) such that

𝔉^{q}HF^{p}(L, L; Λ_{0,nov})/𝔉^{q+1}HF^{p}(L, L; Λ_{0,nov}) ≅ E_{∞}^{p,q}.

and that

HF (L, Φ; Λ_{nov}) ≅ HF (L, L; Λ_{0,nov}) ⊗_{Λ_{0,nov}} Λ_{nov}.

Here HF (L, Φ; Λ_{nov}) ≅ HF (L, Φ(L); Λ_{nov}) is the Floer homology with Λ_{nov} coefficient. (6.14.3) We define the subgroup K_{i} ⊂ E_{i} by :

K_{2} = P D(Ker(H_{k}(L; ℚ) → H_{∗}(M; ℚ))) ⊗ gr_{∗}(Λ_{0,nov}),
K_{r+1} = \frac{K_{r} ∩ Ker δ_{r}}{K_{r} ∩ Im δ_{r}} ⊂ E_{r+1}.

Then we have Im δ_{r} ⊆ K_{r} for every r. (6.14.4) Suppose that M(β) = ∅ for all β ∈ G(L) with μ(β) > 0. We consider the homology classes P D([L]) ∈ H^{0}(L; ℚ) and P D([pt]) ∈ H^{n}(L; ℚ) as the terms in E_{2} = H(L; ℚ) ⊗ gr_{∗}(Λ_{0,nov}) then we have

δ_{r}(P D([L])) = δ_{r}(P D([pt])) = 0 for every r

and hence they determine elements in E_{∞} and so in HF (L, Φ) from (6.14.2). Furthermore, they are linearly independent and in particular

HF (L, Φ) ≠ {0}.

We remark that Theorem E follows from Theorem 6.13 by changing the coefficient to Λ_{nov}.

A similar spectral sequence was constructed by the second named author in [Oh4,5] for the case of monotone Lagrangian submanifolds. We will give the outline of the construction of the spectral sequence E_{r}^{p,q} and the proof of (6.14.1) and (6.14.3) in this section and postpone the proof of (6.14.4) to the next section and (6.14.2) to §16 in Chapter 4. We need to study general properties of on the module over our universal Novikov ring to construct and study our spectral sequence. (Since our universal Novikov ring is not Noetherian a careful treatment is required to perform the construction.) We postpone the detailed argument to §A4.

For the purpose of constructing E_{r}^{p,q}, we need to take a chain complex representing a cohomology theory of L, which should be done carefully to handle the transversality problem. For this purpose we take a cochain complex \overline{S}(L, ℚ) of the currents represented by (rational) singular chains. (More precisely we consider distribution-valued sections of the bundle Λ^{k}(M) of differential forms rather than currents. Namely we consider a sub-complex of De-Rham complex of the sheaf of distribution-valued sections. So we have cochain complex rather than chain complex. See §A1 more on this point.) In other words we regard a codimension k chain as a k dimensional cochain.

We take a countably generated sub-complex C(L; ℚ) of \overline{S}(L, ℚ) which consists of the current realized by singular chains satisfying various transversality properties we need. See §13.2 and §A1 for it. Here in Chapter 2, we describe essential ideas without going into these technical problems.

We also remark that to construct and study the spectral sequence in Theorem 6.13, we work over Λ_{0,nov}. On the other hand, the invariance property of HF (L; Φ) we prove in Chapter 4 holds only with the coefficient ring Λ_{nov}. We will illustrate the difference between these two homologies in Example 6.19. Before doing so, let us mention the definitions of some other Novikov rings. We already defined the Novikov ring Λ(L_{1}, L_{2}) in Definition 5.6. We define Λ(L) in a similar way as follows.

Λ(L) = { ∑_{g∈G(L)} a_{g}[g] | ♯{g|a_{g} ≠ 0, E(g) < C} is finite for each C } .

(We defined the group G(L) in Definition 2.11.) The homomorphisms G(L_{i}) → G(L_{0}, L_{1}) introduced in §2 induces a ring homomorphism

i : Λ(L_{0}) ⊗_{ℚ} Λ(L_{1}) → Λ(L_{0}, L_{1}).

Let ϕ : M → M be a Hamiltonian isotopy. By the result of 6.1, it induces a homomorphisms ϕ_{∗} : G(L) → G(ϕ(L)), ϕ_{∗,0} : G(L_{0}, L_{1}) → G(ϕ(L_{0}), L_{1}) which preserve energy and Maslov index. Therefore it induces maps ϕ_{∗} : Λ(L) → Λ(ϕ(L)), ϕ_{∗,0} : Λ(L_{0}, L_{1}) → Λ(ϕ(L_{0}), L_{1}). The relation of these Novikov rings to our universal Novikov ring is as follows.

Lemma 6.15. There exist canonical inclusions i_{L} : Λ(L) → Λ_{nov}, i_{L_{0},L_{1}} : Λ(L_{0}, L_{1}) → Λ_{nov} with the following properties:

(6.16.1) Let i : Λ(L_{0}) ⊗_{ℚ} Λ(L_{1}) → Λ(L_{0}, L_{1}) as above, then i_{L_{0},L_{1}} ∘ i = i_{L_{0}} · i_{L_{1}}. (6.16.2) Let ϕ_{∗} : Λ(L) → Λ(ϕ(L)) be as above. Then i_{ϕ(L_{0})}ϕ_{∗} = i_{L_{0}}. (6.16.3) If ϕ_{∗,0} : Λ(L_{0}, L_{1}) → Λ(ϕ(L_{0}), L_{1}) is as above, then i_{ϕ(L_{0}),L_{1}}ϕ_{∗} = i_{L_{0},L_{1}}. Similarly we have i_{L_{0},ϕ(L_{1})}ϕ_{∗,0} = i_{L_{0},L_{1}}.

Proof. For each ∑ a_{g}[g] ∈ Λ(L), we define

(6.17)  i_{L_{0}} (∑ a_{g}[g]) = ∑ a_{g}T^{λ_{g}}e^{n_{g}/2},

where λ_{g} = E(g) is the energy and n_{g} = μ(g) is the Maslov class. (Since L is orientable, Maslov index μ(g) is even.) For ∑ a_{g}[g] ∈ Λ(L_{0}, L_{1}) we define i_{L_{0}} (∑ a_{g}[g]) by the same formula as (6.17). They satisfy the required properties because of the proof of Lemma 6.3. □

We put

(6.18)  Λ_{+}(L) = i_{L}^{−1}(Λ_{0,nov}) ⊆ Λ(L).

We can also construct a spectral sequence with gr_{∗}(𝔉Λ_{+}(L)) as its coefficient in place of gr_{∗}(𝔉Λ_{0,nov}).

Example 6.19. Consider a circle S^{1} in ℂ. We have G(S^{1}) = ℤ, G_{+}(S^{1}) = ℤ_{≥0} (independent of the choice of J). Therefore

Λ(S^{1}) = ℚ[[T]][T^{−1}], Λ_{+}(S^{1}) = ℚ[[T]]

and in particular, Λ(S^{1}) is the localization of Λ_{+}(S^{1}). gr_{∗}(Λ_{+}(S^{1})) ≅ Λ_{+}(S^{1}). In this case, our spectral sequence is described as follows:

\begin{aligned}E_{2} & = H^{∗}(S^{1}; ℚ) ⊗ Λ_{+}(S^{1}); \\ d_{2}(P D([pt])) & = T · P D([S^{1}]), \\ d_{2}(P D([S^{1}])) & = 0, d^{3} = ⋯ = 0. \end{aligned}

Therefore,

E^{3} = ⋯ = E^{∞} = \frac{ℚ[[T]] · P D([S^{1}])}{(T)P D([S^{1}])} ≅ \frac{Q[[T]]}{(T)} · P D([S^{1}]) ≅ ℚ · P D([S^{1}]).

This group is nontrivial. However we have

E^{∞} ⊗_{Λ_{+}(S^{1})} Λ(S^{1}) = 0

since T is invertible in Λ(S^{1}) = Q[[T]][T^{−1}]. This is consistent with (6.14.2) and with the fact that the Floer homology HF (S^{1}, id) must be zero since we know that HF (S^{1}, id) ≅ HF (S^{1}, Φ) for any Φ = {ϕ^{t}} and ℂ allows a Hamiltonian isotopy Φ such that ϕ^{1}(S^{1}) ∩ S^{1} = ∅.

Thus Floer homology HF (S^{1}, Φ) is independent of Φ if we consider it over Λ(S^{1}) coefficient but is not independent of Φ if we consider it over Λ_{+}(S^{1}) coefficient.

Now, we start with the construction of boundary operators.

Definition 6.20. We define an operator δ_{β:β_{i_{1}} ,... ,β_{im} ,β_{j_{1}} ,... ,β_{j_{ℓ}}} on C^{∗}(L; Λ_{+}(L)) by

δ_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} ([P, f])
= ±(M_{m+ℓ+2}(β − β_{i_{1}} − ⋯ − β_{i_{m}} − β_{j_{1}} − ⋯ − β_{j_{ℓ}})
(ev_{1},ev_{2},··· ,ev_{m+1},ev_{m+2},··· ,ev_{m+ℓ+1})×_{f×···}

(P × ∏_{d=1}^{m} B(β_{i_{d}}) × ∏_{e=1}^{ℓ} B(β_{j_{e}})), ev_{0})

The precise sign on the right hand side will be given in §25.3.

Figure 6.1

Lemma 6.21. If [P, f] ∈ C^{k}(L; ℚ), then

δ_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [P, f] ∈ C^{k−μ(β)+1}(L; ℚ).

Proof. We will prove only the grading formula postponing the transversality discussion to Chapter 5. First, we compute

\begin{aligned}dim M_{m+ℓ+2} & (β − β_{i_{1}} − ⋯ − β_{i_{m}} − β_{j_{1}} − ⋯ − β_{j_{ℓ}}) \\ & = n + μ(β) − ∑_{d} μ(β_{i_{d}}) − ∑_{e} μ(β_{i_{e}}) + m + ℓ + 2 − 3 \end{aligned}

and

dim B(β_{i_{d}}) = n + μ(β_{i_{d}}) − 1.

Note that the dimension of the chain (P, f) is n − k. Therefore dimension of the chain δ_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [P, f] defined as in Definition 6.20, becomes

n + μ(β) − ∑_{d} μ(β_{i_{d}}) − ∑_{e} μ(β_{i_{e}}) + m + ℓ + 2 − 3

+ ∑_{d}(n + μ(β_{i_{d}}) − 1) + ∑_{e}(n + μ(β_{j_{e}}) − 1) + (n − k) − (m + ℓ + 1)n

= n − (k − μ(β) + 1)

which finishes the proof. □

Now, we define our boundary operator δ by

(6.22)  \begin{aligned}δ([P, f] ⊗ α) & = ∑_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} \frac{1}{m!ℓ!}δ_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [P, f] \\ & ⊗ (α + β). \end{aligned}

Lemma 6.23. With the definition (6.22), we have

δ([P, f] ⊗ α) ∈ C(L; Λ_{+}(L))

and

δ(C^{k}(L; Λ_{+}(L))(C)) ⊂ C^{k+1}(L; Λ_{+}(L))(C).

The proof of this lemma is similar to that of Proposition 5.8 and so will be omitted. We remark that

α ∈ G_{+}(L) if δ_{α:β_{i_{1}} ,... ,i_{m};β_{j_{1}} ,... ,β_{j_{ℓ}}} ≠ 0.

Theorem 6.24. δ satisfies δ ∘ δ = 0.

Proof. Let [P, f] ⊗ α ∈ C(L; Λ_{0}(L)). We consider the (α + β)-component of δ ∘ δ([P, f] ⊗ β). By definition, this component is

(6.25)  \begin{aligned}∑_{α_{1}+α_{2}=α ; {β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}} \atop β_{i′_{1}} ,... ,β_{i{′ \atop m′}} ;β_{j′_{1}} ,... ,β_{j{′ \atop ℓ′}}}} \frac{1}{m!ℓ!m′!ℓ′!} & δ_{α_{1}:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} \\ & ∘ δ_{α_{2};β_{i′_{1}} ,... ,β_{i{′ \atop m′}};β_{j′_{1}} ,... ,β_{j{′ \atop ℓ′}}} [P, f] \end{aligned}

A typical term in (6.25) can be described by the following figure.

Figure 6.2.

We consider the following moduli space

(6.26)  \begin{matrix}M_{m+ℓ+m′+ℓ′+2}(α − ∑_{d} β_{i_{d}} − ∑_{d′} β_{i′_{d′}} − ∑_{e} β_{j_{e}} − ∑_{e′} β_{j′_{e′}}) \\ _{(ev_{1},ev_{2},... ,ev_{m+m′+1},... ,ev_{m+m′+ℓ+ℓ′+1})}×_{f×···} \\ [P × ∏_{d=1}^{m}(B(β_{i_{d}}) × ∏_{d′=1}^{m′} B(β_{i′_{d′}}) & × ∏_{e=1}^{ℓ} B(β_{j_{e}}) × ∏_{e′=1}^{ℓ′} B(β_{j′_{e′}})] \end{matrix}

which defines a rational geometric cycle in M via the evaluation map ev_{0}. The boundary of (6.26) has various components. For example, one of them is obtained by replacing P by ∂P in (6.26). This corresponds to the case when α_{2} = 0 i.e., the term δ_{α:···} ∘ δ_{0} in (6.25).

General boundary components arise from the following phenomena:

(6.27.1) One of the marked point is sent to ∂B(β) = (−1)^{n+1}o(β).

(6.27.2) The disc splits into two.

The terms in (6.27.1) is pictured in Figure 6.3.

Figure 6.3

We can divide (6.27.2) into the following two sub-cases:

(6.27.2.1) The 0-th and the (m + m′ + 1)-th marked points are on the same disc after splitting. (6.27.2.2) The 0-th and the (m + m′ + 1)-th marked points are on different discs after splitting.

Here we note that the 0-th marked point is the one where we take the evaluation ev_{0} and the (m + m′ + 1)-th marked point is the one which we assume is sent to P. Figure 6.3 also corresponds to the case (6.27.2.1). Therefore (6.27.1) and (6.27.2.1) cancel each other. On the other hand, Figure 6.2 corresponds to the case (6.27.2.2) and so (6.27.2.2) gives the typical terms in (6.25). Therefore, we conclude that the (α+β)-component (6.25) of δ ∘δ([P, f]⊗α) defines the zero chain via ev_{0} for any β. We refer Proposition 25.10 in §25.3 for the precise argument about the orientations. This finishes the proof of Theorem 6.24. □

We will prove the following theorem in §16 Chapters 4 and Chapter 5.

Theorem 6.28. We have

H^{∗}(C(L; Λ_{+}(L)) ⊗_{Λ_{+}(L)} Λ_{nov}, δ) = HF (L, L; Λ_{nov}).

Theorem 6.28 will imply (6.14.2).

Now we use the energy filtration to construct our spectral sequence as follows. In this section, we consider only the case where the following additional assumption is satisfied. The general case will be discussed in §A4.

Condition 6.29. Let L satisfy that the subgroup

E(G(L)) = {E(g)|g ∈ G(L)} ⊂ ℝ

is discrete. Such a Lagrangian submanifold is sometimes called rational.

We then use the following Novikov ring.

Λ̂_{+}(L) = {∑ a_{i}T^{λ_{i}}e^{n_{i}} ∈ Λ_{0,nov} | λ_{i} ∈ E(G(L))} .

Note that Λ̂_{+}(L) is Noetherian if Condition 6.29 is satisfied. (In fact its degree 0 part Λ̂_{+}(L)^{(0)} is a discrete valuation ring.) This simplifies the study of our spectral sequence.

We assume Condition 6.29 in the rest of this section unless otherwise mentioned. Assuming G(L) ≠ {0}, we put

E(G(L)) = ℤ · λ_{0}, λ_{0} > 0.

(Then we indeed have

Λ̂_{+}(L) ≅ ℚ[[S]][e, e^{−1}]

where S = T^{λ_{0}}.)

The filtration F^{λ}Λ_{0,nov} induces a filtration on Λ̂_{+}(L). We then put

𝔉^{m}Λ̂_{+}(L) = F^{mλ_{0}} Λ̂_{+}(L).

Then obviously we have

gr_{q}(𝔉Λ̂_{+}(L)) ≅ ℚ[e, e^{−1}].

Note

i_{L}(Λ_{+}(L)) ⊆ Λ̂_{+}(L).

We consider the chain complex

C(L; Λ̂_{+}(L)) = C(L; Λ_{+}(L)) ⊗_{Λ_{+}(L)} Λ̂_{+}(L).

The filtration 𝔉^{∗}Λ̂_{+}(L) induces a filtration 𝔉^{∗}C(L; Λ̂_{+}(L)) on C(L; Λ̂_{+}(L)). This filtration defines the spectral sequence. We remark that Λ_{0,nov} is flat as Λ̂_{+}(L) module. Therefore a spectral sequence in Theorem 6.13 is induced.

Now we start with the proof of (6.14). We first note that there is an isomorphism

gr_{∗}(𝔉C(L; Λ̂_{+}(L))) ≅ C(L; ℚ) ⊗ gr_{∗}(𝔉Λ̂_{+}(L))

as gr_{∗}(F^{∗}Λ̂_{+}(L)) module. By construction the differential induced on it is (−1)^{n}δ_{0}⊗ id where δ_{0} is the classical boundary operator on C(L; ℚ). (6.14.1) is an immediate consequence of this.

(6.14.2) follows from Theorem 6.28. (In fact we need to discuss convergence properties of the spectral sequence to derive (6.14.2) from Theorem 6.28. We will do it in §A4 in much more delicate situation.)

We are now going to verify (6.14.3). We do not assume Condition 6.29 for a moment, (since the result below will be used also in §A4 to study the general case). We define \overline{S}(M; ℚ) in a similar way as \overline{S}(L; ℚ) and take an appropriate countably generated sub-complex

C^{k}(L; Λ̂_{+}(L)) ⊆ (\overline{S}(M; ℚ) ˆ⊗ Λ̂_{+}(L))^{k}.

We define an operator b_{β} : C(L; Λ̂_{+}(L)) → C(M; Λ̂_{+}(L)) by the sum

b_{β} = ∑_{β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} \frac{1}{m!ℓ!}b_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}}

where each map in the summand is defined by

b_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [P, f]

= ±(M_{m+ℓ+1,1}(β − β_{i_{1}} − ⋯ − β_{i_{m}} − β_{j_{1}} − ⋯ − β_{j_{ℓ}})

(ev_{1},ev_{2},... ,ev_{m+ℓ+1}) ×_{f×···} (P × ∏_{d=1}^{m} B(β_{i_{d}}) × ∏_{e=1}^{ℓ} B(β_{j_{e}})), ev^{+})

The sign on the right hand side is the same as in Definition 6.15. (We recall that M_{k,1}(⋯) denotes the moduli space of stable map from discs with k marked points on the boundary and one marked point at interior.

Figure 6.4

Lemma 6.30. If [P, f] ∈ C^{k}(L; ℚ), then

b_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [P, f] ∈ C^{k−μ(β)}(M; ℚ)

Lemma 6.31. The element

b([P, f] ⊗ α) = ∑_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} \frac{1}{m!ℓ!}b_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [P, f] ⊗ (α + β)

is well-defined as an element in C(M; Λ̂_{+}(L)).

The proofs of these two lemmas are repetitions of that of Lemma 6.21 and so omitted.

Proposition 6.32. Let δ_{0} be the standard coboundary operator on C^{∗}(M; Λ̂_{+}(L)). Then we have

(6.33.1)  δ_{0}b ± bδ = 0
(6.33.2)  (id − b) ≡ 0 mod T^{λ_{1}}, where λ_{1} > 0.

The proof of (6.33.1) again is similar to those of Theorem 6.24 and Proposition 4.21, and so will be omitted. (6.33.2) is immediate from construction.

Now we assume Condition 6.29. (6.14.3) then follows from Proposition 6.32 via the standard construction of spectral sequence. (See §A4 for detail.) □

Remark 6.34. In case L ⊂ M is monotone, our spectral sequence coincides with the one constructed by the second named author [Oh4]. There are two differences between these two constructions: One is that the Morse-Witten complex is used there while the singular chain complex (or its image in the chain complex of currents) are used here. Although this is rather a technical point, because of various “higher order” terms appearing in our situation here, it seems more difficult to work with Morse-Witten complex as done in [Oh4].

The other is that the Morse index was used to define a filtration in [Oh4], while we use the energy here. However in case of monotone Lagrangian submanifolds, the two filtrations coincide by the definition of monotonicity.

§7. Obstruction class of the top dimension.

We first remark that the obstruction class o(β) with μ(β) > 0 could be nonzero only when μ(β) = 2. In case μ(β) = 2, the corresponding obstruction class lies in the top dimensional homology H_{n}(L; ℚ). The class o(β) with μ(β) = 2 has some distinguished properties which are not shared by others with μ(β) ≤ 0. So far, we have not used these properties. In this section, we study them. First, we slightly improve the condition under which the k-th obstruction class o(β_{k}) be defined.

Proposition 7.1. The class o(β_{k}) is well-defined if we assume that o(β_{i}) is defined and becomes zero in homology for i < k only for those β_{i}’s with μ(β_{i}) ≤ 0.

Proof. We put G_{+++}(L) = {β ∈ G_{++}(L) | μ(β) ≤ 0} and re-define our obstruction class only using G_{+++}(L)

(7.2)  o_{k}(L) = ∑_{{m=0,1,2,... \atop i_{1},... ,im<k} ; β_{k}−β_{i_{1}} −···−β_{im} ∈G_{+++}(L)} \frac{1}{m!}(M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{m}}), ev_{0})

in place of Definition 4.8. We will prove ∂o_{k}(L) = 0 with this modified definition by induction on k. The proof of §4 shows that in the boundary ∂o_{k}(L), only the terms corresponding to Figure 4.4 with μ(β_{i_{j}}) = 2 could survive.

For the convenience of our exposition, we assume without loss of generalities that L is connected. Since o(β_{i_{j}}) with μ(β_{i_{j}}) = 2 is n-dimensional, it must be a multiple of the fundamental cycle (not only as a homology class but as a chain (current)). We set o(β_{i_{j}}) = c_{i_{j}} · L and compute c_{i_{j}} by counting the order (counted with sign) of the moduli space described by the following Figure 7.1.

Figure 7.1.

Note that the condition that the j-th marked point be mapped to a point in L, is an empty condition and so the geometric chain corresponding to Figure 7.1 has the same support as that of the chain corresponding to the following Figure 7.2.

Figure 7.2.

We now calculate the (virtual) dimension of the moduli space described by Figure 7.2,

m + 1 + μ(β_{k} − β_{i_{1}} − ⋯ − β_{i_{m+1}}) − 3 − ∑_{d≠j} codim B(β_{i_{d}})

= m − 2 + μ(β_{k}) − ∑_{j}^{m} μ(β_{i_{j}}) − ∑_{d≠j}^{m}(−μ(β_{i_{d}}) + 1)

= μ(β_{k}) − μ(β_{i_{j}}) − 1 = μ(β_{k}) − 3..

This is strictly smaller than dim o(β_{k}) = μ(β_{k})−2. Therefore, the terms corresponding to Figure 7.1 become zero as (μ(β_{k}) − 2)-chains and so the obstruction chain given in (7.2) has zero boundary i.e., ∂o_{k}(L) = 0. We however note that we need to take care of one delicate point to make the current proof completely rigorous. See Remark 7.9 and Section 20 for complete explanation on this point. □

Proposition 7.3. The Floer cohomology HF (L; Φ) for a Hamiltonian isotopy Φ = {ϕ^{t}}_{0≤t≤1} and the spectral sequence defined in Theorem 6.13 are well-defined, even if we assume that o(β_{k}) = 0 only for μ(β_{k}) ≤ 0. They also satisfy the properties in Theorem E and Theorem 6.13.

The proof of Proposition 7.3 will be completed in Chapter 4. The construction of the chain complex (C^{∗}(L; Λ_{+}(L))⊗_{Λ_{0}(L)} Λ(L), δ) and proof of δ ∘δ = 0, assuming Proposition 7.3, goes in the same way as the proof in §6 and of Proposition 7.1. So we omit it. The proof that the resulting spectral sequence converges to the Floer cohomology is the same as the proof of (6.14.2) which will be given in Chapter 4.

To illustrate the statement (6.14.4) in Theorem 6.13, let us look at the following example.

Example 7.4. Consider an embedded circle in S^{2} ≃ ℂP^{1}. There are two holomorphic discs which bound the circle. We denote them by D_{u} and D_{ℓ}. In the case they have the same area, they belong to the same class β in G(S^{1}) ≃ ℤ. Otherwise they belong to two different classes in G(S^{1}) ≃ ℤ ⊕ ℤ. We denote the classes by β_{u} and β_{ℓ} respectively. It is easy to see that

μ(β) = μ(β_{u}) = μ(β_{ℓ}) = 2.

We write by S_{eq}^{1} for the first circle and S_{neq}^{1} for the second.

7.4.1. The case for S_{eq}^{1} : HF (S_{eq}^{1}; Φ) ≃ HF (S_{eq}^{1}, S_{eq}^{1}).

We will check in Proposition 11.9 that these two holomorphic discs have the same contribution to the obstruction class o(ℂP^{1}, S_{eq}^{1}; β) ∈ H_{1}(S^{1}; ℚ) and hence we have

o(ℂP^{1}, S_{eq}^{1}; β) = 2[S_{eq}^{1}] in H_{1}(S^{1}; ℚ).

Since there is no holomorphic discs with negative Maslov indices (in fact this is the monotone case), the Floer homology is well-defined by Proposition 7.3. We now perturb S_{eq}^{1} by a Hamiltonian isotopy such that ϕ^{1}(S_{eq}^{1}) ∩ S_{eq}^{1} = {p, q}. (Note that #(ϕ^{1}(S_{eq}^{1}) ∩ S_{eq}^{1}) ≥ 2). The moduli space of connecting orbits from p to q (or q to p) with minimal area has two components, respectively. We will prove that the two have different contribution cancelling each other (See Proposition 11.5 and 11.9 in Chapter 3). One can prove the same cancellation for the components with nonminimal areas. Hence the boundary operator is trivial and the well-defined Floer homology

HF (S_{eq}; Φ) ≅ ℚ[[T]][T^{−1}] ⊕ Q[[T]][T^{−1}].

7.4.2. The case for S_{neq}^{1} : HF (S_{neq}^{1}; Φ).

We first note that there exists a large Hamiltonian isotopy Φ = {ϕ^{t}}_{0≤t≤1} with ϕ^{1}(S_{neq}^{1}) ∩ S_{neq} = ∅. Therefore if the Floer homology HF (S_{neq}^{1}; Φ) is well-defined and invariant under the change of Φ, then it must be zero. To check this, we again consider the case where

ϕ^{1}(S_{neq}^{1}) ∩ S_{neq}^{1} = {p, q}.

Figure 7.3

Note that in Figure 7.3, the two connecting orbits from q to p will be in different orbits under the deck transformation G(S_{neq}^{1}, Φ), while those from p to q cancels each other as in the above case. This proves that the boundary ∂ maps

[p] ↦ 0
[q] ↦ T_{u}[p] − T_{ℓ}[p] + “higher” = (T_{u} − T_{ℓ})[p] + “higher”

where T_{u} = [β_{u}] and T_{ℓ} = [β_{ℓ}]. From this, we conclude that HF (S_{neq}; Φ) is indeed zero, because T_{u} − T_{ℓ} is invertible in the Novikov ring Λ(S_{neq}^{1}).

7.4.3. The case of (S_{eq}^{1}, S_{neq}^{1}).

We consider the case where S_{1}^{1} is not Hamiltonian isotopic to S_{2}^{1}. For example, one could take S_{1}^{1} = S_{eq}^{1} and S_{2}^{1} = S_{neq}^{1}. Again by considering the case ϕ^{1}(S_{eq}^{1}) ∩ S_{neq}^{1} = {p, q}, we derive that the boundary δ maps

[p] ↦ [q] − T_{u}^{−1}T [q] + “higher”
[q] ↦ T_{ℓ}T^{−1}[p] − T_{u}T^{−1}[p] + “higher”.

From this, we prove that δ ∘ δ ≠ 0 and so the Floer homology is not defined.

We now go back to modifying the construction of boundary operators δ_{β} given in §6 under the weaker hypotheses of Proposition 7.3. We define δ_{β;β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} by the same formula as in Definition 6.20, but summing over β_{i_{d}}’s and β_{j_{e}}’s with μ(β_{i_{d}}) ≠ 2 ≠ μ(β_{j_{e}}). Every term involving β with μ(β) = 2 in the proof of δ ∘ δ = 0 will become zero by the argument as in Proposition 7.1. Other terms in δ ∘ δ will be cancelled away as in the proof of Theorem 6.24. □

We note that by the same argument, one can prove that the operators defined in Definition 7.2 and 4.8 coincide in case both are defined.

We finally prove (6.14.4) of Theorem 6.11. We would like to remark that the assumption in (6.14.4) that M(β) is empty for any β with μ(β) > 0 is different from those in Proposition 7.1 and 7.3. Unlike Proposition 7.1 and 7.3, for the proof of (6.14.4), we really need the assumption on M(β) for β with μ(β) > 2 too.

For the case of Propositions 7.1 and 7.3, the assumption that o(β) = 0 is automatic for μ(β) > 2. Although the obstruction class and the extra boundary operators δ_{β;β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} both are related to the same moduli space M(β), they behave differently. For example as we mentioned in Example 7.4, the upper and lower hemispheres have the same sign for defining obstruction classes but have the different signs for defining (modified) boundary operators.

Now we start with the proof of (6.14.4). We first remark from the hypothesis (6.4.4) that the hypothesis in (6.4.4) implies that the operator δ_{β} is zero unless μ(β) ≤ 0. This implies that the Floer’s degree itself also provides a filtration on our cochain complex C^{∗}(L; Λ_{+}(L)) = C^{∗}(L; ℚ) ˆ⊗ Λ_{+}(L). More precisely, it follows from Lemma 6.21

(7.5)  δ(C^{k}(L; ℚ) ˆ⊗ Λ̂_{+}(L)) ⊂ ⨁_{ℓ≥k+1} C^{ℓ}(L; ℚ) ˆ⊗ Λ̂_{+}(L).

(Note that C^{k}(L; ℚ) ˆ⊗ Λ̂_{+}(L) is different from the degree k part of C(L; ℚ) ˆ⊗ Λ̂_{+}(L).) (7.5) immediately implies that the (Poincaré dual of) the fundamental cycle [L] will never become a boundary i.e., P D[L] will never be in Im δ_{r}. It also implies that δ_{r}(P D([pt])) = 0. for every r.

Therefore the main part of the proof will be to show

72 FUKAYA, OH, OHTA, ONO

Proposition 7.6. δ_{r}(P D([L])) = 0 for every r. Proof. We write [L] in stead of P D([L]) from now on. It suffices to show that δ_{β}([L]) = 0 for every β. We recall

(7.7)  \begin{matrix}δ_{β:β_{i_{1}} ,... ,β_{im}:β_{j_{1}} ,... ,β_{j_{ℓ}}} (L) \\ = (M_{m+ℓ+2}(β − β_{i_{1}} − β_{i_{2}} − ⋯ − β_{i_{m}} − β_{j_{1}} − ⋯ − β_{j_{ℓ}}) \\ _{(ev_{1},ev_{2},... ,ev_{m+ℓ+1})} ×_{f×···} (L × ∏_{d=1}^{m} B(β_{i_{d}}) × ∏_{e=1}^{ℓ} B(β_{j_{e}})), ev_{0}), \end{matrix}

We use the following Figure 7.5 to describe the right hand side of (7.7)

Figure 7.5

Since the condition that ev_{1}(z_{1}) ∈ L is an vacuous condition, the corresponding geometric chain has the same support as the one represented by the following Figure 7.6.

Figure 7.6

Now, we calculate the (virtual) dimension of the moduli space described by Figure 7.6. It is

(7.8)

dimM_{m+ℓ+1}(β − β_{i_{1}} − ⋯ − β_{i_{m}} − β_{j_{1}} − ⋯ − β_{j_{ℓ}})

− ∑_{d} codim B(β_{i_{d}}) − ∑_{e} codim B(β_{j_{e}})

= (m + ℓ + 1) − 3 + n + μ(β − β_{i_{1}} − ⋯ − β_{j_{1}} − ⋯ − β_{j_{ℓ}})

+ ∑_{d=1}^{m}(μ(β_{i_{d}}) − 1) + ∑_{e=1}^{ℓ}(μ(β_{j_{e}}) − 1)

= n + μ(β) − 2.

Hence its codimension is 2 − μ(β). On the other hand, we have the degree of δ_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [L] which is supposed to be 1 − μ(β). Therefore

δ_{β:β_{i_{1}} ,... ,β_{im};β_{j_{1}} ,... ,β_{j_{ℓ}}} [L] = 0.

More precisely, we have the following commutative diagram: (7.9)

M_{m+ℓ+1}(β − β_{i_{1}} − ⋯ − β_{i_{ℓ}}) × (L × ∏_{d=1}^{m} B(β_{i_{d}}) × ∏_{e=1}^{ℓ} B(β_{j_{ℓ}})) ⟶^{ev_{0}} L
↓  ^{=}↓
M_{m+ℓ}(β − β_{i_{1}} − ⋯ − β_{i_{ℓ}}) × (∏_{d=1}^{m} B(β_{i_{d}}) × ∏_{e=1}^{ℓ} B(β_{j_{ℓ}})) ⟶^{ev_{0}} L

Moreover the virtual dimension of the fiber of the vertical arrow is positive. Therefore

[M_{m+ℓ+1}(β − β_{i_{1}} − ⋯ − β_{i_{ℓ}}) × (L × ∏_{d=1}^{m} B(β_{i_{d}}) × ∏_{e=1}^{ℓ} B(β_{j_{ℓ}})), ev_{0}] = 0

as a current. This finishes the proof of Proposition 7.6. □

We next remark that [pt] is not zero in E_{∞}. This is an immediate consequence of (6.14.3) because the natural homomorphism H_{0}(L) → H_{0}(M) is injective. Finally the independence of [L] and [pt] over Λ(L) follows from the fact that the filtration by the degree is preserved by the differential. This finally finishes the proof of (6.14.4).

Remark 7.9. We would like to remark that the proofs of Proposition 7.1 and 7.6 have some delicate points to be refined, which we suppress at this point and postpone its full discussion until §20. In both proofs, we strongly use the fact that the forgetful map M_{m+1}(β) → M_{m}(β) is compatible with the evaluation maps. It turns out that this compatibility fails after perturbation to achieve transversality. More specifically, we may not assume the diagram (7.9) commutes. See §20 for detailed explanations on this.

Chapter 3. Applications

§8. Estimate of the number of intersection points.

In this section, we prove Theorem F when Condition 6.29 is satisfied. The proof of Theorem F in the general case will be given in §A4. We assume that L is unobstructed in this section.

We consider the Novikov ring

Λ̂(L) = {∑ a_{i}T^{λ_{i}}e^{n_{i}} ∈ Λ_{nov} | λ_{i} ∈ E(G)}

where E is the symplectic area and G is the group defined as in §6, and study the Floer homology HF^{p}(L, L; Λ̂(L)) with this coefficient ring. (Λ̂(L) is a localization of the ring Λ̂_{+}(L) introduced in §6.) More precisely, we need to specify b ∈ M(L) and to write the Floer homology as HF^{p}((L, b), (L, b); Λ̂(L)). We omit b from our notation when no confusion can occur. Λ̂(L) has a filtration 𝔉^{m}Λ̂(L) such that its associated graded ring becomes

gr_{q}(𝔉Λ̂(L)) ≅ ℚ[e, e^{−1}].

We consider a countably generated complex C(L; Λ̂(L)) and its filtration 𝔉^{∗}C(L; Λ̂(L)) which we introduced in §6. Then, in place of the spectral sequence in Theorem E, we obtain a spectral sequence E_{r}^{p,q} such that

(8.1)  E_{2}^{p,q} = ⨁_{s=0}^{n} H^{s}(L; ℚ) ⊗ gr_{q}(𝔉Λ̂(L)^{(p−s)}).

Here Λ̂(L)^{(p−s)} denotes the degree p − s part. We can also show that there exists a filtration 𝔉^{∗}HF^{p}((L, L; Λ̂(L)) such that

(8.2)  E_{∞}^{p,q} ≅ 𝔉^{q}HF^{p}(L, L; Λ̂(L))/ 𝔉^{q+1}HF^{p}((L, L; Λ̂(L)).

Note that the multiplication by T^{λ_{0}} gives an isomorphism between 𝔉^{q}HF^{p}(L, L; Λ̂(L)) and 𝔉^{q+1}HF^{p}(L, L; Λ̂(L)). Also the multiplication by e gives an isomorphism between 𝔉^{q}HF^{p}(L, L; Λ̂(L)) and 𝔉^{q}HF^{p+2}(L, L; Λ̂(L))

Let D be a graded Λ̂(L) module. We assume that it has a filtration 𝔉^{∗}D such that

𝔉^{ℓ}Λ̂(L) · 𝔉^{k}D ⊆ 𝔉^{k+ℓ}D.

Let Λ̂(L)^{(0)} be the degree 0 part of Λ̂(L). We have

Λ̂(L)^{(0)} ≅ ℚ[[T^{λ_{0}}]][T^{−λ_{0}}].

Hence Λ̂(L)^{(0)} is a field. Therefore, the degree p part D^{p} of D is a vector space over Λ̂(L)^{(0)}.

Definition 8.3 We define

rank D = dim_{Λ̂(L)^{(0)}} D^{p} + dim_{Λ̂(L)^{(0)}} D^{p+1}.

Since the multiplication by e defines an isomorphism D^{p} ≅ D^{p+2}, it follows that the right hand side is independent of p.

Lemma 8.4. For any q, we have :

rank HF (L, L; Λ̂(L)) = dim_{ℚ} 𝔉^{q}HF^{p}(L, L; Λ̂(L))/𝔉^{q+1}HF^{p}(L, L; Λ̂(L))

+ dim_{ℚ} 𝔉^{q}HF^{p+1}(L, L; Λ̂(L))/𝔉^{q+1}HF^{p+1}(L, L; Λ̂(L))

= dim_{ℚ} E_{∞}^{p,q} + dim_{ℚ} E_{∞}^{p+1,q}.

The proof is easy and is omitted.

The following lemma follows easily from Theorem 6.28 and Theorem C.

Lemma 8.5. If ϕ is a Hamiltonian diffeomorphism and if L is transversal to ϕ(L) then

rank HF ((L, b), (L, b); Λ̂(L)) ≥ ♯(L ∩ ϕ(L)).

for any b ∈ M(L).

We now estimate dim_{ℚ} E_{∞}^{p,q} + dim_{ℚ} E_{∞}^{p+1,q} from below. Let δ_{r} : E_{r} → E_{r} be the boundary operator of the spectral sequence. We have δ_{r}δ_{r} = 0 and

E_{r+1} = Ker δ_{r}/ Im δ_{r}.

E_{r} is a gr_{∗}(𝔉Λ̂(L)) module and gr_{∗}(𝔉Λ̂(L)) ≅ Λ̂(L). Hence we can define rank E_{r} by Definition 8.3.

Lemma 8.6.

rank E_{r} − 2 rank Im δ_{r} = rank E_{r+1} ≤ rank E_{r}.

Proof. There exist short exact sequences of Λ̂(L) modules :

0 → Ker δ_{r} → E_{r} → (Im δ_{r})[−1] → 0,
0 → Im δ_{r} → Ker δ_{r} → E_{r+1} → 0.

We obtain

rank E_{r} = rank Ker δ_{r} + rank Im δ_{r}

from the first exact sequence. The second exact sequence then implies the lemma. □

Now we give the proof of Theorem F, when Condition 6.29 is satisfied. It follows from (8.1) and the fact that e is of degree 2 ,we have

rank E_{2} = ∑ rank_{ℚ} H_{∗}(L; ℚ).

We put K_{2} = P D(Ker(H_{∗}(L; ℚ) → H_{∗}(M; ℚ))) ⊗_{ℚ} gr_{∗}(𝔉^{∗}Λ̂(L)),

K_{r+1} = \frac{K_{r} ∩ Ker(δ_{r})}{K_{r} ∩ Im(δ_{r})}

as in (6.14.3). Then Im(δ_{r}) ⊂ K_{r}. Therefore

rank K_{r+1} ≥ rank K_{r} − rank Im(δ_{r}).

Hence using Lemma 8.6 we obtain

rank E_{r} = rank E_{2} − 2 ∑_{2≤i<r} rank Im(δ_{r})
= ∑ rank_{ℚ} H^{∗}(L; ℚ) − 2 ∑_{2≤i<r} rank Im(δ_{r})
≥ ∑ rank_{ℚ} H^{∗}(L; ℚ) − 2(rank K_{2} − rank K_{r}).

Therefore

rank HF (L, L) = rank E_{∞} ≥ ∑ rank_{ℚ} H^{∗}(L; ℚ) − 2 rank K_{2}.

The proof of Theorem F is complete when Condition 6.29 is satisfied. □

§9. Automorphism groups of marked discs.

So far we have assumed that L is relatively spin. This is because the general stable compactification scheme involves rational cycles and so requires the orientations on the various moduli spaces of stable maps. However these restrictions on L are too strong for the applications and so we will attempt to relax it.

Note that the only reason why we need to use rational cycles in the stable map compactification comes from the fact that the automorphism groups of stable maps we use could be non-trivial. A simple examination of the constructions in [FOn2] or in Chapter 5 later of this paper in fact shows that all the construction would work with ℤ_{2}-coefficient without assuming any condition except vanishing of the obstruction classes (with ℤ_{2}-coefficient), as long as all the stable maps involved in the construction have trivial automorphism groups.

In this respect, the following observation is important, although its proof is immediate from the structure of P SL(2; ℝ).

Lemma 9.1. The automorphism group of a semi-stable open curve (Σ, z⃗) ∈ M_{m}^{b} is torsion free if m ≥ 1 and if it has no sphere components. In particular, if a stable map ((Σ, z⃗), w) ∈ M_{m}(J : β) with m ≥ 1 has no sphere bubbles, then the automorphism group of ((Σ, z⃗), w) is trivial.

(We specify J, the almost complex structure, in the notation M_{m}(J : β) above, since we use perturbation of J in this section and so J-dependence of the moduli space is a crucial issue here.) An immediate corollary is the following

Corollary 9.2. If any element ((Σ, z⃗), w) ∈ M_{m}(J : β) does not have sphere components, then any ((Σ, z⃗), w) ∈ M_{m}(J : β) with m ≥ 1 has trivial automorphism group. In particular, this is the case when (M, ω, J) does not allow any pseudoholomorphic sphere.

Definition 9.3. A Lagrangian submanifold L ⊂ (M, ω) is called semi-positive if ω(β) ≤ 0 for any β with

3 − n ≤ μ_{L}(β) < 0.

Remark 9.4.

(1) Note that if n ≤ 3, the semi-positivity automatically holds.

(2) If L ⊂ (M, ω) is semi-positive, then (M, ω) itself must be semi-positive in the sense of [Mc2].

(3) All minimal Lagrangian submanifolds, in particular special Lagrangian submanifolds, on Calabi-Yau manifold are semi-positive. (See [Da].)

Note that if M has a semi-positive Lagrangian submanifold then M is semipositive. (See [Oh2].) McDuff [Mc2] proved that there exists a compatible almost complex structure such that all somewhere injective pseudo-holomorphic spheres are transversal. Therefore if M has a semi-positive Lagrangian submanifold then there exists no pseudo-holomorphic sphere v : S^{2} → M such that c^{1} ∩ v_{∗}[S^{2}] < 0. In fact if v : S^{2} → M is a pseudo-holomorphic sphere with c^{1} ∩ v_{∗}[S^{2}] < 0 then there exists a somewhere injective v′ : S^{2} → M with the same property. On the other hand, semi-positivity implies that the virtual dimension of moduli space of such v′ : S^{2} → M is negative.

Hereafter we fix a compatible almost complex structure such that exists no pseudo-holomorphic sphere v : S^{2} → M with c^{1} ∩ v_{∗}[S^{2}] < 0.

The following Theorem 9.5 is the main result in this section. (We refer the notations, Kuranishi structure, obstruction bundle etc. to §A2 or [FOn2].)

Theorem 9.5. Let L ⊂ (M, ω) be semi-positive. Let P_{1}, ⋯, P_{k} and β ∈ π_{2}(M, L) be chains given. We consider the moduli space

M_{1}(β : P_{1}, ⋯, P_{k}) := M_{k+1}(J : β) ×_{(ev_{1},··· ,ev_{n})} (P_{1} × ⋯ × P_{k}).

Then there exists a family of single valued sections 𝔰_{ϵ} of the obstruction bundle in a Kuranishi neighborhood of M_{k+1}(β) and a decomposition

(9.6)  \begin{matrix}M_{1}^{𝔰_{ϵ}}( & β : P_{1}, ⋯, P_{k}) : \\ = M_{k+1}^{𝔰_{ϵ}}(β) ×_{(ev_{1},···ev_{n})} (P_{1} × ⋯ × P_{k}) \\ = M_{1}^{𝔰_{ϵ}}(β : P_{1}, ⋯, P_{k})_{reg} ∪ M_{1}^{𝔰_{ϵ}}(β : P_{1}, ⋯, P_{k})_{sing} \end{matrix}

of the perturbed moduli space such that :

(9.7.1) M_{1}^{𝔰_{ϵ}}(β : P_{1}, ⋯, P_{k})_{reg} is a smooth manifold. 𝔰_{ϵ}_{ϵ} is transversal to zero and the fiber product of (9.6) is smooth there. (9.7.2) M_{1}^{𝔰_{ϵ}}(β : P_{1}, ⋯, P_{k}) has a triangulation such that M_{1}^{𝔰_{ϵ}}(β : P_{1}, ⋯, P_{k})_{sing} is contained in a sub-complex of dimension

dim M_{1}^{𝔰_{ϵ}}(β : P_{1}, ⋯, P_{k})_{reg} − 2,

and that the triangulation is compatible with the smooth structure on M_{1}^{𝔰_{ϵ}}(β : P_{1}, ⋯, P_{k})_{reg}. (9.7.3) lim_{ϵ→0} 𝔰_{ϵ} = s, where s is the Kuranishi map.

Proof. We are going to find a single valued perturbation (in other words, a section of the obstruction bundle) 𝔰_{ϵ} so that any element in the moduli spaces (9.6) with trivial automorphism group is transversal. We first define automorphism group. Let (Σ, v) be an element of (9.6). Let Aut(Σ, v) be the group of all biholomorphic maps u : Σ → Σ such that v ∘ u = v. The definition of stable map implies that the group Aut(Σ, v) is of finite order.

We remark that the genus of Σ is zero. Hence if u ∈ Aut(Σ, v) then there exists a component S_{i}^{2} ≅ S^{2} ⊂ Σ such that u acts as z ↦ e^{2π\sqrt{−1}ℓ/k}z on S_{i}^{2}, and on the other components, u acts trivially or exchanges the components. As a consequence, the quotient space \overline{Σ} = Σ/Aut(Σ, v) is again a (pointed) open Riemann surface of genus zero. v induces a map \overline{v} : Σ/Aut(Σ, v) → M on it. We call (\overline{Σ}, \overline{v}) the reduced model of (Σ, v).

Let (Σ, v) ∈ M_{1}(β : P_{1}, ⋯, P_{k}) and (\overline{Σ}, \overline{v}) be its reduced model. It is contained in M_{1}(\overline{β} : P_{1}, ⋯, P_{k}) for some \overline{β}. Since there is no pseudo-holomorphic sphere of negative Chern number, it follows that (9.8.1) vir dim M_{1}(β : P_{1}, ⋯, P_{k}) ≥ vir dim M_{1}(\overline{β} : P_{1}, ⋯, P_{k}) (9.8.2) If (Σ, v) ≠ (\overline{Σ}, \overline{v}) in addition, then \overline{Σ} has a sphere component and hence is contained in the stratum of codimension 2.

One important remark is that in case when Aut(Σ, v) is trivial it is easy to find a single valued perturbation which is transversal to 0 near (Σ, v).

We are going to construct a section 𝔰_{ϵ} of our Kuranishi structure on M_{1}(β : P_{1}, ⋯, P_{k}) by induction on E(β). We construct it so that it satisfies a compatibility condition described below.

Let (\overline{Σ}, \overline{v}) be a reduced model of (Σ, v). We consider a Kuranishi neighborhood (\overline{U}, \overline{E}, \overline{s}, \overline{ψ}) and (U, E, s, ψ) of (\overline{Σ}, \overline{v}), (Σ, v) respectively. Note that U a sub-orbifold of V/Aut(Σ, v), and V is an open subset of an Euclidean space. Since Aut(\overline{Σ}, \overline{v}) is trivial it follows that \overline{V} is smooth. (Namely \overline{V} is a manifold.) It is easy from the construction of the Kuranishi structure (see [FOn2] and Chapter 5 of present paper), we may assume the following.

(9.9.1) \overline{V} is a closed submanifold of V. Let I_{\overline{V} ,V} : \overline{V} → V be the closed embedding. (9.9.2) \overline{E} can be identified to a subbundle of I_{\overline{V} ,V}^{∗} E. (Here we identify E with its pull back to V.) We write I_{\overline{E},E} be the inclusion. (9.9.3) 𝔰 ∘ I_{\overline{V} ,V} = I_{\overline{E},E} ∘ \overline{𝔰}. (9.9.4) If \overline{𝔰}(x) = 0 then \overline{ψ}(x) is the reduced model of ψ(I_{\overline{V} ,V} (x)).

Now we require our perturbation 𝔰_{ϵ} to satisfy

(9.10)  𝔰_{ϵ} ∘ I_{\overline{V} ,V} = I_{\overline{E},E} ∘ \overline{𝔰}_{ϵ}.

Now we start the construction of the single valued perturbation 𝔰_{ϵ} by induction on E(β).

When β = 0, we refer to §19 in Chapter 5: In this case, the moduli space M_{k+1}(β) ≅ M_{k+1} × M and is transversal. We have only to choose a Kuranishi neighborhood so that the evaluation map

ev_{1} × ⋯ × ev_{k} : M_{k+1}(β) → L × ⋯ × L

is weakly submersive, which is explained in §19.

Now suppose that we have found a single valued section 𝔰_{ϵ} such that the theorem holds for any β ∈ π_{2}(M, L) with ω(β) < E_{0}, and consider β with ω(β) = E_{0}.

Let M_{1}(β : P_{1}, ⋯, P_{k})_{sing} be the set of all elements of M_{1}(β : P_{1}, ⋯, P_{k}) with nontrivial automorphism. For each element (Σ, v) of it, we consider its reduced model (\overline{Σ}, \overline{v}). By induction hypothesis, we constructed a perturbation 𝔰_{ϵ} in a neighborhood of (\overline{Σ}, \overline{v}). We remark that this perturbation is transversal to zero in a neighborhood of (\overline{Σ}, \overline{v}).

We pull back the perturbation to a neighborhood of (Σ, v). In other words we define 𝔰_{ϵ} on M_{1}(β : P_{1}, ⋯, P_{k})_{sing} so that (9.10) is satisfied.

We can prove using (9.10), that they can be patched together to define a perturbation on a neighborhood of M_{1}(β : P_{1}, ⋯, P_{k})_{sing}.

Then we can extend this perturbation so that it is transversal outside M_{1}(β : P_{1}, ⋯, P_{k})_{sing}. (This follows from usual transversality theorem since our spaces are manifold there.) Now (9.8) implies that M_{1}(β : P_{1}, ⋯, P_{k})_{sing} is of dimension smaller than dim M_{1}^{𝔰}(β : P_{1}, ⋯, P_{k})_{reg} − 2. Thus to complete the proof we only need to show that dim M_{1}^{𝔰}(β : P_{1}, ⋯, P_{k}) has a triangulation. The proof of it will be given in §A3. □

We can use Theorem 9.5 to construct the fundamental chain (which is an integral current) of M_{1}^{s}(β : P_{1}, ⋯, P_{k}). Thus all the construction of Chapter 2 can be carried out over ℤ or ℤ_{2} coefficient, in case the Lagrangian submanifolds are semipositive.

§10. Application to the Maslov class

In this section, we will study the image of the Maslov index homomorphism

I_{μ} : π_{2}(M, L) → ℤ.

We denote

Im I_{μ} = Σ_{L}ℤ

with Σ_{L} a nonnegative integer. (Σ_{L} = 0 if and only if I_{μ} is zero.) Note that in the case of (M, ω) with c_{1}(T M) = 0 the Maslov index homomorphism

I_{μ} : π_{2}(M, L) → ℤ

depends only on the homotopy class of the boundary loop of the discs and so defines a cohomology class

μ_{L} ∈ H^{1}(L; ℤ)

which we call the Maslov class of L ⊂ M. It is also known (see [Da]) that on a Calabi-Yau manifold with a Ricci-flat Kähler metric, the mean curvature vector H⃗ of the Lagrangian submanifold L represents its Maslov class. More precisely, one has the formula

(10.1)  μ_{L} = \frac{1}{π} (H⃗⌋ω).

Note that the condition Σ_{L} = 0 (or more generally that Σ_{L} is even) automatically implies that L is orientable (see [AG]).

Theorem 10.2. If L is relatively spin and unobstructed, then the Floer homology HF ((L, b), (L, b)) is well-defined, and non-trivial if Σ_{L} = 0. In particular, if

L ∩ ϕ(L) = ∅

for some Hamiltonian diffeomorphism ϕ, then Σ_{L} ≠ 0.

Proof. The first half is an immediate consequence of Theorem 6.13 (6.14.4). We prove L∩ϕ(L) = ∅ by contradiction. We assume Σ_{L} = 0. Then the Floer homology HF ((L, b), (L, b)) is well-defined for some choice of b. On the other hand, Theorem C (which will be proved in the next chapter) implies that there exists b′ such that

HF ((L, b), (L, b)) ≃ HF ((L, b), (ϕ(L), b′)).

If L ∩ ϕ(L) = ∅, the definition implies that HF ((L, b), (ϕ(L), b′)) is zero for any choice of b, b′. On the other hand, the assumption Σ_{L} = 0 implies by Theorem 6.13 (6.14.4) that HF ((L, b), (L, b)) ≠ 0 which would give rise to a contradiction. This finishes the proof. □

In the rest of this section, we will generalize this theorem in two directions: one is to relax the condition on L and the other is to obtain more information on the Maslov class of L. For these generalizations, it is crucial to use ℤ_{2}-coefficients and so the materials in §9. One point that we should be careful in imitating the constructions in Chapter 2, without assuming the orientability of L which we abandon in this chapter, is the appearance of pseudo-holomorphic discs with odd Maslov indices, especially those with Maslov index 1. But examination of the construction in Chapter 2 shows that all of the constructions can be repeated with ℤ_{2}-coefficient except that we need to study the effect of discs with Maslov index 1 in §7.

An immediate consequence of Theorem 9.5, using ℤ_{2}-coefficients, is the following whose proof will be postponed. In fact, we will prove a stronger theorem later in this section.

Theorem 10.3. Suppose that L ⊂ (M, ω) is semi-positive, satisfies H^{2}(L; ℤ_{2}) = 0, and allows a Hamiltonian diffeomorphism ϕ such that

(10.4)  L ∩ ϕ(L) = ∅.

Then Σ_{L} ≠ 0.

Since ℂ^{n} obviously allows Hamiltonian diffeomorphisms satisfying (10.4) for any compact Lagrangian submanifold L, we immediately have the following partial result on the Maslov class conjecture.

Corollary 10.5. Let L ⊂ ℂ^{n} be a Lagrangian submanifold with H^{2}(L; ℤ_{2}) = {0}. Then its Maslov class μ_{L} is not zero.

Proof. Suppose that μ_{L} is zero. Then obviously L ⊂ ℂ^{n} is semi-positive. The proof immediately follows from Theorem 10.3. □

Recall that this Maslov class conjecture has been proved for the Lagrangian tori in ℂ^{n} [Vi] and for the two dimensional case [Po1]. Some results on the Maslov class for monotone Lagrangian embedding have also been obtained in [Po1], [Oh4].

In fact, we can prove a stronger theorem if we assume that L is unobstructed.

Theorem 10.6. Assume that L ⊂ (M, ω) is semi-positive and that L is unobstructed with respect to ℤ_{2}-coefficients. Suppose that there is a Hamiltonian diffeomorphism ϕ that satisfies (10.4). Then we have

(10.7)  1 ≤ Σ_{L} ≤ n + 1.

Corollary 10.8. Let L ⊂ ℂ^{n} be a Lagrangian submanifold that is unobstructed. Then we have

1 ≤ Σ_{L} ≤ n + 1.

We would like to compare this with [Vi2] where the same inequality was proven for the case of (arbitrary) Lagrangian tori in ℂ^{n}.

The remaining section will be occupied for the proof of Theorem 10.6.

Proof of Theorem 10.6. If Σ_{L} = 1, we are already done and so we assume that Σ_{L} ≠ 1 in the proof below. By the remarks made right before Theorem 10.3, this assumption allows to use the constructions made in Chapter 2 using ℤ_{2}-coefficient.

Now we will prove (10.7) by contradiction. Suppose that (10.7) does not hold. Then we have either Σ_{L} = 0 or

(10.9)  Σ_{L} ≥ n + 2 (≥ 3).

We consider two cases separately. First suppose Σ_{L} = 0. The vanishing of the obstruction classes implies that the Floer homology HF ((L, b), (L, b)) with ℤ_{2}- Novikov ring as its coefficient, is well-defined and is invariant under the Hamiltonian isotopy. The invariance property of the Floer homology and (10.4) implies

HF ((L, b), (L, b)) = {0}.

On the other hand, Theorem 6.13 (6.14.4) implies that

HF ((L, b), (L, b)) ≠ {0}.

This give rise to a contradiction and so Σ_{L} ≠ 0. Incidentally, this proof also proves Theorem 10.3 since the assumption H^{2}(L; ℤ_{2}) = {0} would automatically imply vanishing of the obstruction classes.

Now suppose that (10.9) holds. Recall the argument used in the proof of Proposition 7.6 (when applied to the case of ℤ_{2}-coefficients) implies that

(10.10)  δ[L] = 0

where [L] is the fundamental class of L and δ is the boundary operator induced from the bounding chain b as defined in §6. We will show that [L]⊗1 ∈ C(L; Λ̂_{+}(L))⊗_{Λ̂_{+}(L)} Λ_{nov} is not in the image of δ. Suppose the contrary

(10.11)  \begin{aligned}[L] ⊗ 1 & = δ(∑_{β} c_{β} ⊗ [β]), c_{β} ∈ C(L; Λ̂_{+}(L)) \\ & = ∑_{β} δ([c_{β}] ⊗ [β]) = ∑_{β} ∑_{γ} δ_{γ}[c_{β}] ⊗ [β + γ]. \end{aligned}

Here

δ_{γ} := ∑_{γ_{i_{1}} ,··· ,γ_{im};γ_{j_{1}} ,··· ,γ_{j_{k}}} \frac{1}{m!k!}δ_{γ:γ_{i_{1}} ,··· ,γ_{im};γ_{j_{1}} ,··· ,γ_{j_{k}}}

where δ_{γ:γ_{i_{1}} ,··· ,γ_{im};γ_{j_{1}} ,··· ,γ_{j_{k}}} is the operator defined in (6.19). (m!, k! in the right hand side may look strange when we work over ℤ_{2} coefficient. Dividing by m!k! means that we use only main component of the moduli space. See the remark right after the proof of Lemma 4.11.)

By comparing two sides of (10.11), we get

(10.12)  [L] = ∑_{β+γ=0} δ_{γ}[c_{β}] = ∑_{β} δ_{−β}[c_{β}] = δ_{0}[c_{0}] + ∑_{β≠0} δ_{−β}[c_{β}],

where δ_{0} is the classical boundary operator and the second term is one coming from quantum contributions. Obviously we have

deg (δ_{0}[c_{0}])) ≤ n − 1.

On the other hand, the assumption (10.9) implies that no term from ∑_{β≠0} δ_{−β}[c_{β}] can possibly have its degree n. Combining these two, we have shown that the right hand side of (10.12) cannot have any term of degree n which gives rise to a contradiction. This finishes the proof of Theorem 10.6. □

Remark 10.13. Rigorously speaking, the above proof is not complete in that the proof is based on the assumption that the fundamental class e = [L] is a unit in our A_{∞} algebra. However as we already mentioned before in Remark 7.9, e is only a homotopy unit, rather than a unit. Because of this, there are some technical points in the above proof which we have to be careful about. As in the case of Remark 7.9, we hide these technicalities at this point but refer to §20 for a complete discussion and proof.

Proof of Corollary 10.8. First note that Σ_{L} cannot be zero, for otherwise it would imply the semi-positivity of L and then contradict to Theorem 10.6. On the other hand, if Σ_{L} ≥ n + 2, it particularly would imply the semi-positivity again and contradict to Theorem 10.6. □

§11. Fixed points of anti-holomorphic involutions

The fixed points of anti-holomorphic involutions naturally arise in the study of real algebraic geometry as the real points of complex algebraic variety. We will call the set of fixed points of some anti-symplectic involution symmetric (τ-symmetric if the involution τ is specified, respectively). It was also studied by the second named author for the case of compact Hermitian symmetric spaces in relation to the Arnold-Givental conjecture.

Arnold-Givental conjecture. Let τ be an anti-symplectic involution of (M, ω), i.e., be a diffeomorphism with τ^{∗}ω = −ω. Then if L = Fix τ is non-empty and if ϕ : M → M be a Hamiltonian diffeomorphism such that provided L intersects ϕ(L) transversely, then we have

(11.1)  #(L ∩ ϕ(L)) ≥ rank H^{∗}(L; ℤ_{2}).

In [Oh3], the second named author proved this conjecture for the real forms of compact Hermitian symmetric spaces with suitable condition on the Maslov indices (e.g., irreducible Hermitian symmetric spaces satisfy the condition). Let τ be an anti-symplectic involution of (M, ω) and L = Fix τ be non-empty. Let j_{ω}^{τ} be the set of almost complex structures J that is compatible with ω and satisfies

(11.2)  τ_{∗}J = −J.

The following lemma is not difficult to show.

Lemma 11.3. Let τ be as above. Then the space j_{ω}^{τ} is non-empty and becomes an infinite dimensional (Frechet) manifold.

Proof. For given J ∈ j_{ω}^{τ}, its tangent space T_{J}j_{ω}^{τ} consists of sections Y of the bundle End(T M) whose fiber at p ∈ M is the space of linear maps Y : T_{p}M → T_{p}M such that

Y J + JY = 0, ω(Y v, w) + ω(v, Y w) = 0, τ^{∗}Y = −Y.

Note that the second condition means that JY is a symmetric endomorphism with respect to the metric g_{J} = ω(·, J·). It immediately follows that j_{ω}^{τ} becomes a manifold.

The fact that j_{ω}^{τ} is non-empty follows from the polar decomposition theorem by choosing a τ-invariant Riemannian metric on M. □

For each such J ∈ j_{ω}^{τ}, we have one to one correspondence

(11.4)  \begin{matrix}τ_{∗} : & M_{k+1}(J : β) → M_{k+1}(J : τ_{∗}β) \\ (w, (z_{0}, z_{1}, ⋯, z_{k})) ↦ (w̃, (\overline{z}_{0}, ⋯, \overline{z}_{k})) \end{matrix}

Here we denote β = [w] and τ_{∗}β = [w̃], where

w̃(z) = τ ∘ w(\overline{z}).

11.1. Orientation

Although we do not need this material for the proof of an Arnold-Givental conjecture, we include the study on how the map (11.4) affects orientation.

In Chapter 6, we will show that a choice of relative spin structure of L naturally induces an orientation on M_{k+1}(β) for any given β ∈ π_{2}(M, L). We denote this orientation on M_{k+1}(β) by o_{sp}(β) and by M_{k+1}(β) the manifold with this orientation as an oriented manifold.

Now we would like to study when the diffeomorphism (11.4) preserves the orientation. We first consider the case of no markings.

Proposition 11.5. The map

(11.6)  τ_{∗} : M̃(β) → M̃(τ_{∗}β)

is orientation preserving if and only if μ_{L}(β) ≡ 0 mod 4. The same is true for the moduli space of unparameterized maps M̃(β)/P SL(2; ℝ).

Proof. Let w ∈ M̃(β) and w̃ ∈ M̃(τ_{∗}β) be the corresponding element. With respect to the orientations given on M̃(β) and M̃(τ_{∗}β), we need to study whether the tangent map

T τ_{∗} : T_{w}M̃(β) → Tw̃M̃(τ_{∗}β)

respects the orientation or not. Assuming w (and so w̃) is regular, we identify

T_{w}M̃(β) = Ker D\overline{∂}_{w}

where the D\overline{∂}_{w} is the map

D\overline{∂}_{w} : W^{1,p}(D, ∂D : w^{∗}T M, (∂w)^{∗}T L) → L^{p}(D; w^{∗}T M).

And we do similarly for w̃. We have the commutative diagram of bundle pairs

(w^{∗}T M, (∂w)^{∗}T L) ⟶^{T τ} (w^{∗}T M, (∂w̃)^{∗}T L)
↓  ↓
(D^{2}, ∂D^{2})  ⟶^{c}  (D^{2}, ∂D^{2})

A trivialization

Φ : (w^{∗}T M, (∂w)^{∗}T L) → (D^{2}, ∂D^{2}; ℂ^{n}, Λ)

naturally induces a trivialization

Φ̃ : (w^{∗}T M, (∂w̃)^{∗}T L) → (D^{2}, ∂D^{2}; ℂ^{n}, Λ̃)

where Λ : S^{1} ≃ ∂D^{2} → Λ(ℂ^{n}) is a loop of Lagrangian subspaces given by Λ(z) := T_{w(z)}L in the trivialization and Λ̃ is defined by

Λ̃(z) = Λ(\overline{z}).

With respect to these trivializations, we have the commutative diagram

(D^{2}, ∂D^{2}; ℂ^{n}, Λ) ⟶^{Φ̃∘T τ∘Φ^{−1}} (D^{2}, ∂D^{2}; ℂ^{n}, Λ̃)
↓  ↓
(D^{2}, ∂D^{2})  ⟶^{c}  (D^{2}, ∂D^{2})

Note that τ_{z} := Φ̃ ∘ T τ ∘ Φ^{−1}(z, ·) : (ℂ^{n}, Λ(z)) → (ℂ^{n}, Λ(z)) is an involution with the Lagrangian subspace Λ(z) fixed. Now we deform the metric on D^{2} and the trivialization Φ so that Λ(z) ≡ ℝ^{n} as in the proof of Proposition 21.3 in Chapter 6. Recall that we assume L is orientable and so the bundle (∂w)^{∗}T L → S^{1} is trivial. After deforming further the Cauchy-Riemann operator on (D^{2}, ∂D^{2}; ℂ^{n}, Λ), we are reduced to considering the case

C : Hol (D^{2}, ∂D^{2} : ℂ^{n}, ℝ^{n}) × Hol (ℂP^{1} : E)
→ Hol (D^{2}, ∂D^{2} : ℂ^{n}, ℝ^{n}) × Hol (ℂP^{1} : E)

where E is a holomorphic vector bundle whose topology is determined by Λ (see the proof of Proposition 21.3) and C is the natural map induced from the conjugation on ℂ^{n} and E. In particular, we have dim_{ℂ} Hol (ℂP^{1} : E) = \frac{1}{2}μ(Λ). The first factor is invariant under the conjugation because Hol (D^{2}, ∂D^{2} : ℂ^{n}, ℝ^{n}) ≃ ℝ^{n}. For the second factor, we have Hol (ℂP^{1} : E) ≃ ℂ^{\frac{1}{2} μ(Λ)} with C is reduced to the standard conjugation on ℂ^{\frac{1}{2} μ(Λ)}. Therefore it boils down to considering the conjugation

C : ℂ^{\frac{1}{2} μ(Λ)} → ℂ^{\frac{1}{2} μ(Λ)}.

It is easy to see that this map is orientation preserving if and only if \frac{1}{2}μ(Λ) ≡ 0 mod 2, i.e., μ(Λ) ≡ 0 mod 4. This finishes the proof. □

Example 11.7. (1) Consider the case of M = ℂP^{n}, L = ℝP^{n}. In this case, each Maslov index μ(β) has the form

μ(β) = ℓ_{β}(n + 1)

for some integer ℓ_{β}. We know that when n is even, L is not orientable and so we consider only the case where n is odd. On the other hand, when n is odd, L is relatively spin : The class st will be the generator of H^{2}(ℂP^{n}; ℤ_{2}). And from the above formula for the Maslov index, we conclude from Proposition 11.5 that the map (11.6) is orientation preserving if and only if ℓ_{β} is even.

(2) Let M be a Calabi-Yau 3-fold and L ⊂ M be a Real manifold (i.e., the fixed point set of anti-holomorphic involutive isometry). In this case, L is orientable (because it is special Lagrangian) and spin (because any orientable 3-manifold is spin). Further more μ_{L}(β) = 0 for any β ∈ π_{2}(M, L). Therefore Proposition 11.5 implies that the map (11.6) is orientation preserving.

Now let us study the effect of markings. Consider the moduli space M_{k+1}(L, J : β) of (k +1)-marked (stable) maps and the map (10.4) which is induced by the map

(11.8)  (w, (z_{0}, z_{1}, ⋯, z_{k})) ↦ (w̃, (\overline{z}_{0}, \overline{z}_{1}, ⋯, \overline{z}_{k})).

Note that the map (11.8) naturally projects down to M_{k+1}(J : β). We have the following commutative diagram,

S^{1} × ⋯ × S^{1} ⟶^{c} S^{1} × ⋯ × S^{1}
↓  ↓
M_{k+1}(J : β) ⟶^{(11.4)} M_{k+1}(J : τ_{∗}β)
𝔣𝔬𝔯𝔤𝔢𝔱↓  ^{𝔣𝔬𝔯𝔤𝔢𝔱}↓
M(J : β) ⟶^{(11.6)} M(J : τ_{∗}β)

where c is the conjugation c(z_{0}, ⋯, z_{k}) = (\overline{z}_{0}, ⋯, \overline{z}_{k}) and 𝔣𝔬𝔯𝔤𝔢𝔱 are the forgetful maps of marking points. From this diagram, we conclude the following proposition.

Proposition 11.9. (11.9.1) When k +1 is even, (11.4) is orientation preserving if and only if (11.6) is so. (11.9.2) When k + 1 is odd, (11.4) is orientation preserving if and only if (11.6) is not so.

Example 11.7 (2) continued. Note that the obstruction chain corresponds to the case k + 1 = 1. For the case of real manifolds in Calabi-Yau 3-folds considered in Example 11.7 (2), it follows from Proposition 11.5 and 11.9 that the obstruction classes vanish.

11.2. Arnold-Givental Conjecture

In this section, we will prove Arnold-Givental conjecture for semi-positive Lagrangian submanifolds. The essential idea behind the proof is that when J is τ- (anti)invariant, i.e., satisfies (11.2), non-trivial pseudo-holomorphic disc with boundary on L = Fix τ comes in a pair (w, w̃). Therefore all the quantum contributions counted with ℤ_{2}-coefficient vanish. In particular, the ℤ_{2}-obstruction classes all vanish and the boundary operator δ defined in §6 reduces to the ordinary boundary operator δ_{0}, and hence its homology is isomorphic to H^{∗}(L; ℤ_{2}). This would finish the proof.

However to make this argument rigorous, we have to make sure that the transversality result proven in §9 and in Chapter 5 can be proved using only those J’s satisfying τ^{∗}J = −J. Furthermore we also have to make sure that the perturbation 𝔰^{ϵ} in Theorem 9.5 can be chosen to be τ-equivariant.

We first prove the following Proposition 11.10. Let M be a semi-positive symplectic manifold with anti holomorphic involution τ. Then there exists a τ invariant compatible almost complex structure J, such that every J holomorphic sphere v : S^{2} → M satisfy v_{∗}[S^{2}] ∩ c^{1} ≥ 0.

Proof. Let J be a τ invariant compatible almost complex structure.

Definition 11.11. A J-holomorphic sphere v : S^{2} → M is said to be τ-somewhere injective if there exists z ∈ S^{2} such that

(11.12.1) v is an immersion at z, (11.12.2) v(z) ∉ v(S^{2}\{z}), (11.12.3) v(z) ∉ τ(v(S^{2})).

Let w : (D^{2}, ∂D^{2}) → (M, L) a J-holomorphic disc. We say that it is τ-somewhere injective if there exists z ∈ Int D^{2} such that

(11.13.1) w is an immersion at z, (11.13.2) w(z) ∉ v(D^{2}\{z}), (11.13.3) w(z) ∉ τ(v(D^{2})).

Lemma 11.14. For generic τ invariant almost complex structure J, all τ-somewhere injective spheres and τ-somewhere injective discs are transversal.

Proof. The proof is in fact almost the same as that of [Mc1]. Let v : S^{2} → M be τ-somewhere injective J holomorphic sphere. We suppose that it is not transversal. Let

D\overline{∂}_{J} : Γ(Ω_{J}^{(0,0)}(v^{∗}T M)) → Γ(Ω_{J}^{(0,1)}(v^{∗}T M))

be the linearization of J holomorphic curve equation. By differentiating \overline{∂}_{J} with respect to J we obtain a map

D_{J} \overline{∂}_{J} : T_{J}j_{ω}^{τ} → Γ(Ω_{J}^{(0,1)}(v^{∗}T M)).

It suffices to show that the union of the images of D\overline{∂}_{J} and D_{J} \overline{∂}_{J} is Γ(Ω_{J}^{(0,1)}(v^{∗}T M)).

Let us take z as in (11.12). Let U be a small neighborhood of v(z) in M and W be a small neighborhood of z in S^{2} such that U ∩ v(S^{2}\W) = ∅, U ∩ τ(v(S^{2})) = ∅. Let j_{ω}(W) be the set of all compatible almost complex structures J′ which coincides with J outside U. (Note that the element of j_{ω}(W) may not be τ invariant.) It is proved in [Mc1] that

(11.15)  D_{J} \overline{∂}_{J}(T_{J}j_{ω}(W)) + Im(D\overline{∂}_{J}) = Γ(Ω_{J}^{(0,1)}(v^{∗}T M)).

We define a map S : j_{ω}(W) → j_{ω}^{τ} as follows. Let J′ ∈ j_{ω}(W), we put

S(J′) = \{ \begin{matrix}τ(J′) & on τ(W), \\ J′ & outside τ(W). \end{matrix}

Since τ(W) ∩ v(S^{2}) = ∅ it follows that D_{J} \overline{∂}_{J} ∘ DS = D_{J} \overline{∂}_{J}. Therefore

D_{J} \overline{∂}_{J}(T_{J}j_{ω}(W)) ⊇ D_{J} \overline{∂}_{J}(T_{J}j_{ω}^{τ}).

Thus the we have proved the statement on τ-somewhere injective spheres. The argument for τ-somewhere injective discs is similar. □

Remark 11.16. McDuff proved that every J holomorphic sphere is a covering space of somewhere injective sphere. On the other hand, if we define somewhere injectivity of J holomorphic disc by (11.13.1) and (11.13.2), then it is not true in general that every J holomorphic discs are covering space of somewhere injective discs. To analyze the structure of the image of J holomorphic discs we need more delicate result as in [KO]. (See [La] also.) We do not use those deep structure theorems in the proof of Proposition 11.10.

Now we go back to the proof of Proposition 11.10. Let us take a compatible almost complex structure satisfying the conclusion of Lemma 11.14. Then the semi positivity of M implies that there exists no τ somewhere injective J holomorphic sphere v : S^{2} → M with v_{∗}[S^{2}] ∩ c^{1} < 0. Note that not all J holomorphic sphere is a covering space of τ somewhere injective one. So we need more argument to prove Proposition 11.10.

Let v : S^{2} → M be a J holomorphic sphere with v_{∗}[S^{2}] ∩ c^{1} < 0. Then v can not be τ somewhere injective. If v is not somewhere injective, we can always find one such that v is a (branched) covering of it [Mc1]. Evidently we have \overline{v}_{∗}[S^{2}] ∩ c^{1} < 0 and so we may and will assume that v : S^{2} → M is somewhere injective (but not τ somewhere injective).

Since v is somewhere injective and is not τ somewhere injective, it follows that τ : M → M induces an anti holomorphic involution τ̃ : S^{2} → S^{2} such that v ∘ τ̃ = τ ∘ v. We consider the deformation complex

(11.17)  D\overline{∂}_{J}(v) : Γ(Ω^{0}(v^{∗}T M)) → Γ(Ω_{J}^{(0,1)}(v^{∗}T M)).

Using τ and τ̃, we define an involution I on Γ(Ω^{0}(v^{∗}T M)) by

(I(u))(z) = Dτ^{−1}(u(τ̃z)).

An involution I on Γ(Ω_{J}^{(0,1)}(v^{∗}T M)) is defined in the same way. It is easy to see that D\overline{∂}_{J}(v) commutes with I. Let Γ(Ω^{0}(v^{∗}T M))^{+} ⊕ Γ(Ω^{0}(v^{∗}T M))^{−} be the decomposition of Γ(Ω^{0}(v^{∗}T M)) to I invariant and I anti-invariant parts. We define Γ(Ω_{J}^{(0,1)}(v^{∗}T M))^{±} in a similar way. Then the complex (11.17) is split to

(11.18.1)  D\overline{∂}_{J}(v)^{+} : Γ(Ω^{0}(v^{∗}T M))^{+} → Γ(Ω_{J}^{(0,1)}(v^{∗}T M))^{+}.
(11.18.2)  D\overline{∂}_{J}(v)^{−} : Γ(Ω^{0}(v^{∗}T M))^{−} → Γ(Ω_{J}^{(0,1)}(v^{∗}T M))^{−}.

Lemma 11.19.

Index D\overline{∂}_{J}(v)^{+} = Index D\overline{∂}_{J}(v)^{−}.

Proof. J induces involutions on Γ(Ω^{0}(v^{∗}T M)) and Γ(Ω_{J}^{(0,1)}(v^{∗}T M)) respectively. We denote them also by J. It is easy to see IJ = −JI. Hence J exchanges D\overline{∂}_{J}(v)^{+} and D\overline{∂}_{J}(v)^{−}, and also J exchanges Γ(Ω_{J}^{(0,1)}(v^{∗}T M))^{+} and Γ(Ω_{J}^{(0,1)}(v^{∗}T M))^{−}.

Moreover J commutes with the symbol of the operator D\overline{∂}_{J}(v). (Note J may not commutes with D\overline{∂}_{J}(v).) The lemma then follows from the fact that index depends only on the symbol. □

Now we have

Lemma 11.20. We assume that J satisfies the conclusion of Lemma 11.4. Let v be somewhere injective but not τ somewhere injective sphere. We assume that v_{∗}[S^{2}] ∩ c^{1} < 0. We define τ̃ as above. Then τ̃ is fixed point free.

Proof. Suppose that τ̃ has a fixed point. Then it is easy to see that τ̃ : S^{2} → S^{2} can be identified to the map z ↦ \overline{z} by appropriate coordinate change of S^{2}. Hence the restriction of v to upper hemi-sphere is a J holomorphic discs w : D^{2} → M such that w(∂D^{2}) ⊂ L = the fixed point set of τ. Since v is somewhere injective it follows that w is τ somewhere injective. Hence w is transversal.

On the other hand, it is easy to see that the deformation complex of w can be identified to the complex (11.18.1). Note the virtual dimension of the moduli space of the pseudo-holomorphic curve homologous to v is negative, since we assumed M to be semi-positive. Therefore Lemma 11.19 implies that the index of (11.18.1) is negative. This contradicts to the fact that v′ is transversal. □

Now we consider the case when τ̃ is fixed point free.

Lemma 11.21. For a generic element J of j_{ω}^{τ} the following holds: Let v be somewhere injective but not τ somewhere injective sphere. Then the operator (11.18.1) is surjective.

Proof. We use the same notation as in the proof of Lemma 11.4. Let

D_{J} \overline{∂}_{J} : T_{J}j_{ω}^{τ} → Γ(Ω_{J}^{(0,1)}(v^{∗}T M)),

be as before and D_{J} \overline{∂}_{J}^{+} be the composition of D_{J} \overline{∂}_{J} and the projection

Γ(Ω_{J}^{(0,1)}(v^{∗}T M)) → Γ(Ω_{J}^{(0,1)}(v^{∗}T M))^{+}.

It suffices to show

(11.22)  D_{J} \overline{∂}_{J}(T_{J}j_{ω}(W))^{+} + Im(D\overline{∂}_{J})^{+} = Γ(Ω_{J}^{(0,1)}(v^{∗}T M))^{+}.

We have a point z ∈ S^{2} which satisfies (11.12.1) and (11.12.2), (but not necessarily (11.12.3)). Again we have (11.15). On the other hand, it is easy to see that

(11.23)  D_{J} \overline{∂}_{J} ∘ DS = 2D_{J} \overline{∂}_{J}.

(In this case τ(U) ∩ v(S^{2}) is nonempty.) (11.23) and (11.15) implies (11.22). The proof of Lemma 11.21 is now complete. □

Now we assume that J satisfies the conclusion of Lemma 11.21 and Lemma 11.14 and prove that such J satisfies the conclusion of Proposition 11.10. It remains to consider the case where v : S^{2} → M is a somewhere injective but not τ somewhere injective J holomorphic sphere such that τ̃ has no fixed point. Suppose v_{∗}[S^{2}]∩c^{1} < 0. Then, by the semi-positivity, the index of the deformation complex (11.17) is negative. Hence by Lemma 11.19 the index of (11.18.1) is negative. However this contradicts Lemma 11.21. The proof of Proposition 11.10 is now complete. □

Now we are ready to state the analogue to Theorem 9.5 for the τ-equivariant case.

Proposition 11.24. Let L = Fix τ be semi-positive and J ∈ j_{ω}^{τ} be as in Proposition 11.10. Then the same statement as Theorem 9.5 still holds for some single valued section 𝔰_{ϵ}. Moreover we may choose 𝔰_{ϵ} so that is τ-invariant.

Proof. When [β] = 0, we again refer to §19 of Chapter 5 which one can easily see that one can adapt the argument choosing a weakly transverse (single valued) perturbation in a τ-invariant way.

We now consider the case [β] ≠ 0. we note that τ acts freely on the moduli space M_{1}(β : P_{1}, ⋯, P_{k}) provided τ_{∗}[β] = [β]. The quotient space M_{1}(β : P_{1}, ⋯, P_{k})/ℤ_{2} has also a Kuranishi structure. We apply the proof of Theorem 9.5 to this quotient space with quotient Kuranishi structure. Hence (using the fact that the action is free) we have a section satisfying conditions of Proposition 11.24. Then we pull back this section and obtain the desired one. On the other hand, if τ_{∗}[β] ≠ [β], τ induces a free action on M_{1}(β : P_{1}, ⋯, P_{k}) ∐ M_{1}(τ_{∗}β : P_{1}, ⋯, P_{k}) which switches the two factors to each other. The above argument also proves the statement for this case. This finishes the proof. □

Now we are ready to finish up the proof of Arnold-Givental conjecture for semi positive Lagrangian submanifold. By the remark in the beginning of this subsection, a consequence of Proposition 11.24 is that the ℤ_{2}-obstruction classes all vanish, i.e., L = Fix τ is unobstructed. By Theorem C, (which will be proved in the next chapter), ϕ(L) is also unobstructed for any Hamiltonian diffeomorphism ϕ. Therefore the Floer homology HF ((L, b), (ϕ(L), ϕ_{∗}(b))) is well-defined and invariant under the change of ϕ. Here ϕ_{∗}b is the bounding chain naturally induced by b and ϕ as in §18. In particular, we have the isomorphism

HF ((L, b), (ϕ(L), ϕ_{∗}(b))) ≅ HF ((L, b), (L, b)).

Again Proposition 11.24 implies that all the quantum contribution in HF ((L, b), (L, b)) vanishes and so

HF ((L, b), (ϕ(L), ϕ_{∗}(b)); Λ_{nov}) ≅ H(L; ℤ_{2}) ⊗ Λ_{nov}.

This immediately proves :

Theorem 11.25. Let τ be an anti-symplectic involution of (M, ω) and let L = Fix τ be non-empty and semi-positive. Then Arnold-Givental conjecture holds, i.e., the inequality (11.1) holds for any Hamiltonian diffeomorphism ϕ, provided L intersects ϕ(L) transversely.

Since the real forms of compact Hermitian symmetric spaces are all semi-positive we have the following slight generalization of the result [Oh3], in that it eliminates restrictions on the real forms.

Corollary 11.26. Arnold-Givental conjecture holds for the real forms of any compact Hermitian symmetric spaces.

Example 11.27. One special case of semi-positive symmetric Lagrangian submanifold is the case of those in Calabi-Yau manifolds. In fact, it follows that the Maslov class of symmetric Lagrangian submanifolds in Calabi-Yau manifolds vanishes.

Chapter 4. Algebraic Deformation of Lagrangian Submanifolds and Floer cohomology.

§12.Outline of Chapter 4.

In this chapter we study dependence of the obstruction classes and the Floer homology we have defined, on the chains B(β_{i}), on the Hamiltonian deformations of L, and on other parameters involved in the definitions. Through this chapter, we always assume that a Lagrangian submanifold L is orientable and relatively spin, and that a pair of two Lagrangian submanifolds is relatively spin. (See Definition in §1.) We begin with an explanation of the constructions without delving into too much detail.

First we explain the reason why the Floer homology depends on perturbations. We recall that the basic trouble being discussed in this book in defining the Lagrangian intersection Floer homology lies in the fact that in general the moduli space of pseudo-holomorphic discs has codimension one boundary. It means that the fundamental cycle of the moduli space of pseudo-holomorphic discs may not be well-defined in the usual sense.

A similar phenomenon was found by Donaldson [Do] in his study of the moduli space of solutions of ASD equation on 4-manifolds with b_{2}^{+} = 1. In that case, Donaldson observed that if we consider one parameter family of perturbations (especially the perturbation of Riemannian metric), then the homology class of the moduli space may jump at some point. This phenomenon is called the wall crossing. We recall how the wall crossing occurs for the case, for simplicity, where the virtual dimension of the moduli space is 0. The order, counted with sign, of the moduli space is the invariant (Donaldson invariant). If we consider one parameter family of perturbations, we have one parameter family M_{s} of moduli spaces (s ∈ [0, 1]) whose union ∪_{s}{s} × M_{s} is a one dimensional manifold. Independence of the invariant under the perturbation could be proved by a cobordism argument if ∪_{s}{s} × M_{s} were compact for a generic one parameter family, in which case its boundary would become −M_{0} ∪ M_{1}. However in the case where the bubbling phenomenon is of codimension one, we cannot completely avoid bubbling even for a generic one parameter family and so the parameterized moduli space ∪_{s}{s} × M_{s} is not compact in general. Therefore the cobordism argument cannot prove independence of the invariant thus constructed on the perturbation involved.

In the gauge theory, Donaldson introduced the “chamber structure” to handle this trouble. In that case, the trouble has a nice feature that we can control where the wall crossing occurs, in terms of the abelian gauge theory which is nothing but the cohomology theory. Namely the wall crossing occurs at the points where there exists a nonzero anti-self-dual harmonic 2 form whose cohomology class is integral. In particular, if we take a perturbation which does not change the abelian gauge theory (especially the perturbation which does not change the Riemannian metric) then the wall crossing does not occur. So the Donaldson invariant is well-defined, for example, if we fix a Riemannian metric. The problem is similar in the case of Seiberg-Witten invariant of 4-manifolds with b_{2}^{+} = 1.

In our situation, the problem is more nontrivial, because we do not have any control where the wall crossing might occur. So we are forced to consider the whole family of perturbations to obtain something invariant under the perturbations. To carry out this, we will investigate infinitesimal structure of the moduli space of (unobstructed) Lagrangian submanifolds. Let us take and fix a Lagrangian submanifold L of M. (We also fix an almost complex structure J.) We are going to find a formal neighborhood of L in the extended moduli space of Lagrangian submanifolds. For another Lagrangian submanifold L′ near L, we say that L ∼ L′ if there exists a Hamiltonian isotopy ϕ^{s} such that L′ = ϕ^{1}(L). One can see that the set of ∼ equivalence classes of such Lagrangian submanifolds is isomorphic to a neighborhood of 0 in the first cohomology group H^{1}(L; ℝ). (There exists a trouble related to the Hausdorffness of the moduli space of Lagrangian submanifolds. We do not discuss this since we concern only with the local problem here. See [Fu7] §1.)

Because of various reasons, we want to extend this parameter space to H^{odd}(L; ℝ). Only H^{1}(L; ℝ)-direction of this extended space H^{odd}(L; ℝ) has direct geometric meaning. We will see later (Theorem 12.7) that one can still define deformations of “Floer (co)homology” to other directions. As we will explain below, our construction is similar to that of deformations of the cup product using the genus 0 Gromov-Witten invariant: Let M be a symplectic manifold. The space H^{2}(M; ℝ) parameterizes (locally) the moduli space of symplectic structures of M. The Gromov-Witten theory and the Frobenius structure obtained from it is regarded as a family of deformations of the cup product of M parameterized by H^{even}(M; ℝ), which is the “extended moduli space of symplectic structures on M”. The deformations to H^{2}(M; ℝ)-direction have obvious geometric meaning, that is the deformation of symplectic forms. This deformation space is extended to H^{even}(M; ℝ) direction also by using the Gromov-Witten potential. One may extend it to H^{odd}(M; ℝ) direction by regarding this parameter space as a super space (See [ASKZ], [KM], [Ma]).

The above extended moduli space of Lagrangian submanifold is a natural analogue to the extended moduli space ([Ra]) that appeared in the Frobenius structure of Gromov-Witten potentials ([Du], [KM], [Ma]). We consider the formal neighborhood of a Lagrangian submanifold L in this extended moduli space, which is the zero set of formal power series expansion of the defining equation of the moduli space or of the Kuranishi map. (The convergence of this formal power series is not known yet.) In other words, we are studying our moduli space as a “formal scheme”.

One important fact for the Gromov-Witten potential is that the extended moduli space of deformations of the symplectic structure is not obstructed. Namely the full neighborhood of zero in H^{even}(M, ℝ) locally parameterizes the extended moduli space. The group H^{odd}(M, ℝ) in principle is the obstruction group. However the obstruction map or the Kuranishi map is identically zero for the case of Gromov-Witten potential. This fact is consistent with the mirror symmetry in the following way: Let M^{∨} be the mirror of M. Then a part of H^{odd}(M; ℂ) will be isomorphic to H^{2}(M^{∨}, T M^{∨}) which is the vector space where the obstruction to the Kodaira-Spencer deformation theory lives. In other words, this is the space where the Kuranishi map (of the deformation theory of complex structure) takes its value. However the Kuranishi map is automatically 0 for the Calabi-Yau manifold. For the classical direction H^{1}(M^{∨}, T M^{∨}) this fact was proved by Bogomolov-Tian-Todorov [Bo], [Ti], [To]. For the other extended directions H^{k}(M^{∨}, Ω^{odd−k}M^{∨}), this fact was proved by Baranikov-Kontsevich [BK].

On the other hand, our extended moduli space of Lagrangian submanifolds is indeed obstructed in general. The corresponding obstruction map or the Kuranishi map is exactly our obstruction class. Thus, finding a formal neighborhood of extended moduli space of (unobstructed) Lagrangian submanifolds is equivalent to studying the behavior of the obstruction class in a neighborhood of L. Our problem of understanding dependence of the obstruction class and of the Floer (co)homology on the perturbation is closely related to this study.

Let us now describe the formal neighborhood of the extended moduli space of unobstructed Lagrangian submanifolds. We take C(L; ℚ) to be a countably generated sub-complex of all integral currents on L which are represented by singular chains. (See Section 13.2 and Appendix 1 for the reason why we take such sub-complex.) Let C(L; Λ_{0,nov}) = C(L; ℚ) ˆ⊗ Λ_{0,nov} be the completion of C(L; ℚ) ⊗ Λ_{0,nov}. We regard a codimension k chain as a k cochain and C(L; Λ_{0,nov}) as a cochain complex.

Note that Λ_{0,nov} is a graded ring. (See Definition 6.11). We give the grading on C(L; Λ_{0,nov}) by the sum of the cochain degree and the degree of the coefficients in Λ_{0,nov}. We consider the parity change (or suspension) complex C[1](L; Λ_{0,nov}). Namely :

(12.1)  (C[1])^{k}(L; Λ_{0,nov}) = C^{k+1}(L; Λ_{0,nov}).

Let B(C[1](L; Λ_{0,nov})^{∗}) be the free graded associative algebra generated by the dual C[1](M; Λ_{0,nov})^{∗} of C[1](M; Λ_{0,nov}). Namely we put :

(12.2) B(C[1](L; Λ_{0,nov})^{∗}) = ⨁ C[1](M; Λ_{0,nov})^{∗} ⊗ ⋯ ⊗ C[1](M; Λ_{0,nov})^{∗}.

The usual boundary operator on C(M; Λ_{0,nov})^{∗} is extended to B(C[1](L; Λ_{0,nov})^{∗}) as a derivation of degree −1. Let us denote it by d^{0,1}. We are going to construct a degree −1 maps :

(12.3)  d^{k} : C[1](L; Λ_{0,nov})^{∗} → B̂^{k}(C[1](L; Λ_{0,nov})^{∗}) for k ≥ 1.

Here

B^{k}(C[1](L; Λ_{0,nov}))^{∗} = \underbrace{C[1](L; Λ_{0,nov}))^{∗} ⊗ ⋯ ⊗ C[1](L; Λ_{0,nov}))^{∗}}_{k times},

and B̂^{k}(C[1](L; Λ_{0,nov}))^{∗} is its completion. Let B̂(C[1](L; Λ_{0,nov})^{∗}) be the completion of ⨁_{k} B̂^{k}(C[1](L; Λ_{0,nov}))^{∗}, which is isomorphic to the completion of B(C[1](L; Λ_{0,nov})^{∗}).

We will extend the map d^{k} to d^{k} : B̂(C[1](L; Λ_{0,nov})^{∗}) → B̂(C[1](L; Λ_{0,nov})^{∗}) as a derivation. More precisely, we will prove the following in §13.

Theorem 12.4. Suppose that the Lagrangian submanifold L is unobstructed. Then there exist d^{k} for k = 1, 2, ⋯ such that d̂^{2} = 0 where d̂ = d^{1} + d^{2} · · ·. And d^{1} is the Poincaré dual to the differential constructed in §6.

(Note d^{0,1} is included in d^{1}.) Theorem 12.4 implies that there exists a spectral sequence whose E^{2} term is

(12.5)  ⨁_{k} \overbrace{HF^{∗}(L, L; Λ_{0,nov}) ⊗ ⋯ ⊗ HF^{∗}(L, L; Λ_{0,nov})}^{k times}

and that it converges to the cohomology of d̂. We denote its cohomology ring by

R(L) = H(B̂(C[1](L; Λ_{0,nov})^{∗}, d̂).

This is a graded ring.

We note that each element b of C[1]^{0}(L; Λ_{0,nov}) ⊂ C[1](L; Λ_{0,nov}) = B_{1}(C[1](L; Λ_{0,nov})) defines a ring homomorphism φ_{b} : B̂(C[1](L; Λ_{0,nov})^{∗}) → Λ_{0,nov} by

φ_{b}(x_{1} ⊗ ⋯ ⊗ x_{k}) = x_{1}(b) ⊗ ⋯ ⊗ x_{k}(b)

for x ∈ B̂_{k}(C[1](L; Λ_{0,nov})^{∗}). (See Remark 13.33 and Lemma 15.15). Our moduli space of (algebraic) deformations of the Lagrangian submanifold L is defined as

Definition 12.6. Let x be the base point of L and (L, x) be the connected component of L containing x. We define M̂(L, x) to be the set of all elements of b ∈ C[1]^{0}((L, x); Λ_{0,nov}) = C^{1}((L, x); Λ_{0,nov}) such that d̂φ_{b} = 0.

From now on, restricting ourselves to the connected component of x, we may assume that L is connected and suppress the base point x ∈ L. We will define an equivalence relation ∼ on M̂(L) in §15 (see Definition 15.37) and let M(L) be its quotient space. See Theorem 12.12 (12.13.1) for a more precise meaning of this M(L) and §15 fro the precise definition of M(L).

We may regard M(L) as the set of the “Λ_{0,nov}-valued points” of a “noncommutative super scheme” Spec(B̂(C[1](L; Λ_{0,nov})^{∗}), d̂). (We do not know the precise definition of the notion in the quote.) Roughly speaking, M(L) is an algebraic subvariety of H^{odd}(L; Λ_{0,nov}) defined by the equation in the image of d̂.

Now let L′ be another (connected) Lagrangian submanifold and let ℓ_{0} be a path connecting L and L′.

Theorem 12.7. Let b ∈ M̂(L) and b′ ∈ M̂(L′) and ℓ_{0} be the based path connecting L and L′. Then there exists a deformed Floer cohomology HF ((L, b), (L′, b′); ℓ_{0}).

The proof of Theorem 12.7 will be given in §14. We will prove a similar result in the case when L = L′ in §15. We will also prove the following in §15.

Theorem 12.8. Let b_{0}, b_{1} ∈ M̂(L) and b′_{0}, b′_{1} ∈ M̂(L′). Assume that b_{0} ∼ b_{1} and b′_{0} ∼ b′_{1}. Then the deformed Floer cohomology HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}) is canonically isomorphic to HF ((L, b_{1}), (L′, b′_{1}); ℓ_{0})

Thus M(L), the deformation space of the A_{∞} algebra that we attach to L, naturally parameterizes deformations of our deformed Lagrangian intersection Floer homology. The choice of b ∈ M̂(L) is equivalent to that of bounding cochains B(β_{i}) introduced in §4. Namely they are related by

(12.9)  b = ∑ B(β_{i}) ⊗ [β_{i}] = ∑ B(β_{i}) ⊗ T^{ω(β_{i})}e^{\frac{μ(β_{i})}{2}} ∈ C^{1}(L; Λ_{0,nov})

Theorem 12.10. b ∈ M̂(L) if and only if o(β_{i}) = (−1)^{n+1}∂B(β_{i}) as in §4.

We remark that M̂(L) is nonempty if and only if the obstruction class [o(β_{i})] vanishes for all i for some choice of B(β_{i}).

Theorem 12.11. For b = ∑ B(β_{i}) ⊗ [β_{i}] ∈ C^{1}(L; Λ_{0,nov}) and b′ = ∑ B′(β_{i}′) ⊗ [β_{i}′] ∈ C^{1}(L′; Λ_{0,nov}), HF ((L, b), (L′, b′); ℓ_{0}) is canonically isomorphic to the Floer cohomology HF (L, L′; ℓ_{0}) we defined in §5 using the bounding cochains B(β_{i}), B′(β_{i}′).

Theorems 12.10 and 12.11 will be proven in §13 and §14, respectively.

Next we discuss dependence of the Floer cohomology and the obstruction class on Hamiltonian diffeomorphisms and on other perturbations. The main result is the following Theorem 12.12. We remark that the moduli space M(L), a priori, may depend not only on the symplectic manifold M and its Lagrangian submanifold L but also on the compatible almost complex structure J on M and on other parameters related to the perturbation of the moduli space of pseudo-holomorphic discs. We write Ξ for the parameter parameterizing the perturbation of the moduli space of pseudo-holomorphic discs and write M(L; J, Ξ) to specify its dependence of J, Ξ.

Theorem 12.12. Let L, L′ and ℓ_{0} be as above. Then we have the following:

(12.12.1) M(L; J, Ξ) is independent of the choice of J, Ξ. Namely there exists a canonical isomorphism M(L; J, Ξ) ≅ M(L; J′, Ξ′). (In this sense, we omit J, Ξ and write M(L) in case no confusion can occur.)

(12.12.2) Floer cohomology is also independent of J, Ξ. More precisely we have the following : Let b_{0} ∈ M(L; J_{0}, Ξ_{0}), b′_{0} ∈ M(L′; J′_{0}, Ξ′_{0}), and b_{1} ∈ M(L; J_{1}, Ξ_{1}), b′_{1} ∈ M(L′; J′_{1}, Ξ′_{1}) corresponds to them by the isomorphism in (12.13.1) respectively. Then there exists a canonical isomorphism

HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}; (J_{0}, Ξ_{0}), (J′_{0}, Ξ′_{0})) ≅ HF ((L, b_{1}), (L′, b′_{1}); ℓ_{0}; (J_{1}, Ξ_{1}), (J′_{1}, Ξ′_{1})).

In this sense, when no confusion can occur, we can write HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}) for HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}; J_{0}, Ξ_{0}, Ξ′_{0}) suppressing the dependence of J_{0}, Ξ_{0}, Ξ′_{0}.

We next concern how HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}) depends on the Hamiltonian isotopy of Lagrangian submanifolds.

Theorem 12.13. (12.13.1) Any Hamiltonian diffeomorphism ψ : M → M canonically induces an isomorphism

ψ_{∗} : M(L) ≃ M(ψ(L))

which depends only on the isotopy class of the embedding ψ : L → ψ(L). In other words, if Ψ = {ψ^{s}}_{0≤s≤1} is a Hamiltonian isotopy such that ψ^{s}(L) is fixed, then ψ_{∗}^{0} = ψ_{∗}^{1}. (12.13.2) Let Ψ = {ψ^{s}}_{0≤s≤1} and Ψ′ = {ψ^{′s}}_{0≤s≤1} be Hamiltonian isotopies with ψ^{0} = ψ^{′0} = id. Let b_{0} ∈ M(L), and b′_{0} ∈ M(L′). ψ_{∗}^{′1}(b′_{0}) = b′_{1} ∈ M(ψ′^{1}(L′)). Then Ψ, Ψ′ induces an isomorphism

(Ψ, Ψ′)_{∗} : HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}) ≅ HF ((ψ^{1}(L), ψ^{1}(b_{0})), (ψ^{′1}(L′), ψ_{∗}^{′1}(b′_{0}); ψ(ℓ_{0})).

which depends only on the homotopy types of the embeddings L → ψ^{1}(L) and L′ → ψ^{′1}(L)

Theorem 12.12 says, up to ambiguity of the choice of B, that the obstruction class and Floer cohomology are independent of the almost complex structure. On the other hand, Theorem 12.13 describes how the Floer cohomology behaves under the Hamiltonian isotopy.

In §16, we will generalize Theorem 12.12 and 12.13 to the case where L does not necessarily transversely intersect L′ but cleanly intersect in the Bott’s sense. This will imply Theorem 6.11 (6.12.2) whose proof we have postponed since §6.

Some notational remarks are in order. There will be various levels of homotopy of the parameters, especially those of J’s or of Hamiltonian isotopies. We reserve the letter ‘s’ for the first level of homotopy and ‘u’ for the second level of homotopy. We always reserve ‘t’ for the time parameter in the Cauchy-Riemann equation. We will denote by J either time-independent almost complex structure or a t-dependent family of them depending on whether we concern ‘one’ Lagrangian submanifold or ‘two’, i.e., a pair. For example, we denote by j = {J^{s}}_{1≤s≤1} the first level of homotopy of J’s and by \overline{j} = \{j_{u}\}_{0≤u≤1}\} the second level of homotopy.

And from now on we will omit the dependence on the based point x ∈ L and the based path ℓ_{0} from our notation whenever possible, if there is no danger of confusion. One should consider these choices are done implicitly in the rest of this section. We will put them in when we need to emphasize the dependence on them in our discussion.

§13. Filtered A_{∞} algebra

In this section, we prove Theorem 12.4 and 12.10. Firstly, without vanishing of the obstruction class, we consider a Lagrangian submanifold and associate a filtered A_{∞} algebra to it. We remark that the algebra we are constructing in this section is a dual to the one in §12.

13.1. Algebraic framework.

In this subsection, we introduce the algebraic notion of the filtered A_{∞} algebra. So we do not have the Lagrangian submanifold in this subsection. We note that our definition of A_{∞} algebra is a bit different from the standard one in the literature in ([St]) that our coefficient ring is a filtered commutative ring Λ_{0,nov}. So, first of all, we recall the notion of an A_{∞} algebra without filtrations. After that we will introduce our filtered A_{∞} algebra. To distinguish the ordinary (unfiltered) A_{∞} algebra from our filtered A_{∞} algebra, we will put ‘bar’s over the notations for the former.

Let R be a commutative ring with unit (e.g., R = ℚ in our case). Let \overline{C} = ⊕_{m∈ℤ}\overline{C}^{m} be a free graded R module. We put (\overline{C}[1])^{m} = \overline{C}^{m+1} and

B_{k}(\overline{C}[1]) = ⨁_{m_{1},··· ,m_{k}}(\overline{C}[1])^{m_{1}} ⊗ ⋯ ⊗ (\overline{C}[1])^{m_{k}}.

Suppose that we have a sequence of maps \overline{𝔪} = \{\overline{𝔪}_{k}\}_{k≥0} of degree +1

\overline{𝔪}_{k} : B_{k}(\overline{C}[1]) → \overline{C}[1], for k = 0, 1, · · · .

We note that \overline{𝔪}_{0} : R → \overline{C}[1]. The direct sum B(\overline{C}[1]) := ⊕_{k}B_{k}(\overline{C}[1]) has a structure of graded coalgebra. The coproduct Δ is defined by :

Δ(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{k=0}^{n} (x_{1} ⊗ ⋯ ⊗ x_{k}) ⊗ (x_{k+1} ⊗ ⋯ ⊗ x_{n}) .

We can extend \overline{𝔪}_{k} uniquely to a coderivation

\overline{d}_{k} : ⨁_{n} B_{n}(\overline{C}[1]) → ⨁_{n} B_{n−k+1}(\overline{C}[1]),

by the formula

(13.1)  \begin{matrix}\overline{d}_{k}(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{ℓ=1}^{n−k+1} & (−1)^{deg x_{1}+···+deg x_{ℓ−1}+ℓ−1}x_{1} ⊗ ⋯ ⊗ \\ \overline{𝔪}_{k}(x_{ℓ}, ⋯, x_{ℓ+k−1}) ⊗ ⋯ ⊗ x_{n} \end{matrix}

for k ≤ n and \overline{d}_{k} = 0 for k > n. Here and hereafter deg x means the degree of x before we shift it. When k = 0, we put \overline{𝔪}_{0}(1) in place of \overline{𝔪}_{k}(x_{ℓ}, ⋯, x_{ℓ+k−1}). Namely we define \overline{d}_{0} by

\begin{matrix}\overline{d}_{0}(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{ℓ=1}^{n+1} & (−1)^{deg x_{1}+···+deg x_{ℓ−1}+ℓ−1}x_{1} ⊗ ⋯ ⊗ x_{ℓ−1}⊗ \\ \overline{𝔪}_{0}(1) ⊗ x_{ℓ} ⊗ ⋯ ⊗ x_{n}. \end{matrix}

We put ˆ \overline{d} = ∑ \overline{d}_{k}.

Definition 13.2. \overline{𝔪} = \{\overline{𝔪}_{k}\}_{k≥0} defines a structure of A_{∞} algebra over R on \overline{C}, if

ˆ\overline{d} ∘ ˆ \overline{d} = 0.

Remark 13.3. (1) In the later argument, we will use an A_{∞} algebra over R with \overline{𝔪}_{0} = 0. But 𝔪_{0} plays a very important role when we study a filtered A_{∞} algebra and develop our obstruction theory. When \overline{𝔪}_{0} = 0, we find that \overline{𝔪}_{1}\overline{𝔪}_{1} = 0. So we have a complex (\overline{C}[1], \overline{𝔪}_{1}). (See Remark 13.13 (1).)

(2) When we regard a differential graded algebra (DGA) over R as an A_{∞} algebra, we just put \overline{𝔪}_{k} = 0 for k ≥ 3 and \overline{𝔪}_{0} = 0. But we note that the sign of \overline{𝔪}_{2} is different from that of product in the DGA

(13.4)

\overline{𝔪}_{1}(x) = (−1)^{deg x}dx
\overline{𝔪}_{2}(x, y) = (−1)^{deg x(deg y+1)}x · y

Here x · y is the product in the DGA. (See also Remark 13.13 (2) below.)

Now we introduce the notion of the filtered A_{∞} algebra. Let ⊕_{m∈ℤ}C^{m} be a free graded Λ_{0,nov} module. There is a filtration F^{λ}C^{m} on C^{m} (λ ∈ ℝ_{≥0}), such that

(13.5.1) F^{λ}C^{m} ⊂ F^{λ′}C^{m} if λ > λ′. (13.5.2) T^{λ_{0}} · F^{λ}C^{m} ⊂ F^{λ+λ_{0}}C^{m}. (13.5.3) e^{k}C^{m} ⊂ C^{m+2k}. (13.5.4) C^{m} is complete with respect to the filtration. (13.5.5) C^{m} has a basis e_{i} such that e_{i} ∈ F^{0}C^{m} and e_{i} ∉ F^{λ}C^{m} for λ > 0.

We denote by C the completion of ⊕_{m∈ℤ}C^{m} with respect to the filtration. (13.5.3) means that the degree of e is 2. We consider the shifted complex C[1] defined by

(C[1])^{m} = C^{m+1}

and

B_{k}(C[1]) = ⨁_{m_{1},··· ,m_{k}}(C[1])^{m_{1}} ⊗ ⋯ ⊗ (C[1])^{m_{k}}.

Suppose that we have a sequence of maps 𝔪 = {𝔪_{k}}_{k≥0} of degree +1

𝔪_{k} : B_{k}(C[1]) → C[1], for k = 0, 1, · · · .

We note that 𝔪_{0} : Λ_{0,nov} → C[1]. We assume that

(13.5.6)  𝔪_{k} (F^{λ_{1}}C^{m_{1}} ⊗ ⋯ ⊗ F^{λ_{k}}C^{m_{k}}) ⊆ F^{λ_{1}+···+λ_{k}}C^{m_{1}+···+m_{k}−k+2}

and

(13.5.7)  𝔪_{0}(1) ∈ F^{λ′}C[1] for some λ′ > 0 .

As we will explain later (see (13.10) below), Λ_{0,nov} has the maximal ideal

Λ_{0,nov}^{+} = {∑ a_{i}T^{λ_{i}}e^{μ_{i}} ∈ Λ_{nov} | λ_{i} > 0, a_{i} ∈ ℚ} ,

such that Λ_{0,nov}/Λ_{0,nov}^{+} ≅ ℚ. We put

\overline{𝔪}_{k}(x_{1}, …, x_{k}) ≡ 𝔪_{k}(x_{1}, …, x_{k}) mod Λ_{0,nov}^{+}C

for k = 0, 1, 2, . . .. Then \{\overline{𝔪}_{k}\} defines an A_{∞} algebra structure on \overline{C} = C/Λ_{0,nov}^{+}C. We also assume that there exists a constant λ′′ > 0 such that

(13.5.8)  𝔪_{k}(e_{i_{1}}, …, e_{i_{k}}) − \overline{𝔪}_{k}(e_{i_{1}}, …, e_{i_{k}}) ∈ F^{λ′′}C[1].

Here λ′′ is independent of k and e_{i_{1}}, …, e_{i_{k}}. The condition (13.5.8) will be used when we construct a spectral sequence. (See Appendix A4). We call (13.5.8) the gap condition.

Now the direct sum B(C[1]) := ⊕_{k}B_{k}(C[1]) has a structure of graded coalgebra. Note that we are studying the dual of the algebra B̂(C[1])^{∗} introduced in §12. So we regard B(C[1]) as a coalgebra and will construct a coderivation on it. The coproduct Δ is defined by :

(13.6)  Δ(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{k=0}^{n} (x_{1} ⊗ ⋯ ⊗ x_{k}) ⊗ (x_{k+1} ⊗ ⋯ ⊗ x_{n}) .

We can extend 𝔪_{k} uniquely to a coderivation

d_{k} : ⨁_{n} B_{n}(C[1]) → ⨁_{n} B_{n−k+1}(C[1]),

by

(13.7)  \begin{matrix}d_{k}(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{ℓ=1}^{n−k+1} & (−1)^{deg x_{1}+···+deg x_{ℓ−1}+ℓ−1}x_{1} ⊗ ⋯ ⊗ \\ 𝔪_{k}(x_{ℓ}, ⋯, x_{ℓ+k−1}) ⊗ ⋯ ⊗ x_{n} \end{matrix}

for k ≤ n and d_{k} = 0 for k > n. When k = 0, we put 𝔪_{0}(1) in the right hand side. Namely we define d_{0} by

\begin{matrix}d_{0}(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{ℓ=1}^{n+1} & (−1)^{deg x_{1}+···+deg x_{ℓ−1}+ℓ−1}x_{1} ⊗ ⋯ ⊗ x_{ℓ−1}⊗ \\ 𝔪_{0}(1) ⊗ x_{ℓ} ⊗ ⋯ ⊗ x_{n}. \end{matrix}

We want to consider the infinite sum d̂= ∑ d_{k}. Therefore we need to consider a completion B̂(C[1]) of B(C[1]). We define a filtration F^{λ}B_{k}(C[1]) on B_{k}(C[1]) by

F^{λ}B_{k}(C[1]) = ⋃_{λ_{1}+···+λ_{k}≥λ} (F^{λ_{1}}C^{m_{1}} ⊗ ⋯ ⊗ F^{λ_{k}}C^{m_{k}})

Let B̂_{k}(C[1]) be the completion of B_{k}(C[1]) with respect to this filtration.

Definition 13.8. B̂(C[1]) is the set of all formal sum ∑_{k} x_{k} where x_{k} ∈ B̂_{k}(C[1]) such that

x_{k} ∈ F^{λ_{k}} B̂_{k}(C[1])

with lim_{k→∞} λ_{k} → ∞.

Lemma 13.9. If (13.5) is satisfied, then d̂ is well-defined as a map from B̂(C[1]) to B̂(C[1]).

Proof. Let ∑ x_{k} be as in Definition 13.8. The B̂_{k}(C[1])-component of d̂∑ x_{k} is ∑_{ℓ} d_{ℓ}(x_{k+ℓ−1}). By (13.5) and Definition 13.8, we have

d_{ℓ}(x_{k+ℓ−1}) ∈ F^{λ_{k+ℓ−1}} B̂_{k}(C[1])

Therefore the completeness of B̂_{k}(C[1]) implies that d̂∑ x_{k} converges. The proof of Lemma 13.9 is complete. □

We remark that the ring Λ_{0,nov} is a local ring with the maximal ideal

Λ_{0,nov}^{+} = {∑ a_{i}T^{λ_{i}}e^{μ_{i}} ∈ Λ_{nov} | λ_{i} > 0, a_{i} ∈ ℚ} .

We put

(13.10.1)  \overline{C} = C/(Λ_{0,nov}^{+}C)

and

(13.10.2)  \overline{𝔪}_{k}(x_{1}, …, x_{k}) ≡ 𝔪_{k}(x_{1}, …, x_{k}) mod Λ_{0,nov}^{+}C

for k = 0, 1, 2, . . .. Note that the filtration is defined by λ which is the power of T. Thus on the \overline{C}, we do not have filtrations. In this way (\overline{C}, \overline{𝔪} = \{\overline{𝔪}_{k}\}_{k≥0}) becomes an A_{∞} algebra over Λ_{0,nov}/Λ_{0,nov}^{+} ≅ ℚ, which we introduced in the beginning of this section.

Now we introduce the following condition for an element e of C which is relevant to the conclusion of Proposition 7.1. (Compare also [Fu5] §11.)

Condition 13.11. (13.11.1) 𝔪_{k+1}(x_{1}, ⋯, e, ⋯, x_{k}) = 0, k ≥ 2, k = 0. (13.11.2) 𝔪_{2}(e, x) = (−1)^{deg x}𝔪_{2}(x, e) = x.

Definition 13.12. (1) 𝔪 = {𝔪_{k}}_{k≥0} defines a structure of filtered A_{∞} algebra on C if (13.5) are satisfied and if d̂∘ d̂= 0. We call B̂(C[1]) the (completed) bar complex associated to the A_{∞} algebra (C, 𝔪). If a (filtered) A_{∞} algebra has an element e satisfying Condition 13.11, we call the A_{∞} algebra an (filtered) A_{∞} algebra with unit and e a unit.

(2) For a filtered A_{∞} algebra (C, 𝔪), we say that a filtered A_{∞} algebra (C′, 𝔪′) is an A_{∞}-deformation of (C, 𝔪), if (\overline{C}′, \overline{𝔪}′) = (\overline{C}, \overline{𝔪}). Here (\overline{C}′, \overline{𝔪}′) and (\overline{C}, \overline{𝔪}) are defined as in (13.10). We also say that a filtered A_{∞} algebra (C, 𝔪) is an A_{∞}-deformation of an A_{∞} algebra (\overline{C}′′, \overline{𝔪}′′) if (\overline{C}, \overline{𝔪}) = (\overline{C}′′, \overline{𝔪}′′).

Remark 13.13. (1) The equation d̂∘ d̂ = 0 produces infinitely many relations among 𝔪_{k}’s. For example, we have

𝔪_{1}(𝔪_{0}(1)) = 0,

𝔪_{2}(𝔪_{0}(1), x) + (−1)^{deg x+1}𝔪_{2}(x, 𝔪_{0}(1)) + 𝔪_{1}(𝔪_{1}(x)) = 0,

𝔪_{3}(𝔪_{0}(1), x, y) + (−1)^{deg x+1}𝔪_{3}(x, 𝔪_{0}(1), y) + (−1)^{deg x+deg y+2}𝔪_{3}(x, y, 𝔪_{0}(1))

+ 𝔪_{2}(𝔪_{1}(x), y) + (−1)^{deg x+1}𝔪_{2}(x, 𝔪_{1}(y)) + 𝔪_{1}(𝔪_{2}(x, y)) = 0,

· · · · · · · · ·

In general, it is easy to show that d̂∘ d̂= 0 is equivalent to that for each k

\begin{matrix}∑_{k_{1}+k_{2}=k+1} & ∑_{i}(−1)^{deg x_{1}+···+deg x_{i−1}+i−1} \\ 𝔪_{k_{1}}(x_{1}, ⋯, 𝔪_{k_{2}}(x_{i}, ⋯, x_{i+k_{2}−1}), ⋯, x_{k}) = 0. \end{matrix}

We call this identity the A_{∞} formula. If 𝔪_{0} = 0, then 𝔪_{1}𝔪_{1} = 0. So in this case 𝔪_{1} plays the role of a boundary operator.

(2) Suppose 𝔪_{0} = 0 and 𝔪_{3} = ⋯ = 0. Then d̂∘ d̂= 0 implies

(13.14)  𝔪_{2}(𝔪_{2}(x, y), z) + (−1)^{deg x+1}𝔪_{2}(x, 𝔪_{2}(y, z)) = 0.

(13.14) is consistent with (13.4) and the associativity of the product in DGA. One might try to symmetrize the operator 𝔪_{k} to define an L_{∞} algebra in place of A_{∞} algebra. However there is a trouble to do so. For one natural choice of sign would be

(13.15)  𝔪_{2}(x, y) + (−1)^{(deg x+1)(deg y+1)}𝔪_{2}(y, x) = 𝔪_{sym,2}(x, y)

so that 𝔪_{sym,2}(x, y) = (−1)^{(deg x+1)(deg y+1)}𝔪_{sym,2}(y, x) but this formula is inconsistent with (13.4). Namely in case C is graded commutative, the leading term of 𝔪_{2} defined by (13.4) satisfies 𝔪_{sym,2}(x, y) = −(−1)^{(deg x+1)(deg y+1)}𝔪_{sym,2}(y, x) by the symmetrization (13.15).

13.2. Filtered A_{∞} algebras associated to Lagrangian submanifolds.

In this subsection, we show that the construction of Chapter 2 and its generalization lead to an example of the filtered A_{∞} algebra attached to each Lagrangian submanifold L of M. We assume that L is relatively spin. We again emphasize that we do not yet impose the condition o(β_{k}) = 0 on the obstruction classes o(β_{k}).

Let \overline{S}^{k}(L; ℚ) be a free ℚ module generated by all integral k-currents on L which are represented by smooth chains. We denote by C^{k}(L; ℚ) a countably generated submodule of \overline{S}^{k}(L; ℚ). See Lemma A1.2 in Appendix 1. (In the later argument, we use a method of “smooth correspondence”. To do this we need and use the transversality argument and the Baire category theorem. This is why we introduce C^{k}(L; ℚ). See Chapter 5 and Appendix 1.) Since an element in C^{k}(L; ℚ) is represented by a singular chain, we sometimes write it by the chain representative (P, f). We define C^{•}(L; Λ_{0,nov}) by the completion of C^{•}(L; ℚ) ⊗ Λ_{0,nov}. For the convenience of notation, we put

C^{•} = C^{•}(L; Λ_{0,nov}) := (C^{•}(L; ℚ) ⊗ Λ_{0,nov}) .

Recall that the element T^{λ}e^{n} in Λ_{0,nov} has degree 2n. Thus the degree in C^{•}(L; Λ_{0,nov}) is the sum of the degree in C^{•}(L; ℚ) and the degree of the coefficients in Λ_{0,nov}.

Using the filtration on Λ_{0,nov}, we can uniquely define the filtration on C^{k}(L; Λ_{0,nov}) which satisfies the following conditions;

C^{k}(L; ℚ) ⊂ F^{0}C^{k}(L; Λ_{0,nov})

and

C^{k}(L; ℚ) ⊄ F^{λ}C^{k}(L; Λ_{0,nov}) for λ > 0.

We now define the maps

𝔪_{k} : B_{k}(C[1](L; Λ_{0,nov})) → C[1](L; Λ_{0,nov})

for k ≥ 0. To do this, we recall that M_{k+1}(β) is the set of pairs ((Σ, z⃗), w) where w : (Σ, ∂Σ) → (M, L) is a pseudo-holomorphic map which represents the class β. Let M_{k+1}^{main}(β) be the subset of M_{k+1}(β) consisting of elements ((Σ, z⃗), w) where (Σ, z⃗) is in the main component. (See Definition 3.10 (3.10.2) for the definition of the main component.) For given chains

(P_{i}, f_{i}) ∈ C^{g_{i}}(L; ℚ), i = 1, ⋯, k,

we consider

M_{k+1}^{main}(β)_{(ev_{1},··· ,ev_{k})} ×_{f_{1}×···×f_{k}} (P_{1} × ⋯ × P_{k}).

Proposition 13.16. Suppose L is relatively spin. Then

M_{k+1}^{main}(β)_{(ev_{1},··· ,ev_{k})} ×_{f_{1}×···×f_{k}} (P_{1} × ⋯ × P_{k})

has an oriented Kuranishi structure. Its dimension is n − ∑(g_{i} − 1) + μ(β) − 2, where n = dim L.

Proof : Construction of the Kuranishi structure will be given in Chapter 5. As for the orientations on M_{k+1}^{main}(β) and for the fiber product of Kuranishi structures, see Chapter 6. Here we prove only the statement on dimension. The dimension is computed to be

μ(β) + k + 1 − 3 + n − ∑ g_{i} = μ(β) + n − ∑(g_{i} − 1) − 2,

as required. □

Definition 13.17. We define M_{1}^{main}(β; P_{1}, ⋯, P_{k}) by

M_{1}^{main}(β; P_{1}, ⋯, P_{k}) := (−1)^{ϵ}M_{k+1}^{main}(β)_{(ev_{1},··· ,ev_{k})} ×_{f_{1}×···×f_{k}} (P_{1} × ⋯ × P_{k}),

where

ϵ = (n + 1) ∑_{j=1}^{k−1} ∑_{i=1}^{j} deg P_{i}.

(As for the sign, see Section 25.4 in Chapter 6).

Now we define the maps 𝔪_{k}. We recall that we have the element β_{0} = 0 ∈ G_{++}(L) which satisfies μ(β_{0}) = 0 and E(β_{0}) = 0. (See §4).

Definition 13.18. (1) For (P, f) ∈ C^{k}(L, ℚ), we define

𝔪_{0,β}(1) = \{ \begin{matrix}(M_{1}(β), ev_{0}) & for β ≠ β_{0} \\ 0 & for β = β_{0}, \end{matrix}
𝔪_{1,β}(P, f) = \{ \begin{matrix}(M_{1}^{main}(β; P), ev_{0}) & for β ≠ β_{0} \\ (−1)^{n}∂P & for β = β_{0}, \end{matrix}

and e = [L] (the fundamental cycle).

The notation ∂ in the definition of 𝔪_{1,β_{0}} is the usual (classical) boundary operator.

(2) For each k ≥ 2 and (P_{i}, f_{i}) ∈ C^{g_{i}}(L, ℚ), we define 𝔪_{k,β} by

\begin{aligned}𝔪_{k,β}((P_{1}, f_{1}), …, (P_{k}, f_{k})) & = 𝔪_{k,β}((P_{1}, f_{1}) ⊗ ⋯ ⊗ (P_{k}, f_{k})) \\ & = (M_{1}^{main}(β; P_{1}, ⋯, P_{k}), ev_{0}). \end{aligned}

(3) Then we define 𝔪_{k} (k ≥ 0) by

𝔪_{k} = ∑_{β∈π_{2}(M,L)} 𝔪_{k,β} ⊗ [β] = ∑_{β∈π_{2}(M,L)} 𝔪_{k,β} ⊗ T^{ω(β)}e^{\frac{μ(β)}{2}} .

Remark 13.19. (1) We note that 𝔪_{k} depends on the almost complex structure J and perturbation Ξ of Kuranishi map. We will discuss the dependence in §15 later.

(2) We note that the condition (13.5.7) is satisfied if 𝔪_{0,β_{0}}(1) = 0.

(3) Like as in (13.10), we can reduce the coefficient Λ_{0,nov} of (C(L, Λ_{0,nov}), 𝔪) to Λ_{0,nov}/Λ_{0,nov}^{0} ≅ ℚ to get an A_{∞} algebra (\overline{C}, \overline{𝔪}). Because 𝔪_{0,β_{0}} = 0, we note that \overline{𝔪}_{0} = 0 in our case. Furthermore, we should note that (\overline{C}, \overline{𝔪}) is not a DGA, but an A_{∞} algebra. Let us consider the case β = β_{0}(= 0). If P_{1} ≠ P_{2}, M_{1}^{main}(β_{0}; P_{1}, P_{2}) is (as a set) nothing but P_{1} ∩ P_{2}. But an essential point is that we can not define self intersection at the chain level. Similarly we can not expect that \overline{𝔪}_{k} = 0 for k ≥ 3, because we do not have the transversality for the diagonal case. We will discuss this problem in more detail in Chapter 5. In Section 19, for the Lagrangian submanifold L, we construct an A_{∞} algebra (ℤX , 𝔪) over ℤ. We put ℚX = ℤX ⊗ℚ and extend 𝔪 to ℚX. Then in Chapter 5, we will prove that (C(L, Λ_{0,nov}), 𝔪) is an A_{∞} deformation of (ℚX , 𝔪). We also remark one more point about the sign. By (13.4), when β = β_{0}(= 0), we should check

(13.20)  M_{1}^{main}(β_{0}; P_{1}, P_{2}) = (−1)^{deg P_{1}(deg P_{2}+1)}P_{1} ∩ P_{2},

which will be proved in Corollary 25.17 of Section 25.5 in Chapter 6.

Recall that we put deg β = μ(β) and we shift the degree of (P, f) by one. Hence Proposition 13.16 implies that the total degree of 𝔪_{k} on B̂(C[1]) is +1. 𝔪_{k} defines a sequence of maps 𝔪 = {𝔪_{k}}_{k≥0},

(13.21)  𝔪_{k} : B_{k}(C[1](L; Λ_{0,nov})) → C[1](L; Λ_{0,nov}).

Now we have :

Theorem 13.22. Suppose that L is relatively spin. Then (C(L; Λ_{0,nov}), 𝔪) is a filtered A_{∞} algebra (with unit e). Moreover (C(L; Λ_{0,nov}), 𝔪) is an A_{∞} deformation of an A_{∞} algebra (ℚX , 𝔪), which will be constructed in Section 19. Furthermore, (C(L; Λ_{0,nov}), 𝔪) satisfies the gap condition (13.5.8).

Proof. The proof that (C(L; Λ_{0,nov}), 𝔪) is an A_{∞} deformation of the A_{∞} algebra (ℚX , 𝔪) will be given in Chapter 5, Section 19 except the sign in (13.20). The sign will be checked in Corollary 25.17 of Section 25.5. Now we prove that (C(L; Λ_{0,nov}), 𝔪) is a filtered A_{∞} algebra (with unit e). Conditions (13.5) is immediate from the construction. We recall that we have an element β_{1} ∈ G_{++}(L) such that it has the minimal (non zero) area. (See Lemma 4.5). We take λ′′ > 0 such that

λ′′ < ω[β_{1}].

Then we can find that {𝔪_{k}} satisfies the gap condition (13.5.8). We prove d̂∘d̂= 0. This will be a consequence of analyzing the boundary of M_{1}^{main}(β; P_{1}, ⋯, P_{k}). We find that its boundary is the sum of

∑_{i} M_{1}^{main}(β; P_{1}, ⋯, ∂P_{i}, ⋯, P_{k})

and the terms described by the Figure 13.1 below.

Figure 13.1

On the other hand, in order to prove d̂∘ d̂ = 0, we note that it is enough to show that

(13.23) \begin{matrix}∑_{β_{1}+β_{2}=β} ∑_{k_{1}+k_{2}=k+1} & ∑_{i}(−1)^{deg P_{1}+···+deg P_{i−1}+i−1} \\ 𝔪_{k_{1},β_{1}}(P_{1}, ⋯, 𝔪_{k_{2},β_{2}}(P_{i}, ⋯, P_{i+k_{2}−1}), ⋯, P_{k}) = 0. \end{matrix}

(See Remark 13.13 (1).) Here we write P_{i} in place of (P_{i}, f_{i}) for simplicity. We divide the left hand side into 3 terms, corresponding to the cases β_{1} = 0 (= β_{0}) and k_{1} = 1, β_{2} = 0 (= β_{0}) and k_{2} = 1, and the others. Then we can rewrite the left hand side in (13.23) as follows:

(13.24) 𝔪_{1,0}𝔪_{k,β}(P_{1}, ⋯, P_{k})

+ ∑_{i}(−1)^{deg P_{1}+···+deg P_{i−1}+i−1}𝔪_{k,β}(P_{1}, ⋯, 𝔪_{1,0}(P_{i}), ⋯, P_{k})

+ ∑_{{β_{1}+β_{2}=β, k_{1}+k_{2}=k+1; \atop β_{1}≠0 or k_{1}≠1,} ; β_{2}≠0 or k_{2}≠1} ∑_{i}(−1)^{deg P_{1}+···+deg P_{i−1}+i−1}

𝔪_{k_{1},β_{1}}(P_{1}, ⋯, 𝔪_{k_{2},β_{2}}(P_{i}, ⋯, P_{i+k_{2}−1}), ⋯, P_{k}).

By Definition 13.18, we have 𝔪_{1,0} = (−1)^{n}∂, where ∂ is the classical boundary map. Hence the first term in (13.24) is nothing but

(13.25.0)  (−1)^{n}∂(M_{1}^{main}(β : P_{1}, ⋯, P_{k}), ev_{0}),

and the second term in (13.24) is the sum of

(13.25.1)  (−1)^{∑_{j=1}^{i−1}(deg P_{j}+1)}(−1)^{n}(M_{1}^{main}(β : P_{1}, ⋯, ∂P_{i}, ⋯, P_{k}), ev_{0}).

The third term in (13.24) geometrically corresponds to moduli spaces described by Figure 13.1. This is the sum of (13.25.2) (−1)^{∑_{j=1}^{i−1}(deg P_{j}+1)}(M_{1}^{main}(β_{1}; P_{1}, ⋯, M_{1}^{main}(β_{2}; P_{i}, ⋯, P_{i+k_{2}−1}), ⋯, P_{k}), ev_{0}).

Moreover, as for the orientations of these spaces, we can show the following:

(13.26.1) (−1)^{∑_{j=1}^{i−1}(deg P_{j}+1)}(−1)^{n}M_{1}^{main}(β : P_{1}, ⋯, ∂P_{i}, ⋯, P_{k})

⊆(−1)^{n+1}∂M_{1}^{main}(β; P_{1}, ⋯, P_{k})

and (13.26.2)

(−1)^{∑_{j=1}^{i−1}(deg P_{j}+1)}M_{1}^{main}(β_{1}; P_{1}, ⋯, M_{1}^{main}(β_{2}; P_{i}, ⋯, P_{i+k_{2}−1}), ⋯, P_{k})

⊆(−1)^{n+1}∂M_{1}^{main}(β; P_{1}, ⋯, P_{k}).

The proofs of (13.26.1) and (13.26.2) will be given in Section 25.4 (Proposition 25.13). Therefore we find that (13.25.0) and the sum of (13.25.1) and (13.25.2) cancel each other and so (13.24) becomes zero. The proof that (C(L; Λ_{0,nov}), 𝔪) is a filtered A_{∞} algebra is now complete, except the statement about the unit. In fact, strictly speaking, e = [L] does not satisfy Condition 13.11. We refer to §20 for the precise statement on e. □

So far we did not assume that the obstruction class o_{i}(L) vanishes. We now rewrite this condition in terms of the filtered A_{∞} algebra we have just constructed. Let (C, 𝔪) be a filtered A_{∞} algebra and b ∈ (C[1])^{0}. (Note that (C[1])^{0} = C^{1}.) We put

(13.27)  e^{b} := 1 + b + b ⊗ b + ⋯ + \underbrace{b ⊗ ⋯ ⊗ b}_{k times} + ⋯ ∈ B̂(C[1]).

We do not put the factorials here unlike the usual definition of the exponential. This is because we use only the main component among the (k + 1)! components of M_{k+1} to define the map 𝔪_{k,β}.

In the case of our example (C(L; Λ_{0,nov}), 𝔪), we put

(13.28)  b = ∑ B(β_{i}) ⊗ [β_{i}] = ∑ B(β_{i}) ⊗ T^{ω(β_{i})}e^{\frac{μ(β_{i})}{2}} ∈ C[1]^{0}(L; Λ_{0,nov})

As we will explain later, the chains o(β_{i}) and so B(β_{i}) depend on the given almost complex structure J and perturbation Ξ of Kuranishi map.

Definition 13.29. We say that b is a bounding cochain if d̂e^{b} = 0. A filtered A_{∞} algebra is said to be unobstructed if there exists a bounding cochain and obstructed otherwise. We denote by M̂(L; J, Ξ) the set of all bounding cochains b.

Lemma 13.30. The chains B(β_{i}) bound o(β_{i}) inductively if and only if d̂e^{b} = 0.

Proof. First it is easy to prove d̂e^{b} = e^{b}xe^{b} where

x = ∑_{β≠0} x_{β} ⊗ [β]

is defined by

(13.31)  x_{β} = ∑_{k≥1,β_{i_{j}}} 𝔪_{k,β−Σβ_{i_{j}}} (B(β_{i_{1}}), ⋯, B(β_{i_{k}})) + 𝔪_{0,β}(1),

for β ≠ 0. We rewrite this equation as

(13.32)  \begin{matrix}x_{β} = & 𝔪_{1,0}(B(β)) \\ + ( ∑_{k≥2} ∑_{β_{i_{1}} ,··· ,β_{i_{k}}} 𝔪_{k,β−Σβ_{i_{j}}} (B(β_{i_{1}}), ⋯, B(β_{i_{k}})) \\ + ∑_{β′≠0} 𝔪_{1,β′}(B(β − β′)) + 𝔪_{0,β}(1)). \end{matrix}

It follows from our definition that the second term of the right hand side of (13.32) is exactly the chain o(β) and so (13.32) can be written as

\begin{aligned}x_{β} & = 𝔪_{1,0}(B(β)) + o(β) \\ & = (−1)^{n}∂B(β) + o(β). \end{aligned}

(See Definition 13.18 for the definition of 𝔪_{1,β} and note that 𝔪_{1,0} = (−1)^{n}∂, where ∂ is the classical boundary map.) Therefore d̂e^{b} = 0 is equivalent to x_{β} = 0 for all β ≠ 0 i.e.,

o(β) = (−1)^{n+1}∂B(β) for β ≠ 0.

This is nothing but the definition of the orientation of B(β). (See (4.4)). Hence the proof of Lemma 13.30 is complete. □

We remark that Theorem 12.10 immediately follows from Lemma 13.30.

Remark 13.33. (13.32) shows that our equation d̂e^{b} = 0 is an inhomogeneous A_{∞}- version of Maurer-Cartan or Batalin-Vilkovisky master equation [BV]. The relation of Batalin-Vilkovisky master equation to the deformation theory is discussed in [Sch], [ASKZ], [BK], [K5]. (See also [Fu7].) We would like to remark that an element in B̂(C[1](L; Λ_{0,nov})) is of the form e^{b} with b ∈ B_{1}(C[1](L; Λ_{0,nov})) = C[1](L; Λ_{0,nov}) iff it defines a homomorphism φ_{b} : B̂(C[1](L; Λ_{0,nov})^{∗}) → Λ_{0,nov}. (See Lemma 15.9 or rather its dual version). Therefore the moduli space defined in Definition 13.29 coincides with that of Definition 12.6.

Remark 13.34. In this way, we find that our construction in Chapter 2 is similar to the one given by Eliashberg-Givental-Hofer (see [E, EGH]) and by Chekanov [Ch] for the contact homology and for the Floer theory of Legendrian knots. In fact, we can combine them together and define a filtered A_{∞} algebra for (exact) Lagrangian immersions. We will discuss it somewhere else.

Now let (C, 𝔪) be a filtered A_{∞} algebra with bounding cochains b_{1},b_{2}. We define

δ_{b_{1},b_{2}} : C(L; Λ_{0,nov}) → C(L; Λ_{0,nov})

as follows.

Definition 13.35.

δ_{b_{1},b_{2}}(x) = ∑_{k_{1},k_{2}≥0} 𝔪_{k_{1}+k_{2}+1}(\underbrace{b_{1}, ⋯, b_{1}}_{k_{1}}, x, \underbrace{b_{2}, ⋯, b_{2}}_{k_{2}})

Symbolically we may simply write this definition as δ_{b_{1},b_{2}}(x) = 𝔪(e^{b_{1}}xe^{b_{2}}) in an obvious way. It then follows that

(13.36)  d̂(e^{b_{1}}xe^{b_{2}}) = e^{b_{1}}δ_{b_{1},b_{2}}(x)e^{b_{2}} + d̂(e^{b_{1}})xe^{b_{2}} + (−1)^{deg x+1}e^{b_{1}}xd̂(e^{b_{2}}).

The second and the third term vanishes if d̂(e^{b_{1}}) = d̂(e^{b_{2}}) = 0.

112 FUKAYA, OH, OHTA, ONO

Lemma 13.37. If d̂(e^{b_{1}}) = d̂(e^{b_{2}}) = 0, then δ_{b_{1},b_{2}} ∘ δ_{b_{1},b_{2}} = 0. Proof. By (13.36) we have

0 = d̂(d̂(e^{b_{1}}xe^{b_{2}})) = d̂(e^{b_{1}}δ_{b_{1},b_{2}}(x)e^{b_{2}}) = e^{b_{1}}δ_{b_{1},b_{2}} ∘ δ_{b_{1},b_{2}}(x)e^{b_{2}}.

from which the lemma immediately follows. □

Then the following lemma immediately follows from our definitions. Lemma 13.38. If b = b_{1} = b_{2} satisfies (13.29), then δ_{b_{1},b_{2}} coincides with δ defined in (6.19).

Now let b ∈ C[1]^{0}(L, Λ_{0,nov}) be a cochain, which is not necessary a bounding cochain. For the cochain b, we next deform our filtered A_{∞} algebra as follows. Definition 13.39. Using this cochain b, we put

𝔪_{k}^{b}(x_{1}, ⋯, x_{k}) = ∑_{ℓ_{0},··· ,ℓ_{k}} 𝔪_{k+∑ ℓ_{i}}(\underbrace{b, ⋯, b}_{ℓ_{0}}, x_{1}, \underbrace{b, ⋯, b}_{ℓ_{1}}, ⋯, \underbrace{b, ⋯, b}_{ℓ_{k−1}}, x_{k}, \underbrace{b, ⋯, b}_{ℓ_{k}})
= 𝔪(e^{b}x_{1}e^{b}x_{2} ⋯ x_{k−1}e^{b}x_{k}e^{b})

for k = 0, 1, 2 · · ·. We note that 𝔪_{0}^{b}(1) = 𝔪(e^{b}). Proposition 13.40. (C, 𝔪^{b}) is a filtered A_{∞} algebra. In addition, d̂(e^{b}) = 0 is equivalent to 𝔪_{0}^{b} = 0. Proof. We extend 𝔪_{k}^{b} to d̂^{b} as a coderivation on B̂(C[1]) which is given by

\begin{matrix}d̂^{b}(x_{1}, ⋯, x_{k}) = ∑_{ℓ=1}^{k+1} ∑_{ℓ′=ℓ}^{k+1} & (−1)^{deg x_{1}+···+deg x_{ℓ−1}+ℓ−1}x_{1} ⊗ ⋯ ⊗ x_{ℓ−1} \\ ⊗ 𝔪(e^{b}x_{ℓ}e^{b} ⋯ e^{b}x_{ℓ′−1}e^{b}) ⊗ x_{ℓ′} ⊗ ⋯ ⊗ x_{k}. \end{matrix}

When ℓ′ = ℓ, we put 𝔪(e^{b}) in place of 𝔪(e^{b}x_{ℓ}e^{b} ⋯ e^{b}x_{ℓ′−1}e^{b}) in the right hand side. We compare this formula with d̂∘d̂(e^{b}x_{1}e^{b} ⋯ e^{b}x_{k}e^{b}) = 0 and obtain the result that d̂^{b} ∘ d̂^{b} = 0. Moreover by noticing the following identities

d̂(e^{b}) = e^{b}𝔪(e^{b})e^{b} = e^{b}𝔪_{0}^{b}(1)e^{b},

we can find the second assertion. □

Now we are ready to complete the proof of Theorem 12.4. Let us first define

d^{k} : C[1](L; Λ_{0,nov})^{∗} → B̂_{k}(C[1](L; Λ_{0,nov})^{∗}).

Let {P_{i}; i = 1, 2, ⋯ } be the countable basis generating C(L; ℚ). This will be chosen later in Chapter 5 so that all the moduli chains 𝔪_{k}^{b}(P_{i_{1}}, ⋯, P_{i_{k}}; β) lie in the span of C(L; ℚ) over Λ_{0,nov}. (See Theorem 19.10.) Therefore we can find

⟨𝔪_{k}^{b}(i_{1}, ⋯, i_{k}; β), j⟩ ∈ ℚ

so that

𝔪_{k}^{b}(P_{i_{1}}, ⋯, P_{i_{k}}) = ∑_{j,β}⟨𝔪_{k}^{b}(i_{1}, ⋯, i_{k}; β), j⟩P_{j} ⊗ T^{ω(β)}e^{\frac{μ(β)}{2}} .

Then we define

(13.41)  𝔪_{k}^{b∗}(P_{j}^{∗}) = ∑_{j,β}⟨𝔪_{k}^{b}(i_{1}, ⋯, i_{k}; β), j⟩P_{i_{1}}^{∗} ⊗ ⋯ ⊗ P_{i_{k}}^{∗} ⊗ T^{ω(β)}e^{\frac{μ(β)}{2}} .

Here P_{i}^{∗} i = 1, 2, ⋯ is a dual basis. It is easy to see that the left hand side of (13.41) is contained in the completion B̂_{k}(C[1](L; Λ_{0,nov}))^{∗}.

We can extend 𝔪_{k}^{b∗} uniquely to a derivation

d^{k} : B̂(C[1](L; Λ_{0,nov}))^{∗} → B̂(C[1](L; Λ_{0,nov}))^{∗}.

We put d̂^{∗} = ∑_{k≥1} d^{k}. Lemma 13.37 implies d̂^{∗} ∘ d̂^{∗} = 0. It is easy to see from definition that d^{1} is Poincaré dual to the boundary operator in §6. Hence follows Theorem 12.4.

§14. Deformed Floer cohomology

In this section, we deform the standard Floer coboundary operator using the construction described in §13.

14.1. Algebraic framework.

Firstly we prepare some notion algebraically without mentioning Lagrangian submanifolds. Let (C_{1}, 𝔪^{1}), (C_{2}, 𝔪^{2}) be two filtered A_{∞} algebras. Let ⊕_{m∈ℤ}D^{m} be a graded free filtered Λ_{nov} module. Let F^{λ}D^{m} be its filtration. Here λ ∈ ℝ is allowed to be negative. (Compare it with the filtration introduced in Section 13.1. In the previous section, we considered a Λ_{0,nov} module, but in this section we consider a Λ_{nov} module.) We suppose that the filtration F^{λ}D^{m} satisfies the same conditions as (13.5.1)-(13.5.4). We denote by D the completion of ⊕_{m∈ℤ}D^{m} with respect to the filtration. We consider a family of maps

𝔫_{k_{1},k_{2}} : B_{k_{1}}(C_{1}[1]) ⊗_{Λ_{0,nov}} D ⊗_{Λ_{0,nov}} B_{k_{2}}(C_{2}[1]) → D

of degree +1 such that

(14.1)  \begin{matrix}𝔫_{k_{1},k_{2}} (F^{λ_{1}}(C_{1}[1])^{m_{1}} ⊗ ⋯ ⊗ F^{λ_{k_{1}}} (C_{1}[1])^{m_{k_{1}}} ⊗ F^{λ_{0}}D^{m_{0}} \\ ⊗ F^{λ′_{1}}(C_{2}[1])^{m′_{1}} ⊗ ⋯ ⊗ F^{λ′_{k_{2}}} (C_{2}[1])^{m′_{k_{2}}} ) \\ ⊆ F^{Σλ_{i}+λ_{0}+Σλ′_{i}}D^{∑ m_{i}+m_{0}+∑ m′_{i}+1}. \end{matrix}

We now use condition (14.1) to extend 𝔫_{k_{1},k_{2}} to the completion

B̂(C_{1}[1]) ˆ⊗_{Λ_{0,nov}} D ˆ⊗_{Λ_{0,nov}} B̂(C_{2}[1])

of ⨁_{k_{1},k_{2}} B_{k_{1}}(C_{1}[1]) ⊗_{Λ_{0,nov}} D ⊗_{Λ_{0,nov}} B_{k_{2}}(C_{2}[1]). For this purpose we first remark that B̂(C_{1}[1]) ˆ⊗_{Λ_{0,nov}} D ˆ⊗_{Λ_{0,nov}} B̂(C_{2}[1]) is a left B̂(C_{1}[1]) and right B̂(C_{2}[1]) bicomodule. We then extend 𝔫_{k_{1},k_{2}} to a bi-coderivation:

d̂: B̂(C_{1}[1]) ˆ⊗_{Λ_{0,nov}} D ˆ⊗_{Λ_{0,nov}} B̂(C_{2}[1]) → B̂(C_{1}[1]) ˆ⊗_{Λ_{0,nov}} D ˆ⊗_{Λ_{0,nov}} B̂(C_{2}[1]),

by

(14.2) d̂(x_{1,1} ⊗ ⋯ ⊗ x_{1,k_{1}} ⊗ y ⊗ x_{2,1} ⊗ ⋯ ⊗ x_{2,k_{2}})

= ∑_{k′_{1}≤k_{1},k′_{2}≤k_{2}}(−1)^{deg x_{1,1}+···+deg x_{1,k_{1}−k′_{1}} +k_{1}−k′_{1}}x_{1,1} ⊗ ⋯ ⊗ x_{1,k_{1}−k′_{1}}⊗

𝔫_{k′_{1},k′_{2}}(x_{1,k_{1}−k′_{1}+1}, ⋯, y, ⋯, x_{2,k′_{2}}) ⊗ x_{2,k′_{2}+1} ⊗ ⋯ ⊗ x_{1,k_{2}}

+ d̂^{1}(x_{1,1} ⊗ ⋯ ⊗ x_{1,k_{1}}) ⊗ y ⊗ x_{2,1} ⊗ ⋯ ⊗ x_{2,k_{2}}

+ (−1)^{Σ deg x_{1,i}+deg y+k_{1}+1}x_{1,1} ⊗ ⋯ ⊗ x_{1,k_{1}} ⊗ y ⊗ d̂^{2}(x_{2,1} ⊗ ⋯ ⊗ x_{2,k_{2}}).

Here d̂^{i} is defined by 𝔪^{i} (i = 1, 2). (14.1) implies that (14.2) is extended to the completion B̂(C_{1}[1]) ˆ⊗_{Λ_{0,nov}} D ˆ⊗_{Λ_{0,nov}} B̂(C_{2}[1]).

An analogue to Condition 13.9 is the following.

Condition 14.3

(14.3.1)  𝔫_{k_{1},k_{2}}(x_{1} ⊗ ⋯ ⊗ e ⊗ ⋯ ⊗ x_{k_{1}} ⊗ y ⊗ x′_{1} ⊗ ⋯ ⊗ x′_{k_{2}}) = 0

for k_{1} + k_{2} ≥ 1 and

(14.3.2)  𝔫_{k_{1},k_{2}}(x_{1} ⊗ ⋯ ⊗ x_{k_{1}} ⊗ y ⊗ x′_{1} ⊗ ⋯ ⊗ e ⊗ ⋯ ⊗ x′_{k_{2}}) = 0

for k_{1} + k_{2} ≥ 1. And

(14.3.3)  𝔫_{1,0}(e, y) = (−1)^{deg y}𝔫_{0,1}(y, e) = y.

Definition 14.4. We say that 𝔫_{k_{1},k_{2}} defines a left (C_{1}, 𝔪^{1}) and right (C_{2}, 𝔪^{2}) filtered A_{∞} bimodule structure on D if it satisfies (14.1) and d̂∘ d̂= 0. We say that e acts as a unit if Condition 14.3 is satisfied.

We say that D satisfies the gap condition if there exists λ′′ > 0 such that

(14.4.1)  \begin{matrix}𝔫_{k_{1},k_{2}} (F^{λ_{1}}(C_{1}[1])^{m_{1}} ⊗ ⋯ ⊗ F^{λ_{k_{1}}} (C_{1}[1])^{m_{k_{1}}} ⊗ F^{λ_{0}}D^{m_{0}} \\ ⊗ F^{λ′_{1}}(C_{2}[1])^{m′_{1}} ⊗ ⋯ ⊗ F^{λ′_{k_{2}}} (C_{2}[1])^{m′_{k_{2}}} ) \\ ⊆ F^{Σλ_{i}+λ_{0}+Σλ′_{i}+λ′′}D^{∑ m_{i}+m_{0}+∑ m′_{i}+1}. \end{matrix}

Here λ′′ is independent of λ_{i},λ′_{i}, m_{i}, m′_{i} and etc.

14.2. Filtered A_{∞} bimodule structure on C(L, L′; Λ_{nov}).

The main result of this section is the following Theorem 14.5. Let L, L′ be two Lagrangian submanifolds of M. (We do not assume the obstruction class to vanish.) We assume that they are transverse to each other. Let (C(L; Λ_{0,nov}), 𝔪) and (C(L′; Λ_{0,nov}), 𝔪′) be the filtered A_{∞} algebras constructed in Theorem 13.22. To define C(L, L′; Λ_{nov}) precisely, we recall Definition 2.11 and §5 that

CF (L, L′) = CF (L, L′; ℚ) = ˆ⨁_{[ℓ_{p},w]∈Cr(L,L′)}ℚ[ℓ_{p}, w],

where Cr(L, L′) is the set of the equivalence classes [ℓ_{p}, w] which are critical points of our functional A (see §2). The equivalence relation is given in Definition 2.3 and ℓ_{p} the constant path corresponding to an intersection point p ∈ L ∩ L′. Then CF (L, L′; ℚ) is the Λ(L, L′) module (see §5 ), where Λ(L, L′) is the Novikov ring defined in Definition 5.6. On the other hand, we have the injective homomorphism

i_{L,L′} : Λ(L, L′) ⟶ Λ_{nov}

by Lemma 6.27. Via the homomorphism, Λ_{nov} can be regarded as a Λ(L, L′) module. Then we define C(L, L′; Λ_{nov}) by a completion of

CF (L, L′; ℚ) ⊗_{Λ(L,L′)} Λ_{nov}.

Here we take a completion with respect to the energy filtration defined by the following. If T^{λ}e^{μ}[ℓ_{p}, w] ∈ C(L, L′; Λ_{nov}) satisfies

λ + ∫_{w} ω = λ′,

then T^{λ}e^{μ}[ℓ_{p}, w] ∈ F^{λ′}C(L, L′; Λ_{nov}). Moreover we give a grading of an element T^{λ}e^{μ}[ℓ_{p}, w] ∈ C(L, L′; Λ_{nov}) by 2μ + μ([ℓ_{p}, w]). (See §2 for the definition of the Maslov index μ([ℓ_{p}, w]).) Then it turns out that C(L, L′; Λ_{nov}) is a graded free Λ_{nov} module.

Let J = {J_{t}}_{0≤t≤1} be a t-dependent family of almost complex structures as before.

Theorem 14.5. There exists a structure of left (C(L; Λ_{0,nov}), 𝔪) and right (C(L′; Λ_{0,nov}), 𝔪′) filtered A_{∞} bimodule on C(L, L′; Λ_{nov}). It satisfies the gap condition (14.4.1).

Proof. Let [ℓ_{p}, w_{1}], [ℓ_{q}, w_{2}] ∈ Cr(L, L′) and P_{1}, ⋯, P_{k_{1}} ∈ C(L; ℚ), P′_{1}, ⋯, P′_{k_{2}} ∈ C(L′; ℚ). We consider the following conditions. Let Σ be ℝ × [0, 1] possibly with a finite number of sphere components attached. Let u : Σ → M and z_{i} ∈ ∂Σ = ∂(ℝ × [0, 1]).

(14.6.1) On ℝ × [0, 1], u satisfies

\frac{∂u}{∂τ} + J(t, u)\frac{∂u}{∂t} = 0.

If a bubble tree of spheres is rooted at (τ, t), all the spheres are J_{t}-holomorphic. (14.6.2) u(ℝ × {0}) ⊂ L, u(ℝ × {1}) ⊂ L′. (14.6.3) z_{i} = (τ_{i}, 0), for i = 2, ⋯, k_{1}+1. z_{i} = (τ_{i}, 1), for i = k_{1}+2, ⋯, k_{1}+k_{2}+1. (14.6.4) τ_{i} < τ_{i+1}, i = 2, ⋯, k_{1}, τ_{i} > τ_{i+1}, i = k_{1} + 2, ⋯, k_{1} + k_{2}. (14.6.5) u(z_{i}) ∈ P_{i} for i = 2, ⋯, k_{1} + 1. (14.6.6) u(z_{i}) ∈ P′_{i−k_{1}} for i = k_{1} + 2, ⋯, k_{1} + k_{2} + 1. (14.6.7) lim_{τ→−∞} u = p and lim_{τ→∞} u = q. (14.6.8) w_{1}#u ∼ w_{2}.

In (14.6.8), we recall Definition 2.3 for the equivalence relation ∼ and (2.5) for the notation #.

Figure 14.1

We consider the ℝ-action defined by u(·, ·) → u(· + τ_{0}, ·) for τ_{0} ∈ ℝ which induces a natural equivalence relation ∼ among the set of solution u’s of (14.6).

Definition 14.7. M(L, L′; [ℓ_{p}, w_{1}], [ℓ_{q}, w_{2}]; P_{1}, ⋯, P_{k_{1}}; P′_{1}, ⋯, P′_{k_{2}}) is the totality of the ∼ equivalence classes of u satisfying (14.6).

Let g_{i} and g′_{i} be the codimension of P_{i} and P′_{i} respectively. (In other words, g_{i}, g′_{i} are the degree of P_{i} and P′_{i} as distribution valued forms.)

Proposition 14.8. M(L, L′; [ℓ_{p}, w_{1}], [ℓ_{q}, w_{2}]; P_{1}, ⋯, P_{k_{1}}; P′_{1}, ⋯, P′_{k_{2}}) has an oriented Kuranishi structure. Its dimension is given by

(14.9)  μ([ℓ_{q}, w_{2}]) − μ([ℓ_{p}, w_{1}]) − ∑_{i=1}^{k_{1}} g_{i} − ∑_{i=1}^{k_{2}} g′_{i} + k_{1} + k_{2} − 1.

Proof. Construction of the Kuranishi structure will be given in Chapter 5. The precise orientation will be given in Subsection 25.2 in Chapter 6 (Definition 25.6). The dimension can be calculated in a way similar to the one in Chapter 2. □

Definition 14.10. Let the dimension (14.9) be 0. Let T^{λ_{i}}e^{μ_{i}}P_{i} ∈ C(L, Λ_{0,nov}), T^{λ′_{i}}e^{μ′_{i}}P′_{i} ∈ C(L′, Λ_{0,nov}) and T^{λ_{0}}e^{μ_{0}}[ℓ_{p}, w_{1}] ∈ C(L, L′; Λ_{nov}). Then we define 𝔫_{k_{1},k_{2}} by

𝔫_{k_{1},k_{2}}(T^{λ_{1}}e^{μ_{1}}P_{1} ⊗ ⋯ ⊗ T^{λ_{k_{1}}} e^{μ_{k_{1}}} P_{k_{1}} ⊗ T^{λ_{0}}e^{μ_{0}}[ℓ_{p}, w_{1}]

⊗ T^{λ′_{1}}e^{μ′_{1}}P′_{1} ⊗ ⋯ ⊗ T^{λ′_{k_{2}}} e^{μ′_{k_{2}}} P′_{k_{2}})
= ∑_{[ℓ_{q},w_{2}]∈Cr(L,L′)} #M(L, L′; [ℓ_{p}, w_{1}], [ℓ_{q}, w_{2}]; P_{1}, ⋯, P_{k_{1}}; P′_{1}, ⋯, P′_{k_{2}})T^{λ}e^{μ}[ℓ_{q}, w_{2}],

where

\begin{aligned}λ & = ∑_{i=1}^{k_{1}} λ_{i} + λ_{0} + ∫_{w_{1}} ω − ∫_{w_{2}} ω + ∑_{i=1}^{k_{2}} λ′_{i}, \\ μ & = ∑_{i=1}^{k_{1}} μ_{i} + μ_{0} + ∑_{i=1}^{k_{2}} μ′_{i}. \end{aligned}

Then the maps 𝔫_{k_{1},k_{2}} define

\begin{aligned}𝔫_{k_{1},k_{2}} : B_{k_{1}}(C[1](L, Λ_{0,nov})) & ⊗_{Λ_{0,nov}} C(L, L′; Λ_{nov}) \\ & ⊗_{Λ_{0,nov}} B_{k_{2}}(C[1](L′, Λ_{0,nov})) ⟶ C(L, L′; Λ_{nov}), \end{aligned}

and we can easily check that they satisfy (14.1) and (14.3), and have degree +1. (Note that (14.9) is equal to 0 and the proof of (14.3) is similar to one in §7.)

Let d̂be the map obtained from 𝔫_{k_{1},k_{2}} in Definition 14.10. To prove Theorem 14.5 it suffices to show the following. Note that the gap condition follows immediately from Gromov’s compactness theorem.

Lemma 14.11. The homomorphism d̂ defined in (14.2) satisfies d̂∘ d̂= 0.

Proof. Let us consider the moduli space

M(L, L′; [ℓ_{p}, w_{1}], [ℓ_{q}, w_{2}]; P_{1}, ⋯, P_{k_{1}}; P′_{1}, ⋯, P′_{k_{2}})

in the case when its dimension given in (14.9) is 1. The boundary of this moduli space is described by one of the following figures.

Figure 14.2

The terms corresponding to Figure 14.2.1 are

𝔫 (d̂^{1}(P_{1} ⊗ ⋯ ⊗ P_{k_{1}}) ⊗ [ℓ_{p}, w_{1}] ⊗ P′_{1} ⊗ ⋯ P′_{k_{2}}) .

Here d̂^{1} is defined in (C(L, Λ_{0,nov}), 𝔪). The terms corresponding to Figure 14.2.2 are

𝔫 (P_{1} ⊗ ⋯ ⊗ P_{k_{1}} ⊗ [ℓ_{p}, w_{1}] ⊗ d̂^{2}(P′_{1} ⊗ ⋯ P′_{k_{2}})) .

Here d̂^{2} is defined in (C(L′, Λ_{0,nov}), 𝔪′). The terms corresponding to Figure 14.2.3 are

𝔫 (P_{1} ⊗ · · · · · · 𝔫(⋯ ⊗ P_{k_{1}} ⊗ [ℓ_{p}, w_{1}] ⊗ P′_{1} ⊗ ⋯) ⋯ ⊗ P′_{k_{2}}) .

Combing the argument in Section 25.2 and 25.4, we can check the signs in a similar way. Therefore we find that these three terms cancel each other. Hence the proof of Lemma 14.11 is complete. □

Let D be a left (C_{1}, 𝔪^{1}) and right (C_{2}, 𝔪^{2}) filtered A_{∞} bimodule. Let b_{1} and b_{2} be bounding cochains of (C_{1}, 𝔪^{1}) and (C_{2}, 𝔪^{2}) respectively. We define the coboundary operator δ_{b_{1},b_{2}} : D → D, which gives a deformation of the standard Floer coboundary operator, as follows.

Definition 14.12.

δ_{b_{1},b_{2}}(y) = ∑_{k_{1},k_{2}} 𝔫_{k_{1},k_{2}}(b_{1}, ⋯, b_{1}, y, b_{2}, ⋯, b_{2}) = 𝔫(e^{b_{1}}ye^{b_{2}}).

Lemma 14.13. If b_{1},b_{2} are given by (13.28), then the coboundary operator δ_{b_{1},b_{2}} is the same as the one defined in §5.

The proof is immediate from definition. Now by applying Lemma 14.13, we obtain Theorem 12.11 as an immediate consequence.

Lemma 14.14. For given bounding cochains b_{1}, b_{2} of (C_{1}, 𝔪^{1}), (C_{2}, 𝔪^{2}) respectively, we have δ_{b_{1},b_{2}} ∘ δ_{b_{1},b_{2}} = 0.

The proof is the same as that of Lemma 13.37 which is in turn the same as the arguments used in Chapter 2.

Theorem 12.7 is an immediate consequence of Lemma 14.14 and Theorem 14.5.

§15. Homotopy equivalence and deformations.

We will define the notion of the homotopy equivalence between two filtered A_{∞} algebras over the ring Λ_{0,nov}. Using this, we will explain the equivalence relation used in Theorem 12.8. We need to further develop some homological algebra for this purpose.

15.1. Algebraic framework.

Firstly, we consider an A_{∞} algebra over a commutative ring R. After that, we discuss the case of a filtered A_{∞} algebra over Λ_{0,nov}.

Let (\overline{C}_{i}, \overline{𝔪}^{i}), i = 1, 2, be A_{∞} algebras over a commutative ring R with unit. For k = 0, 1, 2, ⋯, let us consider the family of maps

\overline{𝔣}_{k} : B_{k}(\overline{C}_{1}[1]) → \overline{C}_{2}[1]

of degree 0. Note that \overline{𝔣}_{0} : R → \overline{C}_{2}[1] and we put \overline{𝔣}_{0}(1) = 0. These maps induce

\overline{φ}_{k} : B_{k}(\overline{C}_{1}[1]) → B(\overline{C}_{2}[1]),

by

(15.1) \begin{matrix}\overline{φ}_{k}(x_{1} ⊗ ⋯ & ⊗ x_{k}) = ∑_{0≤k_{1}≤···≤k_{n}≤k} \overline{𝔣}_{k_{1}}(x_{1}, ⋯, x_{k_{1}}) ⊗ ⋯ ⊗ \\ \overline{𝔣}_{k_{i+1}−k_{i}}(x_{k_{i}+1}, ⋯, x_{k_{i+1}}) ⊗ · · · · · · ⊗ \overline{𝔣}_{k−k_{n}}(x_{k_{n}+1}, ⋯, x_{k}). \end{matrix}

We note that when \overline{𝔣}_{0} appears in the right hand side of (15.1), we put \overline{𝔣}_{0}(1) = 0 there. Thus, in particular, we have

\overline{φ}_{0}(1) = 1 + \overline{𝔣}_{0}(1) + \overline{𝔣}_{0}(1) ⊗ \overline{𝔣}_{0}(1) + ⋯ = 1.

We put ˆ \overline{φ} = ∑ \overline{φ}_{k}. Then it is easy to see that ˆ \overline{φ} : B(\overline{C}_{1}[1]) → B(\overline{C}_{2}[1]) is a coalgebra homomorphism.

Definition 15.2. We call \overline{𝔣} = \{\overline{𝔣}_{k}\}_{k≥0} an A_{∞} homomorphism from \overline{C}_{1} to \overline{C}_{2} if

ˆ\overline{φ} ∘ ˆ \overline{d}^{1} = ˆ \overline{d}^{2} ∘ ˆ \overline{φ}.

By definition an A_{∞} homomorphism \overline{𝔣} from \overline{C}_{1} to \overline{C}_{2} induces a homomorphism

from H^{∗}(B(\overline{C}_{1}[1]), ˆ \overline{d}^{1}) to H^{∗}(B(\overline{C}_{2}[1]), ˆ \overline{d}^{2}).

Let \overline{𝔣}_{k}^{i} : B_{k}(\overline{C}_{i}[1]) → \overline{C}_{i+1}[1] (i = 1, 2) define an A_{∞} homomorphism.

Definition 15.3. The composition \overline{𝔣}^{2} ∘ \overline{𝔣}^{1} = \{(\overline{𝔣}^{2} ∘ \overline{𝔣}^{1})_{k}\}_{k≥0} of \overline{𝔣}^{1} and \overline{𝔣}^{2} is given by the formula

\begin{matrix}(\overline{𝔣}^{2} ∘ \overline{𝔣}^{1})_{k}(x_{1}, ⋯, x_{k}) \\ = & ∑_{m} ∑_{k_{1}+···+k_{m}=k} \overline{𝔣}_{m}^{2}(\overline{𝔣}_{k_{1}}^{1}(x_{1}, ⋯, x_{k_{1}}), ⋯, \overline{𝔣}_{k_{m}}^{1}(x_{k−k_{m}+1}, ⋯, x_{k})). \end{matrix}

Lemma 15.4. \overline{𝔣}^{2} ∘ \overline{𝔣}^{1} defines an A_{∞} homomorphism from \overline{C}_{1} to \overline{C}_{3}.

Proof. If ˆ \overline{φ}^{1} and ˆ \overline{φ}^{2} are induced from \overline{𝔣}_{k}^{1} and \overline{𝔣}_{k}^{2} by (15.1) respectively, then the composition ˆ \overline{φ}^{2} ∘ ˆ\overline{φ}^{1} is induced from (\overline{𝔣}^{2} ∘\overline{𝔣}^{1})_{k}. Lemma 15.4 follows immediately. □

When \overline{𝔪}_{0} = 0 in an A_{∞} algebra (\overline{C}, \overline{𝔪}), we find that the degree +1 map \overline{𝔪}_{1} satisfies \overline{𝔪}_{1} ∘ \overline{𝔪}_{1} = 0. (See Remark 13.3 (1) and Remark 13.13 (1)). Thus we have the complex (\overline{C}[1], \overline{𝔪}_{1}). An A_{∞} homomorphism induces a homomorphism between the cohomology.

Definition 15.5. Let (\overline{C}_{i}, \overline{𝔪}^{i}) be A_{∞} algebras over R. Suppose that \overline{𝔪}_{0}^{i} = 0, (i = 1, 2). We say that an A_{∞} homomorphism \overline{𝔣} : (\overline{C}_{1}, \overline{𝔪}^{1}) → (\overline{C}_{2}, \overline{𝔪}^{2}) is a weak homotopy equivalence if \overline{𝔣}_{1} induces an isomorphism H^{∗}(\overline{C}_{1}[1], \overline{𝔪}_{1}^{1}) →^{∼} H^{∗}(\overline{C}_{2}[1], \overline{𝔪}_{1}^{2}).

Lemma 15.6. Let (\overline{C}_{i}, \overline{𝔪}^{i}) be A_{∞} algebras with \overline{𝔪}_{0}^{i} = 0 (i = 1, 2). If \overline{𝔣} : (\overline{C}_{1}, \overline{𝔪}^{1}) → (\overline{C}_{2}, \overline{𝔪}^{2}) is a weak homotopy equivalence, then there exists a weak homotopy equivalence \overline{𝔣}′ : (\overline{C}_{2}, \overline{𝔪}^{2}) → (\overline{C}_{1}, \overline{𝔪}^{1}) such that both of the compositions (\overline{𝔣} ∘ \overline{𝔣}′)_{1} and (\overline{𝔣}′ ∘ \overline{𝔣})_{1} induce the identities on \overline{𝔪}_{1}^{i} cohomology.

As for the proof, we refer to Theorem 19.8 and Appendix 5.

Now we consider the filtered A_{∞} algebra case. Let (C_{i}, 𝔪^{i}), i = 1, 2, be filtered A_{∞} algebras over the ring Λ_{0,nov}. For k = 0, 1, 2, ⋯, we consider the family of maps

𝔣_{k} : B_{k}(C_{1}[1]) → C_{2}[1]

of degree 0 such that

(15.7.1)  𝔣_{k}(F^{λ}B_{k}(C_{1}[1])) ⊆ F^{λ}C_{2}[1]

and

(15.7.2)  𝔣_{0}(1) ∈ F^{λ′}C_{2}[1] for some λ′ > 0.

Note that 𝔣_{0} : Λ_{0,nov} → C_{2}[1] and we do not assume 𝔣_{0}(1) = 0 in this case. These maps induce

φ_{k} : B_{k}(C_{1}[1]) → B̂(C_{2}[1]),

by the formula

(15.8) \begin{aligned}φ_{k}(x_{1} ⊗ ⋯ ⊗ x_{k}) = ∑_{0≤k_{1}≤···≤k_{n}≤k} & 𝔣_{k_{1}}(x_{1}, ⋯, x_{k_{1}}) ⊗ ⋯ \\ & ⋯ ⊗ 𝔣_{k_{i+1}−k_{i}}(x_{k_{i}+1}, ⋯, x_{k_{i+1}}) ⊗ ⋯ \\ & ⋯ ⊗ 𝔣_{k−k_{n}}(x_{k_{n}+1}, ⋯, x_{k}), \end{aligned}

and (15.7) implies that ∑ φ_{k} = φ̂ converges. We note that when 𝔣_{0} appears in the right hand side of (15.8), we put 𝔣_{0}(1) there. Thus, in particular, φ_{0} is given by

\begin{aligned}φ_{0}(1) & = 1 + 𝔣_{0}(1) + 𝔣_{0}(1) ⊗ 𝔣_{0}(1) + ⋯ \\ & = e^{𝔣_{0}(1)} \end{aligned}

in our notation. (See (13.27)). Then it is easy to see that φ̂: B̂(C_{1}[1]) → B̂(C_{2}[1]) is a coalgebra homomorphism (see Lemma 15.15).

Definition 15.9. We call 𝔣 = {𝔣_{k}}_{k≥0} a filtered A_{∞} homomorphism from C_{1} to C_{2} if φ̂ ∘ d̂^{1} = d̂^{2} ∘ φ̂.

Let e_{i_{j}} be a basis of C_{1} as in §13. We say that 𝔣 satisfies the gap condition if

(15.9.1)  𝔣_{k}(e_{i_{1}}, …, e_{i_{k}}) − \overline{𝔣}_{k}(e_{i_{1}}, …, e_{i_{k}}) ∈ F^{λ′′}C_{(2)}[1]

where λ′′ > 0 is independent of i_{j} and k.

Let 𝔣_{k}^{i} : B_{k}(C_{i}[1]) → C_{i+1}[1] (i = 1, 2) define a filtered A_{∞} homomorphism.

Definition 15.10. The composition 𝔣^{2} ∘ 𝔣^{1} = {(𝔣^{2} ∘ 𝔣^{1})_{k}} of 𝔣^{1} and 𝔣^{2} is defined by

\begin{matrix}(𝔣^{2} ∘ 𝔣^{1})_{k}(x_{1}, ⋯, x_{k}) \\ = & ∑_{m} ∑_{k_{1}+···+k_{m}=k} 𝔣_{m}^{2}(𝔣_{k_{1}}^{1}(x_{1}, ⋯, x_{k_{1}}), ⋯, 𝔣_{k_{m}}^{1}(x_{k−k_{m}+1}, ⋯, x_{k})). \end{matrix}

Lemma 15.11. 𝔣^{2} ∘ 𝔣^{1} defines a filtered A_{∞} homomorphism from C_{1} to C_{3}.

Proof. By noticing that we have φ_{0}(1) = e^{𝔣_{0}(1)}, the proof is similar to that of Lemma 15.4. □

Let (C_{i}, 𝔪^{i}) (i = 1, 2) be filtered A_{∞} algebras over Λ_{0,nov} and 𝔣 : (C_{1}, 𝔪^{1}) → (C_{2}, 𝔪^{2}) a filtered A_{∞} homomorphism. Then 𝔣 naturally induces an A_{∞} homomorphism \overline{𝔣} : (\overline{C}_{1}, \overline{𝔪}^{1}) → (\overline{C}_{2}, \overline{𝔪}^{2}), where (\overline{C}_{i}, \overline{𝔪}^{i}) are the A_{∞} algebras over ℚ ≅ Λ_{0,nov}/Λ_{0,nov}^{0} defined by (13.10).

Definition 15.12. Let (C_{i}, 𝔪^{i}) (i = 1, 2) be filtered A_{∞} algebras over Λ_{0,nov} such that \overline{𝔪}_{0}^{i} = 0. For these filtered A_{∞} algebras, we say that a filtered A_{∞} homomorphism 𝔣 : (C_{1}, 𝔪^{1}) → (C_{2}, 𝔪^{2}) is a weak homotopy equivalence, if the induced A_{∞} homomorphism \overline{𝔣} : (\overline{C}_{1}, \overline{𝔪}^{1}) → (\overline{C}_{2}, \overline{𝔪}^{2}) is a weak homotopy equivalence in the sense of Definition 15.5.

(We recall that the condition \overline{𝔪}_{0} = 0 is satisfied in our filtered A_{∞} algebra (C(L, Λ_{0,nov}), 𝔪), see Definition 13.18 and Remark 13.19.(1)).

Hereafter we assume that the filtered A_{∞} algebras (C_{i}, 𝔪^{i}) are unobstructed and weakly finite. Here, a filtered A_{∞} algebra (C, 𝔪) is called unobstructed if there exists a bounding cochain b ∈ C[1]^{0} such that d̂(e^{b}) = 0, (see Definition 13.29 for the geometric version), and weakly finite if there exists a finite Λ_{0,nov} module cochain complex (C′, δ′) such that there is a filtered A_{∞} homomorphism 𝔣′ : (C′, δ′) → (C, 𝔪) satisfying the gap condition (15.9.1) which induces an isomorphism between H^{∗}(C′, δ′) and H^{∗}(C[1], 𝔪_{1}^{b}), (see Definition A4.25 in §A4, and Definition 13.39 and Proposition 13.40 for 𝔪^{b}). We also note that our unobstructed filtered A_{∞} algebra (C(L, Λ_{0,nov}), 𝔪) associated to Lagrangian submanifold L is weakly finite, see Theorem A4.28 in §A4). Under these assumptions, we can prove the following lemma which is similar to Lemma 15.6 and will be proved in §A5.

Lemma 15.13. Let (C_{i}, 𝔪^{i}) be unobstructed and weakly finite filtered A_{∞} algebras (i = 1, 2). If a filtered A_{∞} homomorphism 𝔣^{1} is a weak homotopy equivalence and if it satisfies the gap condition (15.9.1), then there exists a filtered A_{∞} homomorphism 𝔣^{2} such that both of the compositions (𝔣^{1} ∘ 𝔣^{2})_{1} and (𝔣^{2} ∘ 𝔣^{1})_{1} induce the identity on the cohomology H^{∗}(\overline{C}_{i}[1], \overline{𝔪}_{1}^{i}).

Definition 15.14. Let (C_{i}, 𝔪^{i}) be filtered A_{∞} algebras over Λ_{0,nov} with \overline{𝔪}_{0}^{i} = 0 (i = 1, 2). We assume that (C_{i}, 𝔪^{i}) are unobstructed and weakly finite. Then (C_{1}, 𝔪^{1}) and (C_{2}, 𝔪^{2}) are said to be weakly homotopy equivalent if there exist filtered A_{∞} homomorphisms 𝔣^{1} and 𝔣^{2} from C_{1} to C_{2} and C_{2} to C_{1} respectively such that the compositions 𝔣^{1} ∘ 𝔣^{2}, 𝔣^{2} ∘ 𝔣^{1} are weakly homotopy equivalences.

Lemma 15.13 implies that if there exists a weak homotopy equivalence 𝔣_{1} : (C_{1}, 𝔪^{1}) → (C_{2}, 𝔪^{2}), then (C_{1}, 𝔪^{1}) and (C_{2}, 𝔪^{2}) are weakly homotopy equivalent.

Now we recall from Lemma 13.37 that two bounding cochains b_{1}, b_{2} on filtered A_{∞} algebra (C, 𝔪) induce a coboundary map

δ_{b_{1},b_{2}} : C → C

as in Definition 13.35. We next prove that a weak homotopy equivalence induces a natural isomorphism between the cohomology of (C, δ_{b_{1},b_{2}}). We first prove the following lemma.

Lemma 15.15. For a non-zero element x of B̂(C[1]), x = e^{b} for some b ∈ B_{1}(C[1]) = C[1] if and only if Δx = x ⊗ x.

Proof. It is easy to check the statement for “only if” and so we will only prove the one for “if”. Write

x = ∑_{k=0}^{∞} x_{k}, x_{k} ∈ B_{k}(C[1])

and assume

(15.16)  Δx = x ⊗ x.

Using the definition (13.6) of the coproduct Δ and comparing the 0th-order term of the two sides of (15.16) we get the equation

(x_{0})^{2} = x_{0}, x_{0} ∈ Λ_{0,nov}

and hence x_{0} = 0 or 1. In case x_{0} = 1, comparison of the 1st-order terms of (15.16) gives rise to the equation

1 ⊗ x_{1} + x_{1} ⊗ 1 = 1 ⊗ x_{1} + x_{1} ⊗ 1.

In other words, the equation gives no restriction on the 1st-order term. We set b := x_{1}. Now for the second order terms in general, it is easy to see that the coproduct satisfies

Δx_{2} = 1 ⊗ x_{2} + x_{2} + x_{2} ⊗ 1.

Therefore the second order term of (15.16) becomes

1 ⊗ x_{2} + x_{2} + x_{2} ⊗ 1 = 1 ⊗ x_{2} + x_{1} ⊗ x_{1} + x_{2} ⊗ 1

and hence x_{2} = x_{1} ⊗ x_{1} = b ⊗ b.

Repeating this straight-forward order-by-order comparisons of the two sides of (15.16), we prove x = e^{b}. In case x_{0} = 0, similar reasoning proves x = 0. This finishes the proof. □

Since φ̂: B̂(C[1]) → B̂(C[1]) is a coalgebra homomorphism, if x satisfies (15.16), so does φ̂(x) and so we have an element φ(b_{0}) such that φ̂(e^{b_{0}}) = e^{φ(b_{0})} by Lemma 15.15. More explicitly, we have

(15.17)  φ(b_{0}) = 𝔣_{0}(1) + 𝔣_{1}(b_{0}) + 𝔣_{2}(b_{0}, b_{0}) + · · · .

Proposition 15.18. Let (C_{0}, 𝔪^{0}) and (C_{1}, 𝔪^{1}) be the filtered A_{∞} algebras such that \overline{𝔪}_{0}^{i} = 0 (i = 0, 1). We assume that (C_{i}, 𝔪^{i}) are unobstructed and weakly finite. Let 𝔣 = {𝔣_{k}}_{k≥0} define a weak homotopy equivalence between them. Let d̂^{0}, d̂^{1} be obtained from 𝔪^{0}, 𝔪^{1}, and φ obtained from 𝔣_{k}. Let b_{0} ∈ (C_{0}[1])^{0}. Then d̂^{0}e^{b_{0}} = 0 if and only if d̂^{1}e^{φ(b_{0})} = 0.

If we assume the gap condition for (C_{0}, 𝔪^{0}), (C_{1}, 𝔪^{1}) and for 𝔣 in addition, then the cohomology of δ_{b_{1},b_{2}}^{0} is isomorphic to that of δ_{φ(b_{1}),φ(b_{2})}^{1}, d̂^{0}e^{b_{0}} = 0.

Proof. The equivalence

d̂^{0}e^{b_{0}} = 0 ⟺ d̂^{1}e^{φ(b_{0})} = 0

is immediate from the definition of φ above. Suppose that this condition is satisfied for b_{1} and b_{2}. If we put

ψ(x) = ∑_{k_{1},k_{2}} 𝔣_{k_{1}+k_{2}+1}(\underbrace{b_{1}, ⋯, b_{1}}_{k_{1}}, x, \underbrace{b_{2}, ⋯, b_{2}}_{k_{2}}),

then we have

φ̂(e^{b_{1}}xe^{b_{2}}) = e^{φ(b_{1})}ψ(x)e^{φ(b_{2})}.

Therefore,

e^{φ(b_{1})}(δ_{φ(b_{1}),φ(b_{2})}ψ(x))e^{φ(b_{2})} = d̂^{1}(e^{φ(b_{1})}ψ(x)e^{φ(b_{2})}) = d̂^{1}(φ̂(e^{b_{1}}xe^{b_{2}}))
= φ̂(d̂^{0}(e^{b_{1}}xe^{b_{2}})) = φ̂(e^{b_{1}}(δ_{b_{1},b_{2}}x)e^{b_{2}})
= e^{φ(b_{1})}ψ(δ_{b_{1},b_{2}}x)e^{φ(b_{2})}.

Hence, ψ is a cochain map from (C_{0}, δ_{b_{1},b_{2}}) to (C_{1}, δ_{φ(b_{1}),φ(b_{2})}). If the gap condition is satisfied, we can use the spectral sequence to prove that φ̂ induces an isomorphism. See §A4 for more details. □

15.2. Homotopy equivalence and deformations of C(L; Λ_{0,nov})

We now apply the above algebraic framework to our problem of deforming Lagrangian submanifolds in the extended moduli space.

The moduli space M_{k+1}^{main}(L, J : β) has an oriented Kuranishi structure and we may assume that ev : M_{k+1}^{main}(L, J : β) → L^{k+1} is weakly submersive for each β and k. (Summary of the notion of Kuranishi structure is given in Appendix 2.) By Sard-Smale transversality argument, we can find a sub-complex of \overline{S}_{∗}(L) generated by countably many currents represented by smooth singular chains, so that its ordinary homology is isomorphic to the singular homology and, after tensoring with Λ_{0,nov}, we get a filtered A_{∞}-algebra (C(L; Λ_{0,nov}), 𝔪(J, Ξ)), where Ξ denotes a system of multi-valued perturbations of Kuranishi maps for the moduli spaces of holomorphic discs and spheres.

We first study perturbation of J and Ξ. The perturbation of L by Hamiltonian isotopy can be reduced to it in many cases, as we will see later.

The following is a main result of this subsection.

Theorem 15.19. Let L be a fixed Lagrangian submanifold and j = {J_{s}}_{0≤s≤1} be a one parameter family of compatible almost complex structures. For filtered A_{∞}- algebras (C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0})) and (C_{(1)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{1})), there exists a filtered A_{∞}-algebra (C_{(2)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{2})), a map

φ̂: (C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0})) → (C_{(2)}(L, Λ_{0,nov}), 𝔪(J_{1}, Ξ_{2})),

that is a weak homotopy equivalence, and a diffeomorphism F : L → L arbitrarily close to the identity such that

(15.19.1) F_{∗}C_{(1)}(L; Λ_{0,nov}) ⊆ C_{(2)}(L, Λ_{0,nov}). (15.19.2) 𝔪_{k}^{2} coincides with F_{∗} ∘ 𝔪_{k}^{1} ∘ (F_{∗}^{−1})^{⊗k} on (F_{∗}C_{(1)}(L; Λ_{0,nov}))^{⊗k}.

Remark 15.20. In Theorem 15.19, we have to extend the chain complex C_{(1)}(L; Λ_{0,nov}) to C_{(2)}(L; Λ_{0,nov}), since the chains in C_{(0)}(L; Λ_{0,nov}) may not be contained in C_{(1)}(L; Λ_{0,nov}) let alone the images of evaluation maps in Definition 15.25 below. (15.19.2) means that Ξ_{2} is basically the same as Ξ_{1}. We use the same J_{1}. We will carefully discuss the way of finding F and extending the chain complex in §A5.

Assuming the extension mentioned above in Remark 15.20 has been carried out, we start with the proof of Theorem 15.19.

Proof of Theorem 15.19. We omit F from our notation and simply write C_{(1)}(L; Λ_{0,nov}) in place of F_{∗}C_{(1)}(L; Λ_{0,nov}) for the sake of simplicity of notations. In fact the role of F is to make the basis of C_{(1)}(L; Λ_{0,nov}) in general position with the basis of C_{(0)}(L; Λ_{0,nov}) and with the various moduli spaces of pseudo-holomorphic discs. We start with the construction of the A_{∞} map φ̂. We need some notations.

Let (Σ, z⃗) be an element of M_{k+1}^{main}. Let 𝔄 be the set of the components of Σ. We define a partial order on 𝔄 as follows. Let α_{1}, α_{2} ∈ 𝔄, and let Σ_{α_{1}}, Σ_{α_{2}} be the corresponding components of Σ. Let Σ_{α_{0}} be the component containing the 0-th marked point. Note that some of the Σ_{α} may contain sphere bubbles. We do not care much about the sphere bubbles since the way of handling with them is by now well understood in the literature.

Definition 15.21. We say α_{1} ≤ α_{2} if every path joining Σ_{α_{1}} to Σ_{α_{0}} intersects with Σ_{α_{2}}.

It is easy to see that Definition 15.21 defines a partial order. Let J_{0}, J_{1} be two generic almost complex structures. Now for each j = {J_{s}}_{0≤s≤1} connecting J_{0}, J_{1}, we consider the system ((Σ, z⃗), (u_{α}), (s_{α})) satisfying the following properties

(15.22.1) u_{α} : (Σ_{α}, ∂Σ_{α}) → (M, L) is a J_{s_{α}} holomorphic map. (15.22.2) If α_{1} ≤ α_{2} then s_{α_{1}} ≤ s_{α_{2}}. (15.22.3) ((Σ, z⃗), (u_{α})) is stable in the sense of Definition 3.8. (15.22.4) The homology class of u_{α} is β(α) and ∑_{α} β(α) = β. (15.22.5) If z ∈ Σ_{α} ∩ Σ_{α′} then u_{α}(z) = u_{α′}(z).

Figure 15.1

We identify two such systems if there exists an isomorphism between them in an obvious sense.

Definition 15.23. Let J_{0}, J_{1} be given generic almost complex structures and j = {J_{s}}_{0≤s≤1} be a family connecting them. We define M_{k+1}^{main}(L, j : β; para) to be the set of all equivalence classes ((Σ, z⃗), (u_{α}), (s_{α})) satisfying (15.22).

We will prove the following result in Chapter 5 by the same way as we do for other similar statements on the moduli space.

Proposition 15.24. M_{k+1}^{main}(L, j : β; para) has an oriented Kuranishi structure.

We denote

M_{1}^{main}(L, j : β; para; P_{1}, ⋯, P_{k}) = M_{k+1}^{main}(L, j : β; para)_{(ev_{1},··· ,ev_{k})} × ∏_{i=1}^{k} P_{i},

where its orientation is defined in a similar way as in Section 25.1. See Section 25.7 for the precise definition of the orientation.

Note the moduli space of J_{0}− or J_{1}-holomorphic discs attached to L is contained in M_{k+1}^{main}(L, j : β; para)|_{s=0, 1}. When we choose a multisection of the Kuranishi structure over M_{k+1}^{main}(L, j : β; para) (a multi-valued perturbation of M_{k+1}^{main}(L, j : β; para)) to construct the fundamental chain of M_{k+1}^{main}(L, j : β; para), we will take one which extends the ones with Ξ_{0} or F_{∗}Ξ_{1} already chosen for the parameters s = 0, 1. This is always possible by taking F to be generic in a suitable sense.

Definition 15.25. Let j be as in Definition 15.23. We define a sequence of currents which are defined by

𝔣_{k,β,j}(P_{1}, ⋯, P_{k}) = (M_{1}^{main}(L, j : β; para; P_{1}, ⋯, P_{k}), ev_{0})

if (k, β) ≠ (1, β_{0}), and

𝔣_{1,β_{0},j}(P) = P.

Since these currents 𝔣_{k,β,j}(P_{1}, ⋯, P_{k}), may not be contained in C_{(1)}(L; Λ_{0,nov}) for P_{1}, ⋯, P_{k} representing chains in C_{(0)}(L; Λ_{0,nov}), we have to extend C_{(1)}(L; Λ_{0,nov}) to a bigger complex C_{(2)}(L; Λ_{0,nov}), which includes all such images. Actually to construct the filtered A_{∞} homomorphism asserted in Theorem 15.19, we need to add more generators to C_{(2)}(L; Λ_{0,nov}). We describe this procedure more in detail in §A5 where we also explain the construction of 𝔣_{k,β_{0}}.

Now we have :

Proposition 15.26. The family of maps 𝔣_{k,β,j} (k = 0, 1, 2, ⋯) induce a filtered A_{∞} homomorphism

φ̂: (C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0})) → (C_{(2)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{2})).

Proof. In §A5 (Main Lemma A5.6), we will discuss construction of the family of maps 𝔣_{k,β_{0}} and prove that it defines a weak homotopy equivalence

φ̂_{(0)} : (C_{(0)}(L; ℚ), \overline{𝔪}(J_{0}, Ξ_{0})) → (C_{(2)}(L; ℚ), \overline{𝔪}(J_{1}, Ξ_{2})).

Here we assume that 𝔣_{k,β_{0}} are already constructed and will study 𝔣_{k,β,j} for β ≠ β_{0}.

To show that φ̂ is a filtered A_{∞}-map, we investigate the boundary of M_{1}^{main}(L, j : β; para; P_{1}, ⋯, P_{k}). Let ((Σ(m), z⃗(m)), (u_{α}(m)), (s_{α}(m))) be a sequence converging to the boundary. (We may assume that the combinatorial type of Σ(m) is independent of m.) Then one of the following cases should occur in the limit:

(15.27.1) One of the components Σ(m)_{α} splits into two components. (15.27.2) lim s_{α}(m) = lim s_{α′}(m), where Σ(m)_{α} ∩ Σ(m)_{α′} ≠ ∅. (15.27.3) lim s_{α}(m) = 0. (15.27.4) lim s_{α}(m) = 1. (15.27.5) The images of ev_{j}(1 ≤ j ≤ k) converge to the boundary ∂P_{j} of P_{j}.

In the same way as in the argument of Chapter 2, we can show that (15.27.1) cancels with (15.27.2). The contribution of (15.27.3) and (15.27.5) give φ̂∘d̂^{0}. (15.27.4) gives the non-classical contribution to d̂^{2} ∘φ. Remember that the classical contribution to d̂^{0} ∘ φ̂ comes from the whole boundary of M_{1}^{main}(L, j : β; para; P_{1}, ⋯, P_{k}) itself. So we have φ̂ ∘ d̂^{0} = d̂^{2} ∘ φ̂. The proof of Proposition 15.26 (modulo the part we postponed to §A5) is complete. □

Figure 15.2

Now we are ready to finish the proof of Theorem 15.19. Note that

φ̂_{(0)} : (C_{(0)}(L; ℚ), \overline{𝔪}(J_{0}, Ξ_{0})) → (C_{(2)}(L; ℚ), \overline{𝔪}(J_{1}, Ξ_{2})).

induces an isomorphism on the classical cohomology group by Main Lemma A5.6 in §A5. It seems to be previously known (see [K5] in the L_{∞}-setting) that out of this one can construct a weak homotopy equivalence in the sense of Definition 15.12 which will finish the proof of Theorem 15.19 (up to the extension mentioned in Remark 15.20). Since we cannot easily locate the proof of this fact in the literature, we will provide the precise statement and the proof concerning this point in Theorem 19.8 and in §A5 respectively. □

We now describe an immediate consequence of Theorem 15.19 and Lemma 15.13. We recall Lemma 15.15 and the identity (15.17).

Corollary 15.28. In the situation of Theorem 15.19 there exists a weak homotopy equivalence

φ′ : (C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0})) → (C_{(1)}(L, Λ_{0,nov}), 𝔪(J_{1}, Ξ_{1})).

Corollary 15.29. There exists a map

φ̂_{∗} : M̂(C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0}))) → M̂(C_{(2)}(L, Λ_{0,nov}), 𝔪(J_{1}, Ξ_{2}))

and

φ′_{∗} : M̂(C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0}))) → M̂(C_{(1)}(L, Λ_{0,nov}), 𝔪(J_{1}, Ξ_{1})).

Corollary 15.29 immediately follows from Theorem 15.19, Corollary 15.28 and Proposition 15.18.

We now recall Definition 12.6 where we define M̂(L; J, Ξ) to be the set of b ∈ C^{1}(L; Λ_{0,nov}) such that d̂φ_{b} = 0. (See Remark 13.33 for relevant remarks.) Note that the dependence of d̂ on J and Ξ is suppressed from our notation. In fact it depends also to the choice of countably generated sub-complex C_{(1)}(L; Λ_{0,nov}). Next we are going to define an equivalence relation ∼ on M̂(L; J, Ξ) (Definition 15.37).

Remark 15.30 The equivalence relation ∼ we will define below is related to the definition of the gauge equivalence of two solution of Maurer-Cartan equation discussed in [K5]. (The L_{∞} structure is studied instead of the A_{∞} structure in [K5] but the arguments are parallel.) In [K5], b(0) is said to be gauge equivalent to b(1) in our terminology, if there exist b(s) and c(s) such that

(15.31)  \frac{∂b(s)}{∂s} = ∂_{b(s)}c(s).

However this definition itself does not work in our situation. In fact in order to make sense of (15.31) we need to assume that b(s) is contained in the cochain complex we work with for every s. As a consequence, we need to take a complex with uncountably many generators. This causes a trouble to apply Baire’s category theorem.

It would be better if we can give a definition of ∼ purely algebraically starting with an arbitrary filtered A_{∞} algebra and can apply it to our geometric situation. However the authors do not know how to do it at the time of writing this book. A difficulty lies in taking a countably generated sub-complex appropriately. Already in the proof of Theorem 15.19 we find that it is more natural to extend the countably generated sub-complex we work with to construct an A_{∞} map we need. So it is in fact rather hard to work with a fixed countably generated sub-complex to define ∼. Our definition given below in not purely algebraic. However we believe that this trouble should be a technical one and purely algebraic definition (which is equivalent to one we will give below) will be found in the future.

The definition of ∼ on M̂(L; J, Ξ) is similar to the construction in the proof of Theorem 15.19. More generally, we will define an equivalence between elements of M̂(L; J_{0}, Ξ_{0}) and of M̂(L; J_{1}, Ξ_{1}) where J_{0}, Ξ_{0}, J_{1}, Ξ_{1} are as in Theorem 15.19.

Let us consider the situation of Theorem 15.19. We assume that b_{i}, i = 0, 1 be two bounding cochains. i.e.,

b_{i} = ∑ b_{i,β} ⊗ [β] ∈ C_{(i)}^{1}(L; Λ_{0,nov}), d̂e^{b_{i}} = 0.

Intuitively, the expected condition would be that there exists a family b(s) such that d̂e^{b(s)} = 0 and b(0) = b_{0}, b(1) = F (b_{1}), where F is as in the conclusion of Theorem 5.19. Hereafter we assume F = id for simplicity.

We might want to consider a chain b_{para,β} in L × [0, 1] which satisfies conditions such as

(15.32.1) b_{para.β} is a current realized by a geometric chain in L × [0, 1]. (15.32.2) We put b(s, β) = b_{para,β} ∩ (L × {s}). Then b(s, β) is a chain for every s. (15.32.3) We put b(s) = ∑ b_{s,β} ⊗ [β]. Then d̂e^{b(s)} = 0 for every s. (15.32.4) b(i, β) = b_{i,β}, i = 0, 1.

However this plan meets an essential trouble. Namely, the operator d̂ is in fact defined only for generic chains. As a consequence the condition d̂e^{b(s)} = 0 would make sense only for almost all s but not for every s and so we need to be more careful to state (15.32.3).

For this purpose we first recall that our operations 𝔪_{k} are induced by the correspondence. We make this statement precise in the following general definition.

Definition 15.33. A filtered A_{∞} structure 𝔪_{k,β} on C(L; Λ_{0,nov}) is said to be weakly motivic if there exists a ℚ-chains

M_{k+1}^{main}(β) → L^{k+1}

for each β, k with (β, k) ≠ (β_{0}, 1) such that

𝔪_{k,β}(P_{1}, ⋯, P_{k}) = (M_{k+1}^{main}(β) ×_{L^{k}} (P_{1} × ⋯ × P_{k}), ev_{0}).

We recall that we defined 𝔪_{1,β_{0}} to be the usual boundary operator in Definition 13.18 (and hence not by the correspondence). It is immediate from this definition that the filtered A_{∞} algebra of our main example is weakly motivic. We recall that the operators 𝔪_{k,β} is defined only for the chains P_{1}, ⋯, P_{k} such that the fiber product M_{k+1}^{main}(β) ×_{L^{k}} (P_{1} × ⋯ × P_{k}) is transversal. Such P_{1}, ⋯, P_{k} are generic. We say that P_{1}, ⋯, P_{k} are transverse to 𝔪_{k} in this case.

Now let j = {J_{s}}_{0≤s≤1} be a family connecting J_{0} and J_{1} and let Q_{1}, ⋯, Q_{k} be geometric chains in L×[0, 1]. We put Q_{i}(0) = Q_{i}∩(L×{0}), Q_{i}(1) = Q_{i}∩(L×{1}). We assume that Q_{1}(0), ⋯, Q_{k}(0) and Q_{1}(1), ⋯, Q_{k}(1) are transverse to 𝔪_{k}(J_{0}, Ξ_{0}) and to 𝔪_{k}(J_{1}, Ξ_{1}) respectively, and define 𝔪_{k,β}^{j}(Q_{1}, ⋯, Q_{k}) as follows:

(15.34)  \begin{matrix}𝔪_{k,β}^{j}(Q_{1}, ⋯, Q_{k}) \\ = (\{(s, x)|x ∈ M_{k}^{main}(J_{s}, β) ×_{L^{k}} (Q_{1}(s) × ⋯ × Q_{k}(s))\}, ev_{0}), \end{matrix}

if (β, k) ≠ (β_{0}, 1) and

𝔪_{1,β_{0}}^{j}(Q) = pr_{2}(∂Q) + Q(1) − Q(0).

Note that we cannot assume that the fiber product

(15.35)  M_{k+1}^{main}(J_{t}, β) ×_{L^{k}} (Q_{1}(s) × ⋯ × Q_{k}(s))

is transversal for every s (even if we assume that Q_{i} are generic.) However we can show that (15.34) (that is the union of (15.35) for s ∈ [0, 1]) is well defined as a chain in L × [0, 1]. To be precise we state the following Lemma 15.36 whose proof is a straightforward analogue of the one given in Chapter 5 and so is omitted. Denote

M_{k+1}^{main}(β; para) = ⋃_{0≤s≤1}{s} × M_{k+1}^{main}(J_{s}, β),

and the corresponding evaluation map is denoted by

ev_{i} : M_{k+1}^{main}(β; para) → [0, 1] × L.

Lemma 15.36. The fiber product

M_{k+1}^{main}(β; para) ×_{[0,1]×L^{k}} (Q_{1} × ⋯ × Q_{k})

has an oriented Kuranishi structure.

Note that in the case when some of Q_{i}’s coincide, our perturbation using the Kuranishi structure is essential (see Chapter 5, §19). Let Q_{β} be a family of geometric chain in L × [0, 1], and define

Q = ∑_{β} Q_{β} · [β]

and put

e^{Q} := 1 + Q + Q ⊗ Q + · · · .

For a given j = {J_{s}}_{0≤s≤1}, we choose a system Ξ̃ of multi-valued perturbations of the Kuranishi maps for M_{k+1}^{main}(β)^{para} and we define d̂^{j} in the same way as d̂ using 𝔪_{k,β}^{j} in place of 𝔪_{k,β}

We now define an equivalence relation among the bounding cochains.

Definition 15.37. Let b_{0} ∈ M̂(L; J_{0}, Ξ_{0}) and b_{1} ∈ M̂(L; J_{1}, Ξ_{1}), where J_{0}, Ξ_{0}, J_{1}, Ξ_{1} are as in Theorem 15.19. We say that b_{0} is equivalent to b_{1} and write b_{0} ∼ b_{1}, if there exist j = {J_{s}}, a system Ξ̃ of perturbations and a chain Q on [0, 1] × L such that Q(1) = b_{1}, Q(0) = b_{0} and

d̂^{j}e^{Q} = 0

as a current in [0, 1] × L.

Example 15.38. In the proof of Theorem 15.19, we eventually constructed Q for b_{0} and φ(b_{0}). Namely, we put Q_{β} to be the union of [0, 1] × (b_{0})_{β} and a chain in [0, 1]×L consisting of (s, x) such that x = ev_{0}(u_{α_{0}}) where u_{α_{0}} is the 0-th component of an element in M_{1}^{main}(L, j : β − β_{i}; para; b_{β_{i}}) and s ≥ s(α_{0}).

Lemma 15.39. ∼ is an equivalence relation. Namely b_{0} ∼ b_{1},b_{1} ∼ b_{2} implies b_{0} ∼ b_{2}.

The proof is easy and is omitted.

Remark 15.40. Even in the case b_{0}, b_{1} ∈ M̂(L; J, Ξ) (in other words in the case when J_{0} = J_{1}, Ξ_{0} = Ξ_{1}) we need to allow j = {J_{s}}_{0≤s≤1} to vary and Ξ̃ to depend on s parameter, in the definition of b_{0} ∼ b_{1}.

Theorem 15.41. Let J_{0} and J_{1} be two almost complex structures and 𝔪(J_{0}, Ξ_{0}) and 𝔪(J_{1}, Ξ_{1}) be the corresponding A_{∞} structures. We assume the same hypotheses as Theorem 15.19. Suppose that b_{0} and b_{1} are bounding cochains with respect to 𝔪(J_{0}, Ξ_{0}) and 𝔪(J_{1}, Ξ_{1}) respectively such that b_{0} ∼ b_{1}.

Then there exists a weak homotopy equivalence

{𝔣_{k}} : (C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0})^{b_{0}}) → (C_{(1)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{1})^{b_{1}}),

such that 𝔣_{0} = 0.

Proof. As in Theorem 15.19 and Corollary 15.28, we may extend the cochain complex and apply Lemma 15.13. (Note in Lemma 15.13 if we assume 𝔣_{0}^{1} = 0, then we may take 𝔣_{0}^{2} = 0.)

So it suffices to construct (C_{(3)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{3})) which contains C_{(2)}(L; Λ_{0,nov}) in Theorem 15.19 and

(15.42) {𝔣′_{k}} : (C_{(0)}(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0})^{b_{0}}) → (C_{(3)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{3})^{b_{1}})

such that 𝔣′_{0} = 0. In fact since b_{1} ∈ C_{(1)}(L; Λ_{0,nov}) ⊆ C_{(3)}(L; Λ_{0,nov}), (15.19.2) implies that the inclusion

(C_{(1)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{1})^{b_{1}}) → (C_{(3)}(L; Λ_{0,nov}), 𝔪(J_{1}, Ξ_{3})^{b_{1}})

is an A_{∞} map with 𝔣_{k} = 0 for k ≠ 1. (We recall we assumed F = id for simplicity.)

We start with the construction of (15.42). Let Q be as in Definition 15.37. We are going to construct a coalgebra homomorphism φ̂ : B̂(C_{(0)}[1](L; Λ_{0,nov})) → B̂(C_{(3)}[1](L; Λ_{0,nov})) such that

φ̂ ∘ d̂^{b_{0}} = d̂^{b_{1}} ∘ φ̂

and φ induces an isomorphism between classical cosmology groups. This will imply that φ̂ is a weak homotopy equivalence from (C_{(1)}(L, Λ_{0,nov}), 𝔪^{b_{0}}) to (C_{(3)}(L, Λ_{0,nov}), 𝔪^{b_{1}}). Here we simply denote 𝔪^{b_{i}} for 𝔪(J_{i}, Ξ_{i})^{b_{i}}, i = 0, 1.

Let (Σ, z⃗) be an element of M_{k+ℓ+1}. We divide {1, ⋯, k + ℓ} into a disjoint union I_{1} ∪ I_{2}. For each i ∈ I_{1} we consider the minimal path joining the i-th marked point to the 0-th marked point. We call a component Σ_{α} of Σ a principal component if it intersects one of those minimal paths and Σ_{α} a bubbling component otherwise. For each bubbling component Σ_{α} there exists the unique principal component Σ_{α′} with the following properties.

(15.43.0) u_{α} is J_{s(α)}-holomorphic. (15.43.1) Σ_{α} ≤ Σ_{α′}. (15.43.2) If Σ_{α′′} is a principal component and if Σ_{α} ≤ Σ_{α′′}, then Σ_{α′} ≤ Σ_{α′′}.

We say that Σ_{α} belongs to Σ_{α′} if (15.43) is satisfied. (A principal component belongs to itself.) Let u_{α} : (Σ_{α}, ∂Σ_{α}) → (M, L) be maps and s(α) ∈ [0, 1] for each principal component Σ_{α}. We consider the following conditions.

(15.44.1) If Σ_{α} belongs to Σ_{α′}, then s(α) = s(α′). (15.44.2) If α_{1} ≤ α_{2}, then s(α_{1}) ≤ s(α_{2}). Here α_{1} ≤ α_{2} is as in Definition 15.21. (15.44.3) ((Σ, z⃗), (u_{α})) is stable in the sense of Definition 3.7. (15.44.4) The homology class of the restriction of u_{α} is β(α). We have ∑_{α} β(α) = β. (15.44.5) If i ∈ I_{2}, z_{i} ∈ Σ_{α}, then u_{α}(z_{i}) ∈ Q(s(α)). (15.44.6) If z ∈ Σ_{α_{1}} ∩ Σ_{α_{2}}, then u_{α_{1}}(z) = u_{α_{2}}(z).

Figure 15.3

We identify two such systems if there exists an isomorphism in the obvious sense.

Definition 15.45. We define M_{k+1}^{main}(j : β; Q; I_{1}, I_{2}) to be the set of all equivalence classes of ((Σ, z⃗), (u_{α}), (s(α))) satisfying (15.44).

Let ♯I_{1} = k and take the order preserving isomorphism i : {1, ⋯, k} → I_{1}. This induces a map ev_{I_{1}} : M_{k+1}^{main}(j : β : Q : I_{1}, I_{2}) → L^{k} by

ev_{I_{1}}((Σ, z⃗), (u_{α}), s(α)) = (u_{α_{1}}(z_{i(1)}), ⋯, u_{α_{k}}(z_{i(k)})).

Here z_{i(j)} ∈ Σ_{α_{j}}. We now define the A_{∞}-map associated to Q by

Definition 15.46.

𝔣_{k,β}^{Q} = ∑_{I_{1},I_{2}:♯I_{1}=k} (M_{k+1}^{main}(j : β; Q; I_{1}, I_{2})_{ev_{I_{1}}} × ∏_{i=1}^{k} P_{i}, ev_{0}) .

(k = 1, 2, · · ·.) If β = β_{0} and k = 1, constant maps do not satisfy the stability condition. So we define 𝔣_{1,β_{0}}^{Q}(P) = P, separately.

Theorem 15.41 will follow from the following.

Lemma 15.47. {𝔣_{k,β}^{Q}}_{k≥1} induces a filtered A_{∞} map (C_{(0)}(L; Λ_{0,nov})), 𝔪^{b_{0}}) → (C_{(3)}(L; Λ_{0,nov})), 𝔪^{b_{1}}).

Proof. Denote the coalgebra map induced from 𝔣_{k,β}^{Q} by φ̂. We again investigate the boundary of M_{k+1}^{main}(j : β; Q; I_{1}, I_{2}). Let (Σ(m), z⃗(m)), (u_{α}(m)), (s(α)(m)) be a divergence sequence. Then one of the following occurs.

(15.48.1) One of the components Σ(m)_{α} splits into two components. The two new components are both principal components. (15.48.2) One of the components Σ(m)_{α} splits into two components. At least one of the two new components is a bubbling component. (15.48.3) lim s(α)(m) = s(α′)(m), where Σ(m)_{α} ∩ Σ(m)_{α′} ≠ ∅ and both of Σ(m)_{α} and Σ(m)_{α′} are principal. (15.48.4) lim s(α)(m) = 0. (15.48.5) lim s(α)(m) = 1. (15.48.6) lim u(m)_{α(m)}(z(m)_{i}) ∈ ∂Q(s). Here i ∈ I_{2}, s = lim s(m)(α′(m)), z(m)_{i} ∈ Σ(m)_{α(m)}, Σ(m)_{α(m)} belongs to Σ(m)_{α′(m)}. (15.48.7) lim_{m→∞} u(m)_{α(m)}(z(m)_{i}) ∈ ∂P_{j}. Here i = i(j) ∈ I_{1} and z(m)_{i} ∈ Σ(m)_{α(m)}.

(15.48.1) and (15.48.3) cancel each other in the same way as in the proof of Proposition 15.26. (15.48.2) cancels (15.48.6) because of Definition 15.37. (d̂^{j}e^{Q} = 0.) (15.48.4) + (15.48.7) correspond to φ̂ ∘ d̂^{b_{0}} and (15.48.5) corresponds to nonclassical contribution to d̂^{b_{1}} ∘φ̂. The classical contribution via the usual coboundary operator (Definition 13.18) to d̂^{b_{1}} ∘ φ̂ comes from the total boundary of M_{k+1}^{main}(j : β; Q; I_{1}, I_{2}). The proof of Lemma 15.47 is now complete. □

Theorem 15.41 enables us to define the (algebraic) deformation space of A_{∞} algebras attached to L, which does not depend on the choice of (J, Ξ) and other parameters involved. This is exactly what was alluded in Definition 12.6

15.3. Deformation of the Floer cohomology.

The main purpose of this subsection is to prove Theorem 12.8 and Theorem 12.12, which describe dependence of our deformed cohomology on the choice of bounding chains (i.e. of b) and of Hamiltonian isotopies. This will correspond to describing the deformation of ‘morphims’ in the A_{∞} category while the previous sections of §15 describe the deformation space of the ‘objects’ in the A_{∞} category.

Proof of Theorem 12.8. The proof is similar to the proof of Theorem 15.41. Let b_{0} ∈ M̂(L; J_{0}, Ξ_{0}), b′_{0} ∈ M̂(L′; J′_{0}, Ξ′_{0}), b_{1} ∈ M̂(L; J_{1}, Ξ_{1}), b′_{1} ∈ M̂(L′; J′_{1}, Ξ′_{1}).

Suppose that b_{0} ∼ b_{1}, b′_{0} ∼ b′_{1}. Let J^{0} = {J_{t}^{0}}_{0≤t≤1}, Ξ be the t-dependent family of almost complex structures and a system of perturbation, and Q ⊂ [0, 1] × L be the chain, whose existence is assumed in Definition 15.37 for b_{0} ∼ b_{1} to hold.

Similarly we have J^{1} = {J_{t}^{1}}, Ξ′, and Q′ for b′_{0} ∼ b′_{1}.

Let j = {J^{s}}_{0≤s≤1} = {J(s, t)}_{(s,t)∈[0,1]×[0,1]} so that J(s, t) is independent of s if s is in a neighborhood of 0 or 1. By composing with a monotonically increasing function χ : ℝ → [0, 1] which are constant at infinity, we will rather consider j as the two parameter family {J(τ, t)}_{ℝ×[0,1]} such that J(τ, ) is independent of τ if |τ| is sufficiently large.

We have Q(0) = b_{0}, Q(1) = b_{1}, Q′(0) = b′_{0}, Q′(1) = b′_{1} and d̂^{j_{0}}e^{Q} = d̂^{j_{1}}e^{Q′} = 0 by Definition 15.37.

Let [ℓ_{p}, w], [ℓ_{q}, w′] ∈ Cr(L, L′). Let Σ be ℝ × [0, 1] be as before. We consider (u, z⃗) such that

(15.49.1) u : Σ → M satisfies

\frac{∂u}{∂τ} + J(τ, t, u)\frac{∂u}{∂t} = 0

on ℝ × [0, 1]. If a bubble tree of spheres is attached at (τ, t) then all the spheres in the tree are J(τ, t)-holomorphic. (15.49.2) u(τ, 0) ⊂ L, u(τ, 1) ⊂ L′. (15.49.3) z_{i} = (τ_{i}, 1), i ∈ 2, ⋯, k_{1} + 1 and z_{i} = (τ_{1}, 0), i ∈ k_{1} + 2, ⋯, k_{1} + k_{2} + 1. (15.49.4) τ_{i} < τ_{i+1}, i ∈ 2, ⋯, k_{1} and τ_{i} > τ_{i+1}, i ∈ k_{1} + 2, ⋯, k_{1} + k_{2}. (15.49.5) u(τ_{i}, 0) ∈ Q(τ_{i}), i ∈ 2, ⋯, k_{1} + 1 and u(τ_{i}, 1) ∈ Q′(τ_{i}), i ∈ k_{1} + 2, ⋯, k_{1} + k_{2} + 1. (15.49.6) lim_{τ→−∞} u = p, lim_{τ→+∞} u = q. (15.49.7) w♯u ∼ w′ Let M(L, L′, \overline{j} : Q, Q′; [ℓ_{p}, w], [ℓ_{q}, w′]) be the set of all such pairs (u, z⃗). Then the moduli space M(L, L′, \overline{j}; Q, Q′; [ℓ_{p}, w], [ℓ_{q}, w′]) has an oriented Kuranishi structure of dimension μ([ℓ_{p}, w]) − μ([ℓ_{q}, w′]).

We put

ψ([ℓ_{p}, w]) = ∑_{[ℓ_{q},w′]} ♯M(L, L′, \overline{j} : Q, Q′; [ℓ_{p}, w], [ℓ_{q}, w′])[ℓ_{q}, w′].

We can prove that ψ : (C(L, L′), δ_{b_{1},b_{2}}) → (C(L, L′), δ_{b′_{1},b′_{2}}) is a cochain map in a similar way to the proof of Theorem 15.41.

Moreover, using the gapped condition, we can construct a spectral sequence whose E_{2} terms are

⨁_{p∈L∩L′} Λ_{nov}[p]

and which converges to the cohomology of (C(L, L′), δ_{b_{1},b_{2}}) and of (C(L, L′), δ_{b′_{1},b′_{2}}) respectively. (See §A4.) ψ induces a morphism of the spectral sequence and it induces an isomorphism on the E_{2} terms. It follows that ψ is an isomorphism. We can prove that the isomorphism induced on the cohomology group is independent of the various choices involved, by using the fact that the space parameterizing the choices (for example the space of compatible almost complex structures) is contractible. The argument for it is an obvious one parameter version of the construction of the map ψ and is omitted. The proof of Theorem 12.8 is complete. □

The remaining section will be occupied by the proof of Theorem 12.12. We start with the following

Theorem 15.50. For a family of compatible almost complex structure j = {J_{s}}_{0≤s≤1} and perturbations Ξ̃ = {Ξ_{s}}_{0≤s≤1}, we have a canonical one-to-one correspondence φ_{∗} between M(L; J_{0}, Ξ_{0}) and M(L; J_{1}, Ξ_{1}).

Proof. By Theorem 15.19 and Corollary 15.28, there exists a map

φ̂_{∗} : M̂(L; J_{0}, Ξ_{0}) → M̂(L; J_{1}, Ξ_{1}).

We are going to prove that φ̂_{∗} induces φ_{∗}. We remark that

(15.51)  b ∼ φ̂_{∗}(b)

by Example 15.38.

Let b_{0}, b_{1} ∈ M̂(L, J_{0}, Ξ_{0}) with b_{0} ∼ b_{1}. Then by (15.49) we have

φ̂_{∗}(b_{0}) ∼ b_{0} ∼ b_{1} ∼ φ̂_{∗}(b_{1}).

Hence φ̂_{∗} induces

φ_{∗} : M(L; J_{0}, Ξ_{0}) → M(L; J_{1}, Ξ_{1}).

We now show that φ_{∗} is a bijection. We first prove that φ_{∗} is surjective. We use Ξ̃′ = {Ξ_{1−s}}_{0≤s≤1}, j′ = {J_{1−s}}_{0≤s≤1} in place of Ξ̃ and j to obtain φ′_{∗}. Let b ∈ M(L; J_{1}, Ξ_{1}). We have

φ′_{∗}(b) ∼ b.

Therefore

φ̂_{∗}(φ′_{∗}(b)) ∼ φ′_{∗}(b) ∼ b,

as required.

We next prove injectivity of φ_{∗}. Let b_{0}, b_{1} ∈ M̂(L; J_{0}, Ξ_{0}) with φ̂_{∗}(b_{0}) ∼ φ̂_{∗}(b_{1}). Then we have

b_{0} ∼ φ̂_{∗}(b_{0}) ∼ φ̂_{∗}(b_{1}) ∼ b_{1}.

The proof of Theorem 15.50 is complete by the following lemma. □

Lemma 15.52. The map φ_{∗} : M(L; J_{0}, Ξ_{0}) → M(L; J_{1}, Ξ_{1}), is independent of the choice of a family of compatible almost complex structure j = {J_{s}} and perturbations Ξ̃ = {Ξ_{s}}.

Proof. Let j′, Ξ̃′ be another choice. And let b_{0} ∈ M̂(L; J_{0}, Ξ_{0}).

Since the set of all compatible almost complex structures is contractible, we can find a family \overline{j} = \{j_{u}\}_{0≤u≤1} = \{J(u, s)\}_{(u,s)∈[0,1]×[0,1]} so that j_{0} = j and j_{1} = j′. In other words, we have J(0, s) = J(s), J(1, s) = J′(s). For a generic parameter u, it induces an A_{∞} homomorphism 𝔣_{k}^{u} from (C_{(0)}(L; Λ_{nov}), 𝔪(J_{0}, Ξ_{0})) to (C_{(2,u)}(L; Λ_{nov}), 𝔪(J_{1}, Ξ_{2,u})) by Theorem 15.19.

In fact 𝔣_{k}^{u} is defined by the correspondence by M_{k+1}^{main}(L, j_{u} : β : para). We put

M_{k+1}^{main}(L, \overline{j} : β : parapara) = ⋃_{u}\{u\} × M_{k+1}^{main}(L, j_{u} : β : para).

It has an oriented Kuranishi structure. Note the Kuranishi structure of M_{k+1}^{main}(L, \overline{j} : β : parapara) at u = 0, 1 is compatible with the Kuranishi structure of M_{k+1}^{main}(L, j : β), M_{k+1}^{main}(L, j′ : β). Hence we can extend the multi-sections Ξ̃, Ξ̃′ given at u = 0, 1 to M_{k+1}^{main}(L, ′j : β : parapara). Thus we can define a fundamental chain of M_{k+1}^{main}(L, j : β : parapara) which is compatible with those of M_{k+1}^{main}(L, j : β), M_{k+1}^{main}(L, j′ : β), and which defines a current on [0, 1] × L^{k+1} via the evaluation map. We put

Q_{β} = ∑_{k,β_{0}β_{1},··· ,β_{k}, ; β=∑ b_{i}}(M_{k+1}^{main}(L, \overline{j} : β_{0} : parapara)×(b_{0}(β_{1})_{ev_{1},··· ,ev_{k}} ×⋯×b_{0}(β_{k})), ev_{0}).

It is easy to see that Q = ∑ Q_{β} ⊗ [β] and the family j : u ↦ J(u, 1) (together with the perturbation Ξ′′ induced from the restriction of the perturbation of M_{k+1}^{main}(L, j : β_{0} : parapara) to s = 1) provides the necessary conditions for φ(b_{0}) ∼ φ′(b_{0}) (Recall Definition 15.37). Here φ is obtained from j, Ξ̃ (corresponding to the parameter u = 0) and φ′ is obtained from j′, Ξ̃′ (corresponding to u = 1). □

Theorem 15.50 immediately implies Theorem 12.12 (12.12.1).

Then (12.12.2) follows from the proof of Theorem 12.8 in the following way: Let b_{0} ∈ M̂(L; J_{0}, Ξ_{0}) ⊂ C_{(0)}^{1}(L; Λ_{nov}), b′_{0} ∈ M̂(L′; J_{0}, Ξ′_{0}) ⊂ C_{(0)}^{1}(L; Λ_{nov}). By Theorem 15.19 and Proposition 15.18, we obtain

\begin{aligned}φ_{∗}(b_{0}) & = b_{1} ∈ M̂(L; J_{1}, Ξ_{1}) ⊂ C_{(2)}^{1}(L; Λ_{nov}), \\ φ_{∗}(b′_{0}) & = b′_{1} ∈ M̂(L′; J_{1}, Ξ′_{1}) ⊂ C_{(2)}^{1}(L′; Λ_{nov}). \end{aligned}

where φ and φ′ are filtered A_{∞} homomorphism constructed in Proposition 15.26. Example 15.38 implies b_{0} ∼ b_{1} and b′_{0} ∼ b′_{1}.

We observe that in the proof of Theorem 12.8 we gave above, we in fact proved a result somewhat stronger than we need. The proof can also be applied to the case where b_{1} and b′_{1} are contained only in the enlarged complexes C_{(2)}^{1}(L; Λ_{nov}) and C_{(2)}^{1}(L; Λ_{nov}) respectively. (12.12.2) follows from this observation.

We next prove Theorem 12.13 (12.13.1). In fact, as we remarked in the beginning of this subsection, this statement can easily follow from Theorem 12.12 (12.12.1). Let ψ be a Hamiltonian diffeomorphism, it naturally induces one to one correspondence between M_{k+1}^{main}((L, x), J : β; P_{1}, ⋯, P_{k}) and

M_{k+1}^{main}((ψ(L), ψ(x)), ψ_{∗}J : β; ψ(P_{1}), ⋯, ψ(P_{k}))

as well as multi-valued perturbations Ξ and ψ_{∗}Ξ of Kuranishi maps. Therefore it induces a natural A_{∞} isomorphism

ψ_{∗} : (C(L; Λ_{0,nov}), 𝔪(J, Ξ)) → (C(ϕ(L); Λ_{0,nov}), 𝔪(ψ_{∗}J, ψ_{∗}Ξ))

preserving the filtration. (Actually 𝔣_{k} = 0 for k ≠ 1 for this A_{∞} isomorphism.) We refer to §13 how 𝔪 is defined. Hence we have (by (12.12.1)).

\begin{aligned}M(C(L; Λ_{0,nov}), 𝔪(J, Ξ)) & ≅ M(C(ϕ(L); Λ_{0,nov}), 𝔪(ψ_{∗}J, ψ_{∗}Ξ)) \\ & ≅ M(C(ϕ(L); Λ_{0,nov}), 𝔪(J′, Ξ′)) \end{aligned}

for any J′ and Ξ′. The proof of Theorem 12.13 (12.13.1) is complete.

We now turn to the proof of Theorem 12.13 (12.13.2). Let Ψ = {ψ^{s}}_{0≤s≤1} and Ψ′ = {ψ^{′s}}_{0≤s≤1} be two Hamiltonian isotopies which are constant near s = 0, 1 and ψ^{0} = ψ^{′0} = id. As before composing with a monotonically increasing function χ : ℝ → [0, 1] such that χ(−∞) = 0, χ(∞) = 1 and χ is constant at infinity, we may parameterize the isotopy by ℝ rather than by [0, 1].

Let b_{0} ∈ M̂(C(L; Λ_{0,nov}), 𝔪(J, Ξ)), b′_{0} ∈ M̂(C(L′; Λ_{0,nov}), 𝔪(J′, Ξ′)), from which we have

\begin{aligned}ψ_{∗}^{1}(b_{0}) & ∈ M̂(C(ψ^{1}(L); Λ_{0,nov}), 𝔪(ψ_{∗}^{1}J, ψ_{∗}^{1}Ξ)) \\ ψ_{∗}^{′1}(b′_{0}) & ∈ M̂(C(ψ^{′1}(L′); Λ_{0,nov}), 𝔪(ψ_{∗}^{′1}J′, ψ_{∗}^{′∞}Ξ′)). \end{aligned}

We will define an isomorphism

(15.53) HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}) ≅ HF ((ψ^{1}(L), ψ_{∗}^{1}(b_{0})), (ψ^{′1}(L′), ψ_{∗}^{′1}(b′_{0})); ℓ_{0}^{ΨΨ′}).

using the isotopies Ψ, Ψ′, where ℓ_{0}^{ΨΨ′} is defined as in (6.1). There is also a canonical isotopy between Ω(L, L′) and Ω(ψ(L), ψ′(L′)): Denote

Ψ_{u} = {ψ^{us}}_{0≤s≤1}, Ψ′_{u} = {ψ^{′us}}_{0≤s≤1}

for u ∈ [0, 1] and consider the family of paths defined by

I_{Ψ_{u}Ψ′_{u}}(ℓ)(t) = ℓ^{Ψ_{u}Ψ′_{u}}(t) = ψ^{′ut} ∘ ψ^{(1−t)u}(ℓ(t))

as in (6.1) for each u ∈ [0, 1].

As usual, we will construct a chain map that induces the isomorphism (15.53). We first choose a t-dependent family J^{0} = {J_{t}^{0}}_{0≤t≤1} connecting J and J′. This defines the boundary map that induces the Floer homology HF ((L, b_{0}), (L′, b′_{0}); ℓ_{0}) for the pair ((L, x), (L′, x′)). Similarly we connect ψ_{∗}^{1}J and ψ_{∗}^{′1}J′ by another family J^{1} = {J_{t}^{1}} which defines the boundary map inducing the Floer cohomology

HF ((ψ^{1}(L), ψ_{∗}^{1}(b_{0})), (ψ^{′1}(L′), ψ_{∗}^{′1}(b′_{0})); ℓ_{0}^{ΨΨ′})

for the pair (ψ^{1}(L), ψ^{′1}(L′)). Now we consider (s, t)-dependent family j = {J^{s}} = {J(s, t)} for (s, t) ∈ [0, 1]^{2} of almost complex structures which satisfy

(15.54.1) J(0, t) = J_{t}^{0}, J(1, t) = J_{t}^{1}. In particular, we have

J(0, 0) = J, J(0, 1) = J′; J(1, 0) = ψ_{∗}^{1}J, J(1, 1) = ψ_{∗}^{′1}J′.

(15.54.2) J(s, 0) = ψ_{∗}^{s}J, J(s, 1) = ψ_{∗}^{′s}J′. (15.54.3) J(s, t) is independent of s when s is sufficiently close to 0 or 1.

Again using a monotonically increasing function χ as before, we reparameterize the isotopies ψ^{s}, ψ^{′s} and J(s, t) using τ ∈ ℝ instead of s ∈ [0, 1].

Now we consider the map u : Σ → M and z⃗ with the following properties where Σ is [0, 1] × ℝ plus sphere components as before.

(15.55.1) On [0, 1] × ℝ, u satisfies

\frac{∂u}{∂τ} + J(χ(τ), t, u)\frac{∂u}{∂t} = 0.

On the sphere bubbles u satisfies a similar condition as (15.50.2). (15.55.2) u(τ, 0) ⊂ ψ^{χ(τ)}(L), u(τ, 1) ⊂ ψ^{′χ(τ)}(L′). (15.55.3) z_{i} = (τ_{i}, 1), i ∈ 2, ⋯, k_{1} + 1 and z_{i} = (τ_{1}, 0), i ∈ k_{1} + 2, ⋯, k_{1} + k_{2} + 1. (15.55.4) τ_{i} < τ_{i+1}, i ∈ 2, ⋯, k_{1} and τ_{i} > τ_{i+1}, i ∈ k_{1} + 2, ⋯, k_{1} + k_{2}. (15.55.5) u(τ_{i}, 0) ∈ ψ^{χ(τ_{i})}b_{1}, i ∈ 2, ⋯, k_{1} + 1 and u(τ_{i}, 1) ∈ ψ^{′χ(τ_{i})}b_{2}, i ∈ k_{1} + 2, ⋯, k_{1} + k_{2} + 1. (15.55.6) lim_{τ→−∞} u(τ) = p, lim_{τ→+∞} u(τ) = q. (15.55.7) w♯((I_{Ψ_{χ}Ψ′_{χ}})^{−1}u) ∼ (I_{ΨΨ′})^{−1}w′

We need to explain the last condition (15.55.7). We should first note that unlike the case of defining the boundary maps where the boundary Lagrangians fixed, u satisfies a moving boundary condition. Furthermore w and w′ satisfy different boundary conditions, which are connected by Ψ and Ψ′. The map

I_{ΨΨ′} : Ω(L, L′) → Ω(ψ^{1}(L), ψ^{′1}(L′))

was defined by

I_{ΨΨ′}ℓ(t) = (ψ^{′t} ∘ ψ^{(1−t)})ℓ(t)

in (6.1). We define the map I_{Ψ_{χ}Ψ′_{χ}} by

I_{Ψ_{χ}Ψ′_{χ}}u(τ, t) = (ψ^{′χ(τ)t} ∘ ψ^{χ(τ)(1−t)})u(τ, t)

which transforms u satisfying the boundary condition with (L, L′) to the one satisfying the moving boundary condition. Therefore both terms in (15.55.7) satisfy the boundary conditions on L at t = 0 and on L′ at t = 1. And it is easy to see that

(I_{ΨΨ′})^{−1}w′(0, t) = ℓ_{0}(t)

since w′(0, t) = ℓ_{0}^{ΨΨ′}(t) = (I_{ΨΨ′}ℓ_{0})(t) by definition, and

(I_{ΨΨ′})^{−1}w′(1, t) = (I_{ΨΨ′})^{−1}ℓ_{q}(t) = (I_{ΨΨ′})^{−1}(u(∞))(t)

by the asymptotic condition of u at τ = ∞. Therefore the equivalence relation asked in (15.55.7) makes sense.

Now we denote by

(15.56)  M((L, Ψ), (L′, Ψ′); ℓ_{0}; j; (ℓ_{p}, w), (ℓ_{q}, w′))

the moduli space of solutions of (15.55) and define the map

(Ψ, Ψ′)_{∗} : (C(L, L′; ℓ_{0}), δ_{b_{0},b′_{0}}) → (C(ψ^{1}(L), ψ^{′1}(L′); ℓ_{0}^{ΨΨ′}), δ_{(ψ^{1})_{∗}(b_{0}),(ψ^{′1})_{∗}(b_{1})})

as (15.57)

(Ψ, Ψ′)_{∗}([ℓ_{p}, w]) = ∑_{[ℓ_{q},w′]} #M((L, Ψ), (L′, Ψ′); ℓ_{0}; j; (ℓ_{p}, w), (ℓ_{q}, w′))[ℓ_{1}, w′].

We can get a cochain homomorphism from the moduli space of (u, z⃗) satisfying (15.55) as in the proof of Proposition 5.8, as long as we make sure that the map (Ψ, Ψ′)_{∗} satisfies the analog of the finiteness condition (5.9). This is not automatic unlike the case where the boundary Lagrangian submanifolds are fixed. What we need to ensure is the following energy estimate. Similar estimate was carried out by the second named author in [Oh2-I] but not in an optimal way.

Proposition 15.58. Let Ψ and Ψ′ be as above. Let [ℓ_{p}, w] ∈ Ω̃_{ℓ_{0}}(L, L′) and [ℓ_{q}, w′] ∈ Ω_{ℓ′_{0}}(ψ^{1}(L), ψ^{′1}(L′)) such that the moduli space (15.56) is non-empty. Then there exists a constant C = C(Ψ, Ψ′) depending only on the isotopies Ψ and Ψ′ (with ψ(L) and ψ′(L′) fixed), but independent of [ℓ_{p}, w], [ℓ_{q}, w′] or u ∈ M([ℓ_{p}, w], [ℓ_{q}, w′]; β) such that

(15.59.1)  0 ≤ ∫ u^{∗}ω ≤ A_{ℓ_{0}^{ΨΨ′}} ([ℓ_{q}, w′]) − A_{ℓ_{0}}([ℓ_{p}, w]) + C.

In particular we have

(15.59.2)  A_{ℓ_{0}^{ΨΨ′}} ([ℓ_{q}, w′]) − A_{ℓ_{0}}([ℓ_{p}, w]) ≥ −C = −C(Ψ, Ψ′).

Proof. First we recall Proposition 6.2 or its proof (see (6.6.1)) from which we derive

(15.60)  A_{ℓ_{0}}([(I_{ΨΨ′})^{−1}ℓ_{q}, (I_{ΨΨ′})^{−1}w′]) = A_{ℓ_{0}^{ΨΨ′}} ([ℓ_{q}, w′]).

Therefore it follows from (15.55.7) and (15.60) that (15.61)

A_{ℓ_{0}^{ΨΨ′}} ([ℓ_{q}, w′]) − A_{ℓ_{0}}([ℓ_{p}, w]) = A_{ℓ_{0}}([(I_{ΨΨ′})^{−1}ℓ_{q}, (I_{ΨΨ′})^{−1}w′]) − A_{ℓ_{0}}([ℓ_{p}, w])
= A_{ℓ_{0}}([(I_{ΨΨ′})^{−1}ℓ_{q}, (I_{Ψ_{χ}Ψ′_{χ}})^{−1}u)]) = ∫_{(I_{ΨχΨ′_{χ}})^{−1}u} ω

It remains to estimate

∫_{(I_{ΨχΨ′_{χ}})^{−1}u} ω = ∫ u^{∗}ω

140 FUKAYA, OH, OHTA, ONO

where for the simplicity of notation, we denote

ũ(τ, t) := (I_{Ψ_{χ}Ψ′_{χ}})^{−1}u(τ, t) = (ψ^{′χ(τ)t} ∘ ψ^{χ(τ)(1−t)})^{−1}(u(τ, t)) =: ψ_{χ(τ),t}(u(τ, t)). Here the diffeomorphisms ψ_{s,t} are defined by

ψ_{s,t} = (ψ^{′st} ∘ ψ^{(1−s)t})^{−1}.

We compute (15.62)

\begin{aligned}∫_{(I_{ΨχΨ′_{χ}})^{−1}u} ω & = ∫_{−∞}^{∞} ∫_{0}^{1} ω(\frac{∂ũ}{∂τ}, \frac{∂ũ}{∂t}) dt dτ \\ & = ∫_{−∞}^{∞} ∫_{0}^{1} ω(\frac{∂}{∂τ} (ψ_{χ(τ),t}(u(τ, t))), \frac{∂}{∂t}(ψ_{χ(τ),t}(u(τ, t))) dt dτ, \end{aligned}

and

(15.63)  \begin{aligned}\frac{∂ũ}{∂τ} = \frac{∂}{∂τ} (ψ_{χ(τ),t}(u(τ, t))) & = χ′(τ)X_{K_{χ(τ),t}}(ũ) + T ψ_{χ(τ),t}(\frac{∂u}{∂τ}(τ, t)) \\ \frac{∂ũ}{∂t} = \frac{∂}{∂t}(ψ_{χ(τ),t}(u(τ, t))) & = X_{H_{χ(τ),t}}(ũ) + T ψ_{χ(τ),t}(\frac{∂u}{∂t}(τ, t)) \end{aligned}

where K_{s,t} and H_{s,t} are the Hamiltonians generating the vector fields \frac{∂ψ_{s,t}}{∂s} ∘ ψ_{s,t}^{−1} \frac{∂ψ_{s,t}}{∂t} ∘ ψ_{s,t}^{−1} respectively. Substituting these into (15.61), if we carry out the calculation as in Section 6.1, we have

(15.64)  \begin{aligned}∫ u^{∗}ω & = ∫_{−∞}^{∞} ∫_{0}^{1} ω(χ′(τ)X_{K_{χ(τ),t}}(ũ) + T ψ_{χ(τ),t} \frac{∂u}{∂t}(τ, t), \\ & X_{H_{χ(τ),t}}(ũ) + T ψ_{χ(τ),t}(\frac{∂u}{∂t}(τ, t)) \\ & = ∫ ω(\frac{∂ũ}{∂τ}, X_{H_{χ(τ),t}}(ũ)) + ∫ ω(χ′(τ)X_{K_{χ(τ),t}}(ũ), \frac{∂ũ}{∂t}) \\ & − ∫ ω(χ′(τ)X_{K_{χ(τ),t}}(ũ), X_{H_{χ(τ),t}}(ũ)) + ∫ u^{∗}ω \end{aligned}

Here we used (15.63) and the identity

(a + b)(c + d) = (a + b)d + b(c + d) − bd + ac

and that ψ_{s,t} are symplectic.

Now we compute the first three terms in the right hand side of (15.64). With the definition of Hamiltonian vector fields, a straight-forward computation yields (15.65)

∫ ω(\frac{∂ũ}{∂τ}, X_{H_{χ(τ),t}}(ũ)) = − ∫_{0}^{1}(H_{1,t}(ũ(∞)) − H_{0,t}(ũ(−∞))) dt

+ ∫_{−∞}^{∞} ∫_{0}^{1} \frac{∂H_{χ(τ),t}}{∂τ} (ũ)

≥ − ∫_{0}^{1}(max_{x} H_{1,t} − min H_{0,t}) dt + ∫_{−∞}^{∞} χ′(τ) ∫_{0}^{1} \frac{∂H_{s,t}}{∂s} |_{s=χ(τ)}(ũ)

Similarly we have

(15.66) \begin{aligned}∫_{−∞}^{∞} ∫_{0}^{1} ω(χ′(τ)X_{K_{χ(τ),t}}(ũ), \frac{∂ũ}{∂t}) ≥ & − ∫_{0}^{1}(max_{x} K_{s,0} − min_{x} K_{s,1}) ds \\ & − ∫_{−∞}^{∞} χ′(τ) ∫_{0}^{1} \frac{∂K_{χ(τ),t}}{∂t} (ũ) dt dτ. \end{aligned}

Finally we have (15.67)

− ∫ ω(χ′(τ)X_{K_{χ(τ),t}}(ũ), X_{H_{χ(τ),t}}(ũ)) = ∫_{−∞}^{∞} χ′(τ) ∫_{0}^{1}{H_{χ(τ),t}, K_{χ(τ),t}} dt dτ

where {·, ·} is the Poisson bracket

{H, K} := ω(X_{H}, X_{K}).

Since H_{s,t}, K_{s,t} depend only on Φ, Ψ and from the expression of (15.65)-(15.67), it is easy to see that we can bound these integrals by a constant C = C(Φ, Ψ). Now (15.58.1) immediately follows from this and (15.61) and (15.64). This finishes the proof. □

Remark 15.68. In fact, we can find the constant C(Ψ, Ψ′) quite explicitly in terms of the isotopy Ψ and Ψ′, if we carry out a more ingenious calculation: Let H and H′ be the Hamiltonians generating the isotopy Ψ and Ψ′ respectively. If we denote by ∥Ψ∥ the Hofer’s length of the isotopy, i.e.,

∥Ψ∥ := ∫_{0}^{1}(max_{x} H − min_{x} H) dt,

it turns out that we can take the constant C(Ψ, Ψ′) in (15.58) to be

C(Ψ, Ψ′) = −(∥Ψ∥ + ∥Ψ′∥).

This is the optimal form of the estimate done in [Section 5.2,Oh2-I]. Since we do not need this detailed estimate in this book, we omit the detail.

Once we have the energy estimate in Proposition 15.58, it follows by the same argument as in the proof of Preposition 5.8 that the map (15.57) defines a cochain homomorphism over the Novikov ring Λ_{nov}, such that the induced homomorphism on the Floer cohomology is independent of various choices involved other than the Hamiltonian isotopy.

By considering the parameterized version of (15.55), we also obtain a chain homotopy when we deform Hamiltonian isotopies by a homotopy with the end points fixed. This proves that the homomorphism on the cohomology induced from (15.57) is independent of the choice of the Hamiltonian isotopies Ψ and Ψ′ as long as the homotopies are homotopic with end points fixed to ψ and ψ′ respectively. (In fact, only the isotopy of Lagrangian submanifolds {ψ^{t}(L)} and {ψ^{′t}(L)} is what matters.)

A cochain homomorphism in the opposite direction is also constructed by considering the Hamiltonian isotopy \overline{Ψ} and \overline{Ψ}′ in a similar way, where \overline{Ψ} = \{(ψ^{s})^{−1}\} and \overline{Ψ}′ = \{(ψ^{′s})^{−1}\}.

Using the standard gluing argument, we can prove that a composition of the Hamiltonian isotopies induce a composition of the cochain homomorphism. The proof of (12.13.4) is now complete. □

Remark 15.69. We recall the way how we defined μ([ℓ_{p}, w]) in §2: The grading of the Floer complex depends on the choice of the based path ℓ_{0} and the choice of a section λ_{0} of ℓ_{0}^{∗}ΛM. Furthermore when the Hamiltonian isotopies Ψ and Ψ′ are given, we can keep track of the based path ℓ_{0} as in §6.1 and the sections λ_{0} over ℓ_{0} in an obvious way. Under this canonical choice induced from the isotopies Ψ and Ψ′, the isomorphism (15.53) preserves the grading.

§16. Clean intersections and Bott-Morse theory

In this section, we are going to generalize Theorem 12.12 and Theorem 12.13 to the case of clean intersections (the Bott-Morse analogue of Lagrangian intersection Floer theory), which implies Theorem 6.11. The study of Floer cohomology in the Bott-Morse setting is initiated in [Fu3] in the context of gauge theory Floer cohomology. It is applied to the Floer cohomology of periodic Hamiltonian system in [PSS], [RuT] and to Lagrangian intersection in [Pz]. We start with the following situation. Let L, L′ be a pair of Lagrangian submanifolds. We do not assume that they intersect transversely but instead assume the following.

Assumption 16.1. We assume that L and L′ cleanly: Let L ∩ L′ = ∪R_{h} be the decomposition into the connected components of L ∩ L′. We assume that R_{h} is a smooth submanifold and N_{R_{h}}L⊕N_{R_{h}}L′⊕T R_{h} = (T L+T L′)|_{R_{h}} for each component R_{h}, where N_{R_{h}}L and N_{R_{h}}L′ are normal bundles of R_{h} in L and L′, respectively.

The case that appeared in §6 is for L = L′. The case of Theorem 12.12 is the one when L ∩ L′ is 0 dimensional. Let d_{h} be the dimension of R_{h}. We first combine the construction of §6 and §14 and define a left (C(L; Λ_{0,nov}), 𝔪(J_{0}, Ξ_{0})) and right (C(L′; Λ_{0,nov}), 𝔪′(J_{1}, Ξ_{1})) filtered A_{∞} bimodule C(L, L′; Λ_{nov}).

In case when L ∩ L′ is disconnected we need to fix some more data to each component. We first fix a base point p_{h} on each R_{h}. We consider the map w : [0, 1] × [0, 1] → M such that

(16.2.1) w(0, t) = ℓ_{0}(t) (16.2.2) w(τ, 0) ∈ L, w(τ, 1) ∈ L′ for all 0 ≤ τ ≤ 1. (16.2.2) w(1, t) ≡ p_{h}.

Here ℓ_{0} is a base point which we fixed for each connected component of Ω(L, L′). (See §2.1.) We define an equivalence relation on w’s by Definition 2.5. Let π̂_{0}(L∩L′) be the set of all equivalence classes of pairs [h, w].

Let us put

π̂_{0}(L ∩ L′)_{h_{0}} = {[h, w] ∈ π̂_{0}(L ∩ L′)|h = h_{0}}.

Let

Λ_{h_{0}} = { ∑_{[h,w_{i}]∈π̂_{0}(L∩L′)_{h_{0}}} a_{i}[h, w_{i}]| ∀C, ♯ {a_{i} ≠ 0 |∫_{w_{i}} ω < C} < ∞} .

We can construct a homomorphism map Λ_{h_{0}} → Λ_{nov} using the energy and the Maslov index in a way similar to Lemma 6.25.

In the Bott-Morse theory in finite dimension, we should take local systems on critical submanifolds into account. The local system used there is the orientation bundle of unstable bundles. In our case, similar data are necessary. If L′ is considered as a cross section of the cotangent bundle of L, then the same local system as in the Bott-Morse case is what we need. In general, we consider the local system in the following way: Denote

V_{h} = {(T L + T L′)/(T L + T L′)^{⊥_{ω}}}|_{R_{h}},

a vector bundle on R_{h}. Denote by U and U′ the image of T L and T L′ in V_{h}. We consider a path γ_{t} of Lagrangian subspaces in V_{h} from U to U′ so that the path γ_{t} ⊕ T R_{h} is a path of oriented Lagrangian subspaces in T M. The we consider the index of the Dolbeault operator on the half disc with Lagrangian boundary condition, which we will discuss more in Chapter 6. In this way, we get a family of operators on the space of paths γ_{t}. The orientation bundle of the family index of these operators descends to a local system on R_{h}, which we denote by Θ_{R_{h}}.

We will define the Maslov index μ[h, w] ∈ ℤ for [h, w] ∈ π̂_{0}(L ∩ L′). Let \overline{S}^{∗−μ[h,w]}(R_{h}; Θ_{R_{h}}) be a cochain complex of Θ_{R_{h}} valued currents realized by the singular chain with coefficients in the local system Θ_{R_{h}}. (Its element is a distribution section of Λ^{∗}⊗Θ_{R_{h}}.) We take a countably generated sub-complex C^{∗−μ[h,w]}(R_{h}; Θ_{R_{h}}) of it. (We assume that it is independent of w up to degree.) We consider

( ⨁_{[h,w]∈π̂_{0}(L∩L′)}(C^{∗−μ[h,w]}(R_{h}; Θ_{R_{h}}) ⊗ ℚ · [h, w])) ⊗_{Λ_{h}} Λ_{nov},

and let C^{∗}(L, L′; Λ_{nov}) be its completion. Here we use the energy

E([h, w]) = ∫_{w} ω

to define a filtration on Λ_{h} and on ⨁_{[h,w]∈π̂_{0}(L∩L′)}(C^{∗−μ[h,w]}(R_{h}; Θ_{R_{h}}) ⊗ ℚ · [h, w]). In other words, C^{∗}(L, L′; Λ_{nov}) is a completion of

(16.3)  ⨁_{[h,w]∈π̂_{0}(L∩L′)}(C^{∗}(R_{h}; Θ_{R_{h}}) ⊗ Λ_{h}) ⊗_{Λ_{h}} Λ_{nov}.

For a singular chain S over the local system Θ_{R_{h}} (belonging to the countable set we have chosen), we obtain an element [S, w] in C^{∗}(L, L′; Λ_{nov}).

Let J = {J_{t}}_{0≤t≤1} be a one parameter family connecting J_{0} and J_{1} and S be a geometric chain in R_{h}, and P_{1,1}, ⋯, P_{1,k_{1}} ∈ C(L), P_{2,1}, ⋯, P_{2,k_{2}} ∈ C(L′).

We now define a moduli space of pseudo-holomorphic map. Let [h, w], [h′, w′] ∈ π̂(L ∩ L′) and let Σ be ℝ × [0, 1] plus sphere components as before. Let u : Σ → M, z_{i} ∈ ∂Σ. We consider the following conditions for them.

(16.4.1) u satisfies

\frac{∂u}{∂τ} + J(t, u)\frac{∂u}{∂t} = 0,

on Σ and u satisfies a condition similar to (15.50.2) on sphere bubbles. (16.4.2) u(ℝ × {0}) ⊂ L, u(ℝ × {1}) ⊂ L′. (16.4.3) z_{i} = (τ_{i}, 0), for i = 2, ⋯, k_{1}+1. z_{i} = (τ_{i}, 1), for i = k_{1}+2, ⋯, k_{1}+k_{2}+1. (16.4.4) τ_{i} < τ_{i+1} for i ∈ 2, ⋯, k_{1} and τ_{i} > τ_{i+1} for i ∈ k_{1} + 2, ⋯, k_{1} + k_{2}. (16.4.5) u(z_{i}) ∈ P_{1,i} for i = 2, ⋯, k_{1} + 1. (16.4.6) u(z_{i}) ∈ P_{2,i−k_{1}} for i = k_{1} + 2, ⋯, k_{1} + k_{2} + 1. (16.4.7) u converges to a point in S as τ → −∞ and u converges to a point in R_{h′} as τ → +∞. (16.4.8) w♯u(Σ) ∼ w′. Here ♯ is an obvious concatenation.

We consider the obvious action of τ-translations and denote by ∼ the corresponding equivalence relation.

Definition 16.5. We define M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}) to be the totality of the ∼ equivalence classes of u satisfying (16.4).

Let g_{1,i}, g_{2,i} be the codimension of P_{1,i}, P_{2,i} respectively.

Proposition 16.6. Each irreducible component of

M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}})

has a Kuranishi structure.

Note that M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}) may not be orientable in the sense of Kuranishi structure. The signs in the A_{∞}-structure and for the boundary operators in the Floer theory are determined by the relation of the orientation bundle of M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}) as well as local systems {Θ_{h}}. More precisely we have the following :

Lemma 16.7. The Kuranishi structure of

M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}})

has a tangent bundle. Let U_{p} = V_{p}/Γ_{p} and E_{p} be as in the definition of Kuranishi structure in §A2 of M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}). Then there exists an isomorphism

det(T U_{p}) ⊗ det(E_{p}) ≅ ev^{∗}Θ_{h′}

which is compatible with the isomorphism

det(T U_{p})|_{I_{pq}} ⊗ det(E_{p})|_{I_{pq}} ≅ det(T U_{q}) ⊗ det(E_{q})

for q ∈ ψ_{p}(s_{p}^{−1}(0)) obtained from the existence of a tangent bundle.

The proof of the next lemma is now standard and omitted.

Lemma 16.8. We can choose μ[h, w] so that the (virtual)dimension of M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}) in the sense of Kuranishi structure is

dim S + d_{h} + μ[h, w] − μ[h′, w′] − ∑_{i=1}^{k_{1}} codim P_{1,i} − ∑_{i=1}^{k_{2}} codim P_{2,i} + k_{1} + k_{2},

if (k_{1}, k_{2}) ≠ (0, 0). If (k_{1}, k_{2}) = (0, 0) then the dimension of M(L, L′, j; [S, w], [h′, w′]) in the sense of Kuranishi structure is

dim S + d_{h} + μ[h, w] − μ[h′, w′] − 1.

We now consider the map

ev : M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}) → R_{h′}.

defined by

ev(u, z⃗) = lim_{τ→∞} u(τ, t).

After compactifing M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}), we obtain a chain

(M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}), ev)

in R_{h′} with Θ_{h′} coefficient.

Definition 16.9. We define 𝔫_{k_{1},k_{2}} by

\begin{matrix}𝔫_{k_{1},k_{2}}(P_{1,1} ⊗ ⋯ ⊗ P_{1,k_{1}} ⊗ [S, w] ⊗ P_{2,1} ⊗ ⋯ ⊗ P_{2,k_{2}}) \\ = ∑_{h′,w′}[(M(L, L′, j; [S, w], [h′, w′]; P_{1,1}, ⋯, P_{1,k_{1}}; P_{2,1}, ⋯, P_{2,k_{2}}), ev), w′]. \end{matrix}

Proposition 16.10. 𝔫_{k_{1},k_{2}} defines a left (C(L, Λ_{0,nov}), 𝔪) and right (C(L′, Λ_{0,nov}), 𝔪′) filtered A_{∞} bimodule structure on C(L, L′; Λ_{nov}).

The proof is a straight-forward generalization of the ones in §6 and §14 and hence is omitted.

Let b_{0} = ∑ B(L; β_{i})[β_{i}], b′_{0} = ∑ B(L′; β_{i})[β_{i}] be bounding cochains of (C(L; Λ_{0,nov}), 𝔪) and (C(L′; Λ_{0,nov}), 𝔪′) respectively. Then by Proposition 16.10 we obtain a cochain complex

(16.11)  (C(L, L′; Λ_{nov}), δ_{b_{0},b′_{0}})

by the same way as in §14. This cochain complex (16.11) coincides with the one defined in §5 in case L is transverse to L′, and with the one defined in §6 in case L = L′.

Now we are going to state a generalization of Theorem 12.12 (12.13.4), which we prove in this section. Let Ψ = {ψ^{s}}_{0≤s≤1} and Ψ′ = {ψ^{′s}} be two Hamiltonian isotopies such that they are constant near s = 0, 1. We assume ψ^{0} = ψ^{′0} = id and that the pairs (L, L′) and (ψ^{1}(L), ψ^{′1}(L′)) satisfy Assumption 16.1.

Let b_{0} ∈ M̂(L; J_{0}, Ξ_{0}), b′_{0} ∈ M̂(L′; J_{1}, Ξ_{1}).

Let J^{0} = {J(0, t)} be a complex structure such that J(0, 0) = J_{0}, J(0, 1) = J_{1}. We find also a family of almost complex structure J^{1} = {J(1, t)} such that

J(1, 0) = ψ_{∗}^{1}J_{0}, J(1, 1) = ψ_{∗}^{′1}J_{1}.

We fix a countably generated sub-complexes C_{(0)}(L, L′; Λ_{nov}) and C_{(1)}(ψ^{1}(L), ψ^{′1}(L′)); Λ_{nov}) of the cochain complexes of currents on L∩L′ and on ψ^{1}(L)∩ψ^{1′}(L′). To specify the dependence of Floer cohomology to almost complex structure J and perturbation Ξ we write

\begin{matrix}(C_{(0)}(L, L′; Λ_{nov}), δ_{b_{0},b′_{0}}; J^{0}, Ξ^{0}). \\ (C_{(1)}(ψ^{1}(L), ψ^{′1}(L′)); Λ_{nov}), δ_{ψ_{∗}^{1}(b_{0}),ψ_{∗}^{′1}(b′_{0})}; J^{1}, Ξ^{1}) \end{matrix}

in place of (16.11).

Theorem 16.12. There exist C_{(2)}(ψ^{1}(L), ψ^{′1}(L′); Λ_{nov}) and a chain map

\begin{matrix}(Ψ, Ψ′)_{∗} : (C_{(0)}(L, L′; & Λ_{nov}), δ_{b_{0},b′_{0}}; J^{0}, Ξ^{0}) \\ → (C_{(2)}(ψ(L), ψ^{′1}(L′); Λ_{nov}), δ_{ψ_{∗}^{1}(b_{0}),ψ_{∗}^{′1}(b′_{0})}; J^{1}, Ξ^{1}), \end{matrix}

with the following properties.

(16.13.1) (Ψ, Ψ′)_{∗} preserves filtration. (16.13.2) (Ψ, Ψ′)_{∗} induces an isomorphism on cohomology. (16.13.3) The natural inclusion

\begin{matrix}(C_{(1)}(ψ^{1}(L), ψ^{′1}(L′) & ; Λ_{nov}), δ_{ψ_{∗}^{1}(b_{0}),ψ_{∗}^{′1}(b′_{0})}; J^{1}, Ξ^{1}) \\ → (C_{(2)}(ψ^{1}(L), ψ^{′1}(L′); Λ_{nov}), δ_{ψ_{∗}^{1}(b_{0}),ψ_{∗}^{′1}(b′_{0})}; J^{1}, Ξ^{1}), \end{matrix}

is a chain map and induces an isomorphism on cohomology. (16.13.4) The chain homotopy type of (Ψ, Ψ′)_{∗} depends only on the homotopy class of the Hamiltonian isotopies Ψ between id and ψ^{1} and Ψ′ between id and ψ^{′1}. Namely it is independent of the choice of particular (Ψ, Ψ′)_{∗} in its homotopy class and of other choices involved in the definition.

Before the proof of Theorem 16.12, we state the following corollary.

Corollary 16.14. The cohomology of (16.11) depends only on the ∼ equivalence classes of b_{0}, b′_{0} and is independent of J the almost complex structure, Ξ the perturbation, the particular representative b_{0}, b′_{0} in its ∼ equivalence classes, and the countably generated sub-complex of \overline{S}(L ∩ L′; ℚ).

Proof of Theorem 16.13 ⇒ Corollary 16.14. We can find Hamilton isotopies Ψ = {ψ^{s}}_{0≤s≤1}, Ψ′ = {ψ^{′s}}_{0≤s≤1} such that ψ^{1}(L) is transverse to ψ^{′1}(L′). Then we apply Theorem 12.12 and Theorem 12.8 to the pair ψ^{1}(L), ψ^{′1}(L′). Corollary 16.14 then follows from Theorem 16.12. □

We remark also Theorem 6.11 follows immediately from Theorem 16.12, by considering the case L = L′.

Proof of Theorem 16.12. The proof is similar to the proof of (12.13.4) given at the end of the last section. We first choose a (s, t)-dependent family of almost complex structure j = {J^{s}}_{s∈[0,1]} = {J(s, t)}_{(s,t)∈[0,1]^{2}} satisfying (15.54). Again reparameterizing s ∈ [0, 1] by τ ∈ ℝ as before, we may assume the family j = {J^{s}} and the isotopies Ψ, Ψ′ are defined over ℝ.

We consider (Σ, u, z⃗) where Σ is as before and assume that u satisfies the following conditions:

(16.15.1)  \frac{∂u}{∂t}(τ, t) + J(χ(τ), t, u)\frac{∂u}{∂s}(τ, t) = 0,

on ℝ × [0, 1] and a condition similar to (15.50.2) is satisfied on the sphere components. (16.15.2) u(τ, 0) ⊂ ψ^{χ(τ)}(L), u(τ, 1) ⊂ ψ^{′χ(τ)}(L′). (16.15.3) z_{i} ∈ ℝ×{0} for i = 2, ⋯, k_{1}+1 and z_{i} ∈ ℝ×{1} for i = k_{1}+2, ⋯, k_{1}+ k_{2} + 1. (16.15.4) τ_{i} < τ_{i+1}, i ∈ 2, ⋯, k_{1} and τ_{i} > τ_{i+1}, i ∈ k_{1} + 2, ⋯, k_{1} + k_{2}. (16.15.5) u(τ_{i}, 0) ∈ ψ^{χ(τ_{i})}b_{1}, i ∈ 2, ⋯, k_{1} + 1 and u(τ_{i}, 1) ∈ ψ^{′χ(τ_{i})}b_{2}, i ∈ k_{1} + 2, ⋯, k_{1} + k_{2} + 1. (16.15.6) u converges to a point in R_{h} as τ → −∞, and u converges to a point in R_{h′} as τ → ∞. (16.15.7) w♯((I_{Ψ_{χ}Ψ′_{χ}})^{−1}u) ∼ (I_{ΨΨ′})^{−1}w′ where I_{Ψ_{χ}Ψ′_{χ}} is the same as in (15.55.7).

We consider an obvious equivalence relation between such (Σ, u, z⃗)’s.

Definition 16.16. M((L, Ψ), (L′, Ψ′), j; [h, w], [h′, w′]; k_{1}, k_{2}) is the moduli space of equivalence classes of (Σ, u, z⃗) satisfying (16.15).

We define the evaluation maps

\begin{aligned}ev_{−∞} : M((L, Ψ), (L′, Ψ′), j; [h, w], [h′, w′]; k_{1}, k_{2}) & → R_{h}, \\ ev_{+∞} : M((L, Ψ), (L′, Ψ′), j; [h, w], [h′, w′]; k_{1}, k_{2}) & → R_{h′}, \end{aligned}

by

ev_{±∞}(Σ, u, z⃗) = lim_{τ±∞} u(τ, t).

Let S ∈ C_{(0)}(L, L′). (Here we identify singular chain with the current realized by it. We remark also that S is a singular chain (or current) with local coefficients.)

Definition 16.17.

φ([S, w]) = ∑_{[h′,w′],k_{1},k_{2}}[(M((L, Ψ), (L′, Ψ′), j; [h, w], [h′, w′]; k_{1}, k_{2})_{ev_{−∞}} × S, ev_{∞}]

To be precise we need to compactify (M((L, Ψ), (L′, Ψ′), j; [h, w], [h′, w′]; k_{1}, k_{2}) in order for Definition 16.17 to make sense. We omit the argument for it since it is quite similar to those we have used many times in this paper. We also need to check that by studying the orientation we can prove a similar result to Lemma 16.7 and can prove the right hand side is a chain in local coefficient Θ_{h′}.

Lemma 16.18. The map (16.17) defined as above is a cochain map.

The proof is similar to the arguments we have used many times in this paper and is omitted.

It is now a routine exercise to show that φ is a cochain homotopy equivalence. The proof of Theorem 16.12 is now complete. □

Finally we note that the cochain homomorphism φ above may be considered as a homomorphism of bimodules, which is determined, up to homotopy equivalence. We do not discuss this in this book.

Chapter 5. Transversality and Gluing.

§17. Construction of the Kuranishi structure I.

In this and the next sections, we are going to construct the Kuranishi structure of the moduli spaces we used in earlier chapters. (See §A2 for the definition of Kuranishi structures.) The constructions of the Kuranishi structures of various moduli spaces are almost the same. So we give a rigorous proof of Proposition 13.16 only. The proof of the other statements are similar. Here is the rigorous form of Proposition 13.16. (See §A2 for the definition of the notion used in Proposition 17.1)

Proposition 17.1. M_{k+1}^{main}(L; β) has a topology with respect to which it is compact and Hausdorff. Furthermore, it has an oriented Kuranishi structure with corners of dimension μ(β) + k − 2 + n with respect to which the evaluation map ev : M_{k+1}^{main}(L; β) → L^{k+1} is a smooth strongly continuous map and is weakly submersive.

More precisely, we need to construct Kuranishi structures in Proposition 13.16 so that it satisfies some compatibility conditions. We describe those compatibility conditions as Proposition 17.2 below. To state the proposition, we need to use the notion of fiber product of Kuranishi structure which is discussed in §A2. We compare the induced Kuranishi structure on \overline{S}_{m}M_{k+1}^{main}(β) with the one obtained as the fiber product of M_{k′+1}^{main}(β′) for β′ · ω < β · ω.

We denote by S_{m}M_{k+1}^{main}(β) the codimension m corner of M_{k+1}^{main}(β). Using the notation of (A2.1) it is the union of ψ_{p}(S_{m}(V_{p}) ∩ s_{p}^{−1}(0)) where S_{m}(V_{p}) is the codimension m corner of V_{p}.

Let ((Σ, z⃗), w) ∈ S_{m}M_{k+1}^{main}(β). Note that an irreducible component can be either disc or sphere, and the union of the disc components is connected still of genus zero and with k + 1 markings. We decompose this latter curve into the irreducible components and Σ accordingly by attaching the tower of sphere components of Σ to the unique irreducible disc component that is rooted. This gives a decomposition Σ = Σ_{1} ∪ ⋯ ∪ Σ_{m+1} where each Σ_{i} intersects the boundary of Σ. (We can show by construction that codimension m corner consists of ((Σ, z⃗), w) such that Σ has m+1 disc components.) Let β_{i} = w_{∗}([Σ_{i}]) ∈ H_{2}(M, L) and Σ_{i} contain b_{i} + 1 singular or marked points on its boundary. Then, the restriction w_{i} of w to Σ_{i} determines an element of M_{b_{i}+1}(β_{i}). We assume z_{0} ∈ Σ_{m+1} and order Σ_{i}’s in the same way as Definition 15.21. For each i ≠ 0, there exists unique j(i) such that Σ_{j(i)} > Σ_{i} and Σ_{j(i)} ∩ Σ_{i} ≠ ∅ (recall that Σ has genus 0). Σ_{j(i)} ∩ Σ_{i} is a marked point of (Σ_{i}, w_{i}) regarded as an element of M_{b_{i}+1}(β_{i}). We order marked points so that this marked point becomes the 0-th one.

We define the map

ev^{+} = (ev_{1}^{+}, ev_{2}^{+}) : ΠM_{b_{i}+1}(β_{i}) → L^{2m} × L^{k+1}

as follows: Take ℓ(i) so that Σ_{j(i)} ∩ Σ_{i} is the ℓ(i)-th marked point of Σ_{j(i)}. Let

ev^{i} : M_{b_{i}+1}(β_{i}) → L^{b_{i}+1}

be the evaluation map. Note that Σ has m singular points on the boundary. The first factor of ev^{+}(w_{1}, ⋯, w_{m}) is defined by

ev_{1}^{+}(w_{1}, ⋯, w_{m}) = (w_{1}(z_{0}^{1}), w_{j(1)}(z_{ℓ(1)}^{j(1)}), ⋯, w_{m}(z_{0}^{m}), w_{j(m)}(z_{ℓ(m)}^{j(m)})) .

Here z_{i}^{j} is the i-marked point of Σ_{j}. (Note z_{0}^{i} and z_{ℓ(i)}^{j(i)} will become the same point of Σ.)

The second factor is the usual evaluation map. Namely if i-th marked point of (Σ, z⃗) corresponds b(i)-th marked point of Σ_{a(i)}, it is given by

ev_{2}^{+}(w_{1}, ⋯, w_{m}) = (w_{m}(z_{0}), w_{a(1)}(z_{b(1)}), ⋯, w_{a(k)}(z_{b(k)})).

(Note a(0) = m, b(0) = 0.) Note that ev^{+} is induced from ev^{i} and hence is a smooth strongly continuous map by Proposition 17.1. Proposition 17.1 also implies that ev_{1}^{+} is weekly submersive.

Let L^{m} → L^{2m} be the map (x_{1}, ⋯, x_{m}) ↦ (x_{1}, x_{1}, ⋯, x_{m}, x_{m}). We consider the fiber product

ΠM_{b_{i}+1}(β_{i})_{ev_{1}^{+}} ×_{L^{2m}} L^{m}.

It is easy to see that this space is homeomorphic to the closure of a component of S_{m}M_{k+1}^{main}(β) and that it carries both S- and L-Kuranishi structures from definition. (See §A2.)

Proposition 17.2. The S-Kuranishi structure on the fiber product ΠM_{b_{i}+1}(β_{i}) ev_{1}^{+} ×_{L^{2m}} L^{m} coincides with the induced Kuranishi structure on (the closure of a component of) S_{m}M_{k+1}^{main}(β).

Furthermore in §19, we need to use a Kuranishi structure M_{k+1}^{main}(β) for which the induced Kuranishi structure on \overline{S}_{m}M_{k+1}^{main}(β) is also compatible with the above L-Kuranishi structure. In fact, we can modify our Kuranishi structures so that this condition is satisfied.

Lemma 17.3. There is a Kuranishi structure on M_{k+1}^{main}(β) such that Proposition 17.1 is satisfied, and that the L-Kuranishi structure on the fiber product ΠM_{b_{i}+1}(L; β_{i})_{ev_{1}^{+}} ×_{L^{2m}} L^{m} coincides with the induced Kuranishi structure.

We prove Lemma 17.3 assuming Propositions 17.1 and 17.2. For this purpose, we compare S- and L-Kuranishi structures on Π_{i}X^{i}_{(f^{i})} ×_{f} P in the situation when both are defined. We remark that since f^{i} are submersions, it follows that V_{((p^{i}),p)}^{L} is the total space of a bundle which is the pull back of N_{P} Y^{n} to V_{((p^{i}),p)}^{S}. Moreover

E_{((p^{i}),p)}^{L} = π^{∗}E_{((p^{i}),p)}^{S} ⊕ π^{′∗}N_{P} Y^{n}.

We now go back to the situation of Lemma 17.3 and prove it assuming Propositions 17.1 and 17.2. We pull back the normal bundle N_{L^{m}}L^{2m} to the submanifold S_{m}M_{k+1}^{main}(L; β) and extend this to its tubular neighborhood in M_{k+1}^{main}(L; β). We then add it both to the Kuranishi neighborhood and to the obstruction bundle. Kuranishi map s in this extended direction is constructed by the isomorphism between the factors added to the Kuranishi neighborhood and to the obstruction bundle. From the description above, it follows that what we have obtained is the one asserted in Lemma 17.3. □

The proof of Propositions 17.1 and 17.2 is a combination of the techniques developed in [FOh], [FOn2] and [Fu3].

To carry out the analytic details required for the proof, we need to choose Riemannian metrics on the semi-stable curves with boundary of genus 0 and use them to define appropriate (weighted) Sobolev norms. We use the metrics introduced in [FOh], which are flat outside a finite number of points and is singular at the points. For the convenience of the reader, we describe the metrics now.

We first recall the notion of metric ribbon trees (See [FOh] §1 for more details). A ribbon tree is a pair (T, i) where T is a tree and i : T → D^{2} is an embedding such that

(17.4.1) no vertex of T has 2 edges, (17.4.2) if v ∈ T is a vertex with one edge then i(v) ∈ ∂D^{2}, (17.4.3) i(T) ∩ ∂D^{2} consists of vertices of one edge.

Let G_{k} be the set of (T, i, v_{1}) where (T, i) is an isotopy class of ribbon tree such that ♯i(T) ∩ ∂D^{2} = k and that v_{1} ∈ i(T) ∩ ∂D^{2}. We call a vertex exterior if it is on ∂D^{2} and interior otherwise. We order the set of exterior vertices so that v_{1} is the first one and that the order is compatible with the cyclic order on ∂D^{2}. We call an edge exterior if it contains an exterior vertex, and interior otherwise. We denote by C_{ext}^{0}(T) the set of all exterior vertices and by C_{int}^{0}(T) the set of all interior vertices. Similarly we denote by C_{ext}^{1}(T), C_{int}^{1}(T) the set of exterior and interior edges respectively.

For each 𝔱 = (T, i, v_{1}) ∈ G_{k}, let Gr(𝔱) be the set of all maps ℓ : C_{int}^{1}(T) → ℝ^{+}. We put Gr_{k} = ∪_{𝔱∈G_{k}}Gr(𝔱) and call its element a metric ribbon tree. We regard ℓ(e) as the length of the edge and set the length of exterior edge to be infinite. We refer to [FOh] for the description of the topology which we put on Gr_{k}. Denote the set of all maps ℓ : C_{int}^{1}(T) → ℝ^{+} ∪ \{∞\} by \overline{Gr}(𝔱) and their unions by \overline{Gr}_{k}. The space \overline{Gr}_{k} is a compactification of Gr_{k}.

We next define a map

Θ : Gr_{k} → M_{k}^{b,main.reg},

following §10 [FOh]. Let (𝔱, ℓ) ∈ Gr_{k}. We take an Euclidean rectangle L_{e} = [0, ℓ(e)] × [0, 1] for each e ∈ C_{int}^{1}(𝔱) and L_{e} = (−∞, 0] × [0, 1] for e ∈ C_{ext}^{1}(𝔱). We remove {0, ℓ(e)} × {1/2} ⊂ [0, ℓ(e)] × [0, 1], {0} × {1/2} ⊂ (−∞, 0] × [0, 1]. For given v ∈ C_{int}^{0}(𝔱), we consider edges e, e′ containing v where e′ is the edge next to e according to the cyclic order we put on the set of edges containing v. Now if both e and e′ are outgoing edges, we glue {0} × (1/2, 1] ⊂ ∂L_{e} and {0} × [0, 1/2) ⊂ ∂L_{e′}. We carry out a similar gluing process for other cases of the orientations put on e and e′. (See Figure 17.1).

Figure 17.1

We have thus obtained a space X_{0}(𝔱, ℓ) together with an (incomplete) flat metric on it. We can fill the holes of it conformally and obtain a space X(𝔱, ℓ) equipped with a complex structure and a singular Riemannian metric. We denote Θ(𝔱, ℓ) := [X(𝔱, ℓ)] which is the isomorphism class of X(𝔱, ℓ). In Theorem 10.4 [FOh], we proved that the map Θ : Gr_{k} → M_{k}^{b,main,reg} is a homeomorphism. We used this homeomorphism to define a smooth structure on Gr_{k}.

We remark that Θ can be extended to the compactification \overline{Gr}_{k} of Gr_{k} and leads to a homeomorphism :

\overline{Θ} : \overline{Gr}_{k} → M_{k}^{b,main}

We thus have chosen a singular metric on each element of M_{k}^{b,main,reg}. We modify this family of metrics in a neighborhood of singular points and obtain a smooth family of smooth Riemaniann metrics. We will use these metrics to define appropriate Sobolev norms.

Now we fix an almost complex structure compatible with ω and denote the corresponding Cauchy-Riemann operator by \overline{∂}. Let L ⊂ (M, ω) be a Lagrangian submanifold.

Let ((Σ, z⃗), w) be a pair such that (Σ, z⃗) ∈ M_{k}^{b,main,reg} and w : (Σ, ∂Σ) → (M, L) be a smooth map. We identify Σ\{z_{1}, ⋯, z_{k}} ≅ X(𝔱, ℓ) and regard w as a map from (X(𝔱, ℓ), ∂X(𝔱, ℓ)). On the other hand, if we are given a map w : (X(𝔱, ℓ), ∂X(𝔱, ℓ)) → (M, L), we need to impose some condition on the asymptotic behavior of w to compactify it to a map : (Σ, ∂Σ) → (M, L). This leads us to considering the following Banach manifold. We take and fix δ > 0 sufficiently small and choose p > 0 sufficiently large.

Definition 17.5 W_{k}^{1,p}(M, L) is the set of all ((𝔱, ℓ), w, p⃗) with the following properties:

(17.6.1) (𝔱, ℓ) ∈ Gr_{k}. (17.6.2) w : (D, ∂D) → (M, L) is a map of W_{loc}^{1,p}, i.e., locally of W^{1,p} class.

(17.6.3) p⃗ = (p_{1}, ⋯, p_{k}), p_{i} ∈ L. (17.6.4) Let e ∈ C_{ext}^{1}(𝔱). We have L_{e} ≃ (−∞, 0] × [0, 1] ⊂ X(𝔱, ℓ). We use τ for the parameter of (−∞, 0] and t for [0, 1]. Then

∫_{L_{e}} e^{δ|τ|}|∇w|^{p}dτdt + ∫_{L_{e}} e^{δ|τ|}dist(w(τ, t), p_{i})^{p}dτdt

is finite. (Here dist is the Riemannian distance and ∇ is the Levi-Civita connection.)

Lemma 17.7. The space W_{k}^{1,p}(M, L) is a Banach manifold. Its tangent space T_{((𝔱,ℓ),w,p⃗)}W_{k}^{1,p}(M, L) at ((𝔱, ℓ), w, p⃗) is identified with the set of (ξ, V, v⃗) such that

(17.8.1) ξ ∈ T_{(𝔱,ℓ)}Gr_{k}. (17.8.2) V is a section of u^{∗}(T M) in W_{loc}^{1,p}. (17.8.3) v⃗ = (v_{1}, ⋯, v_{k}), v_{i} ∈ T_{p_{i}}L. (17.8.4) V (z) ∈ T_{w(z)}L if z ∈ ∂X(𝔱, ℓ) (17.8.5) Let e ∈ C_{ext}^{1}(𝔱). We have L_{e} ≃ (−∞, 0] × [0, 1] ⊂ X(𝔱, ℓ)). We then have

∫_{L_{e}} e^{δ|τ|}(|∇(V − Pal v_{i})|^{p} + |V − Pal v_{i}|^{p})dτdt < ∞.

Here Pal : T_{p_{i}}M → T_{w(τ,t)}M is the parallel transport.

The proof is standard and omitted.

We define a norm on W_{k}^{1,p}(M, L) by

\begin{aligned}∥(ξ, V, v⃗)∥_{1,p}^{p} = ∥ξ∥^{p} & + |v⃗|^{p} + ∑_{e∈C_{int}^{1}(𝔱)} ∫_{L_{e}}(|∇V |^{p} + |V |^{p})dτdt \\ & + ∑_{e_{i}∈C_{ext}^{1}(𝔱)} ∫_{L_{e_{i}}} e^{δ|τ|}(|∇(V − Pal v_{i})|^{p} + |V − Pal v_{i}|^{p})dτdt. \end{aligned}

We next define a Banach bundle E^{0,p} on W_{k}^{1,p}(M, L). Let ((𝔱, ℓ), w, p⃗) be an element of W_{k}^{0,p}(M, L). We consider the section V of Λ^{0,1} ⊗ w^{∗}(T M) such that

(17.9.1) V is in W_{loc}^{0,p}. (17.9.2) Let e ∈ C_{ext}^{1}(𝔱). We have L_{e} ≃ (−∞, 0] × [0, 1] ⊂ X(𝔱, ℓ). Then

∫_{L_{e}} e^{δ|τ|}|V |^{p}dτdt < ∞.

Let E_{((𝔱,ℓ),w,p⃗)}^{0,p} be the totality of such V ’s. It is a Banach space with its norm defined by

∥V ∥_{0,p}^{p} = ∑_{e∈C_{int}^{1}(𝔱)} ∫_{L_{e}} |V |^{p}dτdt + ∑_{e_{i}∈C_{ext}^{1}(𝔱)} ∫_{L_{e}} e^{δ|τ|}|V |^{p}dτdt.

With respect to the obvious topology on

E^{0,p} := ⋃_{((𝔱,ℓ),w,p⃗)} E_{((𝔱,ℓ),w,p⃗)}^{0,p}

it follows that E^{0,p} defines a Banach bundle over W_{k}^{1,p}(M, L) (In fact, this bundle can be extended over W_{k}^{0,p}(M, L)). The following is easy to show.

Lemma 17.10. For each ((𝔱, ℓ), w, p⃗) ∈ W_{k}^{1,p}(M, L), \overline{∂}w ∈ E_{((𝔱,ℓ),w,p⃗)}^{0,p}. Furthermore the assignment ((𝔱, ℓ), w, p⃗) ↦ \overline{∂}w defines a smooth section of E^{0,p}.

Using the parallel transport along the minimal geodesics, we define a local trivialization of E^{0,p}. With this trivialization, we define the covariant derivative of \overline{∂}:

(17.11)  Σ_{((𝔱,ℓ),w,p⃗)}\overline{∂} : T_{((𝔱,ℓ),w,p⃗)}W_{k}^{1,p}(M, L) → E_{((𝔱,ℓ),w,p⃗)}^{0,p}.

Lemma 17.12. If δ > 0 is sufficiently small, then D_{((𝔱,ℓ),w,p⃗)}\overline{∂} becomes a Fredholm operator with Fredholm index μ(β) + k − 3 + n. Here β ∈ π_{2}(M, L) is the homotopy class of w and μ(β) its Maslov index.

The proof is by now standard and omitted. Note that we need a weight e^{δ|τ|} to make D_{((𝔱,ℓ),w,p⃗)} a Fredholm operator. This is because our operator is degenerate at the ends. We also have the following which is a consequence of exponential decay estimates. (See Lemma 18.14 stated later.)

Lemma 17.13. Let ((Σ, z⃗), w) be a pair such that (Σ, z⃗) ∈ M_{k}^{b,main,reg} and w : (Σ, ∂Σ) → (M, L). We identify Σ\{z_{1}, ⋯, z_{k}} with X(𝔱, ℓ) and put p_{i} = w(z_{i}). Then ((𝔱, ℓ), w, p⃗) ∈ W_{k}^{1,p}(M, L) if δ is sufficiently small.

Now we let ((𝔱, ℓ), w, p⃗) ∈ W_{k}^{1,p}(M, L) such that \overline{∂}w = 0. It follows from Lemma 17.13 that this element corresponds to an element of M_{k}(L; β) for some β.

Proposition 17.14. There exists a Kuranishi neighborhood of ((𝔱, ℓ), w, p⃗) in the space M_{k}(L, β).

Proof. We consider the operator (17.11). By Lemma 17.12, we can find a finite dimensional subspace E_{((𝔱,ℓ),w,p⃗)} of E_{((𝔱,ℓ),w,p⃗)}^{0,p} such that the map

(17.15)  π ∘ D_{((𝔱,ℓ),w,p⃗)}\overline{∂} : T_{((𝔱,ℓ),w,p⃗)}W_{k}^{1,p}(M, L) → E_{((𝔱,ℓ),w,p⃗)}^{0,p}/E_{((𝔱,ℓ),w,p⃗)}

is surjective, where π is the obvious projection. In view of the unique continuation theorem [Ar], we may choose E_{((𝔱,ℓ),w,p⃗)} so that each V ∈ E_{((𝔱,ℓ),w,p⃗)} has the following properties:

(17.16.1) V is smooth. (17.16.2) V has compact support.

We next take a small neighborhood U of (𝔱, ℓ) in Gr_{k}. For each (𝔱′, ℓ′) ∈ U we can find a diffeomorphism

I_{(𝔱,ℓ),(𝔱′,ℓ′)} : X(𝔱, ℓ) → X(𝔱′, ℓ′)

with the following properties: (17.17.1) I_{(𝔱,ℓ),(𝔱′,ℓ′)} depends smoothly on (𝔱′, ℓ′). (We use the universal family of open Riemann surfaces over U to define smoothness of I_{(𝔱,ℓ),(𝔱′,ℓ′)} with respect to (𝔱′, ℓ′).) (17.17.2) I_{(𝔱,ℓ),(𝔱′,ℓ′)} is the identity outside a compact subset. Here we remark that X(𝔱, ℓ) minus a compact set is identified with the union of (−∞, −T] × [0, 1] ⊂ L_{e} over e ∈ C_{ext}^{1}(𝔱).

Now let ((𝔱′, ℓ′), w′, p⃗′) ∈ W_{k}^{1,p}(M, L) be in a small neighborhood of ((𝔱, ℓ), w, p⃗). We can find a canonical isomorphism

E_{((𝔱,ℓ),w,p⃗)}^{0,p} ≃ E_{((𝔱′,ℓ′),w′,p⃗′)}^{0,p}

by the parallel transport along the minimal geodesics. Now we consider the equation

(17.18)  \overline{∂}w′ ≡ 0 mod E_{((𝔱,ℓ),w,p⃗)}.

Here the right hand side is identified with a subspace of E_{((𝔱,ℓ),w,p⃗)}^{0,p} by the above isomorphism. Let U((𝔱, ℓ), w, p⃗) be the set of all elements ((𝔱′, ℓ′), w′, p⃗′) satisfying (17.18).

Lemma 17.19. U((𝔱, ℓ), w, p⃗) is a smooth manifold of dimension μ(β) + k − 3 + n + dim E_{((𝔱,ℓ),w,p⃗)}

The proof is immediate from Lemma 17.12 and the implicit function theorem. We now define a map

s_{((𝔱,ℓ),w,p⃗)} : U((𝔱, ℓ), w, p⃗) → E_{((𝔱,ℓ),w,p⃗)}

by

s_{((𝔱,ℓ),w,p⃗)}((𝔱′, ℓ′), w′, p⃗′)) = \overline{∂}w′.

The right hand side is an element of E_{((𝔱,ℓ),w,p⃗)} by the definition of U((𝔱, ℓ), w, p⃗). We recall that the automorphism group of (Σ, z⃗) ∈ M_{k}^{b,main,reg} is torsion free if k > 0. So the automorphism of ((𝔱, ℓ), w, p⃗) is trivial. Hence we can simply take Γ_{((𝔱,ℓ),w,p⃗)} = {1}. (We need nontrivial Γ’s only in case there is a sphere bubble in the interior.) This finishes the proof of Proposition 17.14. □

However we need to put one more condition on our Kuranishi neighborhood in relation to the evaluation map

\begin{matrix}ev : U((𝔱, ℓ), w, p⃗) → L^{k}; \\ ((𝔱′, ℓ′), w′, (p′_{1}, ⋯, p′_{k})) ↦ (w′(p′_{1}), ⋯, w′(p′_{k})). \end{matrix}

Lemma 17.20. We can take our Kuranishi neighborhood in Proposition 17.14 so that ev becomes a submersion.

Proof. We consider the subspace of T_{((𝔱,ℓ),w,p⃗)}W_{k}^{1,p}(M, L) consisting of elements (ξ, V, v⃗) with v⃗ = 0. We denote it by (d ev)^{−1}(0). We consider the restriction:

(17.21)  D_{((𝔱,ℓ),w,p⃗)}\overline{∂}|_{(d ev)^{−1}(0)} : (d ev)^{−1}(0) → E_{((𝔱,ℓ),w,p⃗)}^{0,p}

of the operator (17.15). We remark that (d ev)^{−1}(0) is of finite codimension in T_{((𝔱,ℓ),w,p⃗)}W_{k}^{1,p}(M, L). Hence D_{((𝔱,ℓ),w,p⃗)}\overline{∂}|_{(d ev)^{−1}(0)} is also a Fredholm operator. We can then choose E_{((𝔱,ℓ),w,p⃗)} so that

π ∘ D_{((𝔱,ℓ),w,p⃗)}\overline{∂} : (d ev)^{−1}(0) → E_{((𝔱,ℓ),w,p⃗)}^{0,p}/E_{((𝔱,ℓ),w,p⃗)}

is surjective. It is straightforward to see that for this choice of E_{((𝔱,ℓ),w,p⃗)}. the conclusion of Lemma 17.20 holds. □

§18. Construction of the Kuranishi structure II.

In §17, we constructed a Kuranishi neighborhood of each point in M_{k+1}^{b,main,reg}(L; β). We next construct Kuranishi neighborhoods of points in the set

M_{k+1}^{b,main}(L; β)\M_{k+1}^{b,main,reg}(L; β).

We need to perform Taubes’ type gluing construction for this purpose. We first define a gluing map of graphs

Comp_{k_{1}···k_{m};m′;j_{1},··· ,j_{m}} : \overline{Gr}_{k_{1}+1} × ⋯ \overline{Gr}_{k_{m}+1} × \overline{Gr}_{m+m′+1} → \overline{Gr}_{k_{1}+···k_{m}+m′+1}

where 0 < j_{1} < j_{2} < ⋯ < j_{m} < m + m′ + 1. Let 𝔱_{i} ∈ G_{k_{i}+1}, i = 1, ⋯, m and 𝔱_{0} ∈ G_{m+m′+1}. We identify the 0-th exterior edge of 𝔱_{i} with the j_{i}-th edge of 𝔱_{0} and obtain an element 𝔱 ∈ G_{∑ k_{i}+m′+1}. For given ℓ_{i} ∈ Gr(𝔱_{i}) and ℓ′ ∈ Gr(𝔱_{0}) we define a map ℓ : C_{int}^{1}(𝔱) → (0, ∞] as follows: C_{int}^{1}(𝔱) = ∪C_{int}^{1}(𝔱_{i}) ∪ C_{int}^{1}(𝔱_{0}) ∪ {e_{1}, ⋯, e_{m}} where e_{1}, ⋯, e_{m} are new edges corresponding to the 0-th exterior edge of 𝔱_{i}. We put

ℓ(e) = \{ \begin{matrix}ℓ_{i}(e) & if e ∈ C_{int}^{1}(𝔱_{i}) \\ ℓ_{0}(e) & if e ∈ C_{int}^{1}(𝔱_{0}) \\ ∞ & if e = e_{i}, i = 1, ⋯, m \end{matrix}

and

Comp_{k_{1}···k_{m};m′;j_{1},··· ,j_{m}}((𝔱_{1}, ℓ_{1}), ⋯, (𝔱_{m}, ℓ_{m}); (𝔱_{0}, ℓ_{0})) =: (𝔱, ℓ).

Each such element (𝔱, ℓ) is obtained by an applying Comp iteratively to the elements of Gr_{∗}’s (the interior elements of the moduli space) in a unique way. In the rest of this section, we only consider the case when

(𝔱, ℓ) = Comp_{k;m;j}((𝔱_{1}, ℓ_{1}), (𝔱_{0}, ℓ_{0}))

and (𝔱_{1}, ℓ_{1}) ∈ Gr_{k+1}, (𝔱_{0}, ℓ_{0}) ∈ Gr_{m+1}. (Figure 18.1.)

Figure 18.1

In other words, we only consider the case where under the map \overline{Θ} : \overline{Gr}_{k} → M_{k}^{main}, (𝔱, ℓ) corresponds to a marked semi-stable curve of genus 0 with one singular point (i.e., one nodal point) on the boundary. The other cases are similar except that the notation will become more complicated. We also remark that the way to handle sphere bubble is completely the same as [FOn2] so we do not repeat it.

Let w : (X(𝔱, ℓ), ∂X(𝔱, ℓ)) → (M, L) be a pseudo-holomorphic map that extends to a stable map: (Σ, ∂Σ) → (M, L) when X(𝔱, ℓ) is identified with Σ\{marked points} (see the statement right before Definition 17.5). Then w determines the maps

w_{1} : X(𝔱_{1}, ℓ_{1}) → M, w_{0} : X(𝔱_{0}, ℓ_{0}) → M,

and the points

p⃗^{1} = (p_{0}^{1}, ⋯, p_{k}^{1}) ∈ L^{k+1}, p⃗^{0} = (p_{0}^{0}, ⋯, p_{m}^{0}) ∈ L^{m+1}

such that

((𝔱_{1}, ℓ_{1}), w_{1}, p⃗^{1}) ∈ W_{k+1}^{1,p}(M, L)
((𝔱_{0}, ℓ_{0}), w_{0}, p⃗^{0}) ∈ W_{m+1}^{1,p}(M, L)
p_{0}^{1} = p_{j}^{0}.

We have chosen E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}, E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0}))} as in §17. We define a subspace X of

X^{+} := T_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}W_{k+1}^{1,p}(M, L) ⊕ T_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0}))}W_{m+1}^{1,p}(M, L)

by

X := {((ξ_{1}, V_{1}, v⃗^{1}), (ξ_{0}, V_{1}, v⃗^{0}) ∈ X^{+} | v_{0}^{1} = v_{j}^{0} ∈ T_{p_{0}^{1}}L = T_{p_{j}^{0}}L}.

We consider the direct sum

D_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}\overline{∂} ⊕ D_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0}))}\overline{∂}

and denote the restriction of it to X by

D_{w}\overline{∂} : X → E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}^{0,p} ⊕ E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}^{0,p}.

Lemma 18.1. Let π be the obvious projection map. The composition

π ∘ D_{w}\overline{∂} : X → E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}^{0,p}/E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} ⊕ E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}^{0,p}/E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}

is surjective.

This is a consequence of Lemma 17.20 and the “Mayer-Vietoris principle” of Mrowka [Mr] (Compare it also with Figure 6 [Fu3]).

Using Lemma 18.1, we now perform Taubes’ type gluing construction. To carry out this, we need to modify the pseudo-holomorphic curve equation to one similar to (17.18), and to construct the moduli space of its solutions that is parameterized by the kernel of π ∘ D_{u}\overline{∂} plus a neighborhood of (𝔱, ℓ) in M_{k+m+1}^{b,main}. This neighborhood of (𝔱, ℓ) is parameterized by V × (C, ∞] where V is the neighborhood of 0 in

T_{(𝔱_{1},ℓ_{1})}M_{k+1}^{b,main} ⊕ T_{(𝔱_{0},ℓ_{0})}M_{m+1}^{b,main}.

Here (C, ∞] corresponds to the set of parameters which controls “stretching” of the attaching discs at the boundary singular point. (C is a sufficiently large positive number.)

As in other cases of the gluing construction, we first use partitions of unity to construct approximate solutions (of (18.3) we will define later). Then we study their linearizations and check if they are surjective. Because of the degeneracy at infinity, we need to use the weighted Sobolev norms with some unusual weight to achieve uniformity of the estimates involved (similar to the one used in §8 [Fu3].)

Let (𝔱′_{1}, ℓ′_{1}) ∈ Gr_{k+1} and (𝔱′_{0}, ℓ′_{0}) ∈ Gr_{m+1} be in small neighborhoods of (𝔱_{1}, ℓ_{1}) and (𝔱_{0}, ℓ_{0}) respectively. For each T a sufficiently large positive number and an element ((ξ_{1}, V_{1}, v⃗^{1}), (ξ_{0}, V_{1}, v⃗^{0})) of Ker π ∘ D_{w}\overline{∂}, we are going to construct an approximate solution of pseudo-holomorphic curve equation.

We start with gluing the domains of maps by considering

(𝔱′, ℓ′′) = Comp_{k;m;j}((𝔱′_{1}, ℓ′_{1}), (𝔱′_{0}, ℓ′_{0})).

Note that (𝔱′, ℓ′′) has one interior edge of infinite length, which is the edge where we glued 𝔱′_{0} and 𝔱′_{1}. We write this edge e(new). Replace this edge by an edge of length 10T and denote by

(𝔱′, ℓ′) = Comp((𝔱′_{1}, ℓ′_{1}), (𝔱′_{0}, ℓ′_{0}); T) ∈ Gr_{k+m}

the resulting graph. More precisely, X(𝔱′, ℓ′) is constructed from X(𝔱′_{1}, ℓ′_{1}) and X(𝔱′_{0}, ℓ′_{0}) as follows: Let e_{1,0} be the 0-th exterior edge of 𝔱′_{1} and e_{0,j} be the j-th exterior edge of 𝔱′_{0}. In order to make the exposition easier to understand, we change the parameterizations of L_{e_{1,0}} and L_{e_{0,j}} so that

\begin{matrix}L_{e_{1,0}} ≃ (−∞, 5T] × [0, 1] ⊂ X(𝔱′_{1}, ℓ′_{1}) \\ L_{e_{0,j}} ≃ [−5T, ∞) × [0, 1] ⊂ X(𝔱′_{0}, ℓ′_{0}) \end{matrix}

(in place of identifying L_{e_{1,0}} ≃ (−∞, 0] × [0, 1], L_{e_{0,j}} ≃ (−∞, 0] × [0, 1]). We remove (−∞, 0] × [0, 1] ⊂ L_{e_{1,0}} from X(𝔱′_{1}, ℓ′_{1}) and [0, ∞) × [0, 1] ⊂ L_{e(0)} from X(𝔱′_{0}, ℓ′_{0}) and glue them at {0} × [0, 1]. Noting that

L_{e(new)} ≃ [−5T, 5T] × [0, 1] ⊂ X(𝔱′_{0}, ℓ′_{0}),

it is easy to see that X(𝔱′_{0}, ℓ′_{0}) is obtained this way.

Figure 18.2

By the construction, we have the natural maps :

\begin{aligned}I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)} & : X(𝔱_{0}, ℓ_{0}) − (−∞, −5T) × [0, 1] → X(𝔱′, ℓ′) \\ I_{(𝔱_{1},ℓ_{1}),(𝔱′,ℓ′)} & : X(𝔱_{1}, ℓ_{1}) − (5T, ∞) × [0, 1] → X(𝔱′, ℓ′) \end{aligned}

where (5T, ∞)×[0, 1] is regarded as a subset of L_{e_{0,1}} and (−∞, −5T)×[0, 1] as that of L_{e_{1,j}} respectively. It is obvious that I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)} and I_{(𝔱_{1},ℓ_{1}),(𝔱′,ℓ′)} are diffeomorphisms to their images. Furthermore the union of their images is X(𝔱′, ℓ′) and the intersection of the images is L_{e(new)}.

We need to use cut-off functions of various kinds, which we now introduce. For each S ∈ [−5T + 1, 5T − 1], we take a smooth function χ_{S}^{+} : [−5T, 5T] → [0, 1] such that

χ_{S}^{+}(τ) = \{ \begin{matrix}0 & if τ < S − 1, \\ 1 & if τ > S + 1. \end{matrix}

and

(18.2)  |∇χ_{S}^{+}| < C.

We put χ_{S}^{−} = 1 − χ_{S}^{+}. Regarding them as the functions defined on L_{e(new)} = [−5T, 5T] × [0, 1], we extend χ_{S}^{±} to X(𝔱′, ℓ′) by putting them to be 0 or 1 on the two components of X(𝔱′, ℓ′) − L_{e(new)} respectively. We denote the resulting functions by the same notations.

This time regarding χ_{S}^{−} as a function on L_{e_{1,0}} ≃ (−∞, 5T] × [0, 1], we similarly extend it to X(𝔱_{1}, ℓ_{1}) and denote the resulting function by χ_{S}^{1}. Similarly, regarding χ_{S}^{+} as a function on L_{e_{0,j}} ≃ [5T, ∞) × [0, 1], we extend it to X(𝔱_{0}, ℓ_{0}) and denote the resulting function by χ_{S}^{0}.

Figure 18.3

Note that

\begin{matrix}χ_{S}^{1} = χ_{S}^{−} ∘ I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)}, \\ χ_{S}^{0} = χ_{S}^{+} ∘ I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)}. \end{matrix}

We will use these cut off functions to glue sections on X(𝔱_{0}, ℓ_{0}) and X(𝔱_{1}, ℓ_{1}) to obtain sections on X(𝔱′, ℓ′). Using the exponential convergence of w_{0} ∈ M_{k+1}^{b,main} and w_{1} ∈ M_{m+1}^{b,main} at infinity, we can glue them in a small coordinate neighborhood of p_{0}^{1} = p_{j}^{0} and obtain a map w′′ : X(𝔱′, ℓ′) → M. We now define several maps

I_{0,∗}^{S} : Γ(X(𝔱_{0}, ℓ_{0}), w_{0}^{∗}T M) → Γ(X(𝔱′, ℓ′), w^{′′∗}T M),
I_{1,∗}^{S} : Γ(X(𝔱_{1}, ℓ_{1}), w_{1}^{∗}T M) → Γ(X(𝔱′, ℓ′), w^{′′∗}T M),
I_{∗,0}^{S} : Γ(X(𝔱′, ℓ′), w^{′′∗}T M) → Γ(X(𝔱_{0}, ℓ_{0}), w_{0}^{∗}T M),
I_{∗,1}^{S} : Γ(X(𝔱′, ℓ′), w^{′′∗}T M) → Γ(X(𝔱_{1}, ℓ_{1}), w_{1}^{∗}T M).

which we are going to use in the gluing construction of sections.

We define a map w′_{0} : Im(I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)}) → M by

w′_{0} = w_{1} ∘ I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)}^{−1}

and w′_{1} : Im(I_{(𝔱_{1},ℓ_{1}),(𝔱′,ℓ′)}) → M in a similar way. Let V ∈ Γ(X(𝔱_{0}, ℓ_{0}), w_{0}^{∗}T M). The section χ_{0}^{S}V is supported in X(𝔱_{0}, ℓ_{0})\(−∞, −5T) × [0, 1]. Hence we can use I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)} to obtain a section of Γ(X(𝔱, ℓ), w_{0}^{′∗}T M). We then use the parallel transport along the minimal geodesics to obtain an element I_{0,∗}^{S}(V) in Γ(X(𝔱′, ℓ′), w^{′′∗}T M). The definition of I_{1,∗}^{S} is similar.

Next let V ∈ Γ(X(𝔱′, ℓ′), w^{′′∗}T M). The support of χ_{+}^{S}V is in the image of I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)}. We use the parallel transport along the minimal geodesics to obtain an element of Γ(X(𝔱′, ℓ′), w_{0}^{′∗}T M). We then pull it back by I_{(𝔱_{0},ℓ_{0}),(𝔱′,ℓ′)} to obtain an element I_{∗,0}^{S}(V) of Γ(X(𝔱_{0}, ℓ_{0}), w_{0}^{∗}T M). The definition of I_{∗,1}^{S} is similar.

We also define the maps

J_{0,∗}^{S} : Γ(X(𝔱_{0}, ℓ_{0}), Λ^{0,1} ⊗ w_{0}^{∗}T M) → Γ(X(𝔱′, ℓ′), Λ^{0,1} ⊗ w^{′′∗}T M),
J_{1,∗}^{S} : Γ(X(𝔱_{1}, ℓ_{1}), Λ^{0,1} ⊗ w_{1}^{∗}T M) → Γ(X(𝔱′, ℓ′), Λ^{0,1} ⊗ w^{′′∗}T M),
J_{∗,0}^{S} : Γ(X(𝔱′, ℓ′), Λ^{0,1} ⊗ w^{′′∗}T M) → Γ(X(𝔱_{0}, ℓ_{0}), Λ^{0,1} ⊗ w_{0}^{∗}T M),
J_{∗,1}^{S} : Γ(X(𝔱′, ℓ′), Λ^{0,1} ⊗ w^{′′∗}T M) → Γ(X(𝔱_{1}, ℓ_{1}), Λ^{0,1} ⊗ w_{1}^{∗}T M),

in a similar way. These maps extend to the W^{1,p} or W^{0,p} closures of the spaces involved.

We next recall that elements of E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}, E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})} are of compact support. Using this fact, we can prove that

J_{0,∗}^{S} + J_{1,∗}^{S} : E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} ⊕ E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})} → E_{((𝔱′,ℓ′),w′′,p⃗′)}

is injective. So we may assume

E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} ⊕ E_{(𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})} ⊆ E_{((𝔱′,ℓ′),w′′,p⃗′)}.

We now consider the equation

(18.3)  \overline{∂}w ≡ 0 mod E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} ⊕ E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}.

We first describe an approximate solution of (18.3). For this purpose, we introduce more notations. Define the “small” tangent space

T_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}^{small}W_{k}^{1,p}(M, L) := {(ξ_{1}, V_{1}, v⃗^{1}) ∈ T_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}W_{k+1}^{1,p}(M, L) | ξ_{1} = 0}.

and T_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}^{small}W_{m+1}^{1,p}(M, L) in a similar way. We use them in place of the full tangent space T_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}W_{k+1}^{1,p}(M, L) etc, to obtain X_{small}^{+}, X_{small}.

By changing the choices of E_{(ξ_{1},V_{1},v⃗^{1})}, E_{(ξ_{0},V_{0},v⃗^{0})} if necessary, we may assume that the restriction of π ∘ D_{w′′} \overline{∂} to X_{small} is surjective. We use X_{small} rather than X since we want to handle the moduli parameters ξ_{1}, ξ_{2} separately in a way different from the other parameters V_{1}, v_{0}^{1}, ⋯, v_{k}^{1} etc.

Now by interpolating the sections on X(𝔱_{0}, ℓ_{0}) and X(𝔱_{1}, ℓ_{1}), we define

I : Ker (π ∘ D_{w′′} \overline{∂}|_{X_{small}}) → T_{((𝔱′,ℓ′),w′′,p⃗′)}^{small}W_{k+m}^{1,p}(M, L).

where p⃗′ = (p′_{1}, ⋯, p′_{k+m}) are obtained by re-ordering p_{1}^{1}, ⋯, p_{k}^{1}, p_{0}^{0}, ⋯, p̂_{j}^{0}, ⋯, p_{m}^{0}: For each ((0, V_{1}, v⃗^{1}), (0, V_{0}, v⃗^{0})) ∈ X_{small} we put

I((0, V_{1}, v⃗^{1}), (0, V_{0}, v⃗^{0})) = I_{0,∗}^{0}(V_{0}) + I_{1,∗}^{0}(V_{1}).

We next use the exponential map of M to obtain the natural map

Exp : T_{((𝔱′,ℓ′),w′′,p⃗′)}^{small}W_{k+m}^{1,p}(M, L) → W_{k+m}^{1,p}(M, L).

Now, for each (𝔱′_{0}, ℓ′_{0}),(𝔱′_{1}, ℓ′_{1})), T and ((0, V_{1}, v⃗^{1}), (0, V_{0}, v⃗^{0})) ∈ X_{small}, we take

(18.4)  ((𝔱′, ℓ′), Exp(I(V_{1}, V_{2})))

as our approximate solution where (𝔱′, ℓ′) = Comp((𝔱′_{1}, ℓ′_{1}), (𝔱′_{0}, ℓ′_{0})), T).

The next step is to perturb (18.4) and obtain an exact solution of (18.3). We need to use somewhat nonstandard weighted Sobolev norm for this purpose. The weight function α : X(𝔱′, ℓ′) → [1, ∞) that we use is as follows: Let e ∈ C_{ext}^{1}(𝔱′). We put

α(τ, t) = exp(δ|τ|)

on L_{e} ≃ (−∞, 0] × [0, 1]. Identifying L_{e(new)} with [−5T, 5T] × [0, 1], we put

α(τ, t) = \{ \begin{matrix}1 & if |τ| > 5T − 1 \\ exp(δ(5T − |τ|)) & if 5T − 2 > |τ| > 1 \end{matrix}

In the intermediate regions, there is a constant C such that 1 ≤ α < C for 5T − 1 > |τ| > 5T − 2 and |α − exp(5T δ)| < C for |τ| < 1. (Figure 18.4.)

Figure 18.4

Finally we put α = 1 on other L_{e}’s.

Using this weight function, we now define the Sobolev norm we use. Let w′′ be the map C^{0}-close to Exp(I(0, 0)). We consider the vector space E_{((𝔱′,ℓ′),w′′,p⃗′)} but will put a different norm on it. For V ∈ E_{((𝔱′,ℓ′),w′′,p⃗′)}, we put

(18.5)  ∥V ∥_{0,p,α}^{p} = ∫_{X(𝔱′,ℓ′)} α|V |^{p}.

Since the weight α coincides, outside a compact set L_{e(new)}, with the one we used to define E_{((𝔱′,ℓ′),w′′,p⃗′)}, the right hand side of (18.5) is finite and this norm ∥∥_{0,p,α} is equivalent to the previous norm ∥∥_{0,p}. However the ratio between the norms ∥∥_{0,p} and ∥∥_{0,p,α} on E_{((𝔱′,ℓ′),w′′,p⃗′)} goes to infinity as T goes to infinity.

Next we modify the norm on T_{((𝔱′,ℓ′),w′′,p⃗′)}^{new}W_{k+m}^{1,p}(M, L). Let (0, V, v⃗) = (V, v⃗) be an element of T_{((𝔱′,ℓ′),w′′,p⃗′)}^{new}W_{k+m}^{1,p}(M, L). We remark that V is continuous and so V (0, 1/2) makes sense, if p is sufficiently large. Here (0, 1/2) ∈ L_{e(new)} ≃ [−5T, 5T] × [0, 1]. We put

∥(V, v⃗)∥_{1,p,α}^{p} = ∑_{e∈C_{int}^{1}(𝔱),e≠e(new)} ∫_{L_{e}}(|∇V | + |V |)^{p}

+ ∑_{e_{i}∈C_{ext}^{1}(𝔱)} ∫_{L_{e_{i}}} α · (|∇V | + |V − Pal(v_{i})|)^{p}

+ ∫_{L_{e(new)}} α · (|∇V | + |V − Pal(V (0, 1/2))|)^{p}

+ |V (0, 1/2)|^{p} + ∑ |v_{i}|^{p}

where Pal is an appropriate parallel transport along minimal geodesic.

Proposition 18.6. Assume that the restriction of π ∘ D_{w}\overline{∂} to X_{small} is surjective. Then there exists C > 0 independent of T with the following properties:

For each (𝔱′, ℓ′) = Comp((𝔱′_{1}, ℓ′_{1}), (𝔱′_{0}, ℓ′_{0})); T) and w′′ sufficiently close to Exp(I(0, 0)), there exists a linear map

Q_{((𝔱′,ℓ′),w′′)} : \frac{E_{((𝔱′,ℓ′),w′′,p⃗′)}}{E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} ⊕ E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0}))}} → T_{((𝔱′,ℓ′),w′′,p⃗′)}^{small}W_{k+m}^{1,p}(M, L)

such that

(π ∘ D_{w′′} \overline{∂}) ∘ Q_{((𝔱′,ℓ′),w′′)} = id, (18.7.1) ∥Q_{((𝔱′,ℓ′),w′′)}(V)∥_{1,p,α} < C∥V ∥_{0,p,α}. (18.7.2)

We emphasize that the uniformity of the constant C cannot be achieved if we use ∥∥_{1,p}, ∥∥_{0,p} in place of ∥∥_{1,p,α}, ∥∥_{0,p,α}. Before proving Proposition 18.6, we first state the following lemma whose proof will be postponed to the end of this section.

Lemma 18.8. There exist constants c, C > 0 independent of T such that

∥\overline{∂} Exp(I_{1}(V_{1}, V_{2}))∥_{0,p,α} < C(∥V_{1}∥_{1,p} + ∥V_{2}∥_{1,p} + 1) exp(−cT).

Once we have Proposition 18.6 and Lemma 18.8, it is by now standard to construct a family of solutions of (18.3) parameterized by the kernel of π ∘ D_{w}\overline{∂} plus a neighborhood of (𝔱, ℓ) in \overline{Gr}_{k+m}. Therefore to construct Kuranishi neighborhoods of points at infinity (i.e., at lower dimensional strata), it remains to prove Proposition 18.6 and Lemma 18.8.

Proof of Proposition 18.6. We first remark that there exists C > 0 independent of T such that

(18.9.1)  ∥I_{0,∗}^{S}(V)∥_{1,p,α} < C∥V ∥_{1,p},
(18.9.2)  ∥I_{1,∗}^{S}(V)∥_{1,p,α} < C∥V ∥_{1,p},
(18.9.3)  ∥J_{0,∗}^{S}(V)∥_{0,p,α} < C∥V ∥_{0,p},
(18.9.4)  ∥J_{1,∗}^{S}(V)∥_{0,p,α} < C∥V ∥_{0,p},

These immediately follow from the choice of our norm and weight. We also have

(18.10.1)  ∥I_{∗,0}^{S}(V)∥_{1,p} < C∥V ∥_{1,p,α},
(18.10.2)  ∥J_{∗,0}^{S}(V)∥_{1,p} < C∥V ∥_{1,p,α},

if S ≤ 0 and

(18.10.3)  ∥I_{∗,1}^{S}(V)∥_{1,p} < C∥V ∥_{1,p,α},
(18.10.4)  ∥J_{∗,1}^{S}(V)∥_{1,p} < C∥V ∥_{1,p,α},

if S ≥ 0. (18.10) also follows from the choice of our norm and weight. We next need the following :

Lemma 18.11. There exist constants c > 0, C > 0 independent of T, S such that

∥J_{0,∗}^{S}(D_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}\overline{∂}) (V) − I_{0,∗}^{S}(D_{((𝔱′,ℓ′),w′′,p⃗′)}\overline{∂}) (V)∥_{0,p,α}

< C(min{1, exp(+cS)} + dist(w′′, w′_{0}))∥V ∥_{1,p},

∥J_{1,∗}^{S}(D_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}\overline{∂}) (V) − I_{0,∗}^{S}(D_{((𝔱′,ℓ′),w′′,p⃗′)}\overline{∂}) (V)∥_{0,p,α}

< C(min{1, exp(−cS)} + dist(w′′, w′_{1}))∥V ∥_{1,p}

Proof. We prove only the first inequality. There are two reasons why I and J do not commute with D_{w}\overline{∂}. One is the fact that we use the parallel transport along the minimal geodesics to go from w_{0}^{′∗}T M to w^{′′∗}T M. The terms caused by the parallel transport are estimated by the second term in the right hand side of the lemma.

The second and more essential point is the fact that we used cut-off functions. The support of differential of the cut-off function lies on [S −1, S +1]×[0, 1] ⊂ L_{e_{1,0}}. The weight we put there to define ∥ ∥_{1,p} norm on X(𝔱_{0}, ℓ_{0}) is exp(δ(5T − S)).

On the other hand, the weight α at the corresponding point is exp(δ(5T − S)) if S is positive and is exp(δ(5T + S)) if S is negative. Hence the term caused by the derivative of cut-off functions is estimated by the first term of the right hand side. This proves the lemma. □

Now we go back to the proof of Proposition 18.6. Let

Q_{0} : E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}^{0,p}/E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})} → T_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}^{small}W_{m+1}^{1,p}(M, L)
Q_{1} : E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}^{0,p}/E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} → T_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})}^{small}W_{k+1}^{1,p}(M, L)

be the right inverses to π ∘ D\overline{∂} respectively for (𝔱_{0}, ℓ_{0}) and (𝔱_{1}, ℓ_{1}). We define

R : \frac{E_{((𝔱′,ℓ′),w′′,p⃗′)}}{E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} ⊕ E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0}))}} → T_{((𝔱′,ℓ′),w′′,p⃗′)}^{small}W_{k+m}^{1,p}(M, L)

by

(18.12)  R(V) = (I_{0,∗}^{T} ∘ Q_{0} ∘ J_{∗,0}^{0})(V) + (I_{1,∗}^{−T} ∘ Q_{1} ∘ J_{∗,1}^{0})(V)

We remark that J_{∗,0}^{0} sends E_{((𝔱_{1},ℓ_{1}),w_{1},p⃗^{1})} ⊂ E_{((𝔱′,ℓ′),u′′)} to 0 and E_{((𝔱_{0},ℓ_{0}),w_{0},p⃗^{0}))} ⊂ E_{((𝔱′,ℓ′),w′′)} to E_{(𝔱_{0},ℓ_{0}),w_{0},p⃗^{0})}. Hence (Q_{0} ∘ J_{∗,0}^{0})(V) is well defined. Using (18.9), (18.10), and Lemma 18.11, we obtain

∥(D_{w′′} \overline{∂} ∘ R)(V) − V ∥_{0,p,α} < \frac{1}{2}∥V ∥_{0,p,α}

if T is chosen to be sufficiently large. In particular, D_{w′′} \overline{∂} ∘ R − id is invertible. Then a standard functional analysis argument gives rise a right inverse of D_{w′′} \overline{∂}. The proof of Proposition 18.6 is complete. □

Proof of Lemma 18.8. To prove Lemma 18.8 we use the following lemma. We consider the situation of Lemma 18.8 and use the same notations from there.

Lemma 18.13. Let V be an element of the kernel of D_{((𝔱,ℓ),w,p⃗)}\overline{∂} and e ∈ C_{ext}^{1}(𝔱). We identity L_{e} ≃ (−∞, 0] × [0, 1] as before. Then there exists c > 0, C > 0 independent of T and δ such that

∥V (τ, t)∥ < C exp(−c|τ|).

This lemma is a standard decay estimate which is a part of the proof of Lemma 17.13. The next lemma follows from the removable singularity theorem of pseudoholomorphic disc. See the estimate in p 138 [Oh1] and translate it in terms of the coordinates (−∞, 0] × [0, 1] near the puncture.

Lemma 18.14. Let ((𝔱, ℓ), w, p⃗) ∈ W_{k}^{1,p}(M, L) satisfies (17.14) and e ∈ C_{ext}^{1}(𝔱) be an exterior edge. Identify L_{e} ≃ (−∞, 0] × [0, 1]. Then there exist constants C, C_{ℓ}, c, c_{ℓ} independent of T and δ such that

\begin{matrix}dist(w(τ, t), p_{i}) < C exp(−c|τ|), \\ ∥∇^{ℓ}w(τ, t)∥ < C_{ℓ} exp(−c_{ℓ}|τ|). \end{matrix}

Lemma 18.8 follows immediately from these two lemmas and the construction. □

We thus have constructed Kuranishi neighborhoods of ((Σ, z⃗), w) where Σ has one singular point. The case where there are several singular points on the boundary can be handled in the same way. The case where there are interior singular points can be handled in the same way as in [FOn2] Chapter 3.

We also need to prove “surjectivity”, which means that the Kuranishi neighborhood of ((Σ, z⃗), w) we constructed contains all the elements of the moduli space M_{k+1}(β) close to the given point ((Σ, z⃗), w). The proof of this surjectivity is similar to §14 [FOh] or §9 [FOn2].

Finally, to complete the proof of Propositions 13.15, 17.1, 17.2, we need to glue the Kuranishi neighborhoods constructed so far to produce a global Kuranishi structure. Using the compactness and Hausdorffness of M_{k+1}(β), we may do this gluing only for a finite number of Kuranishi neighborhoods. The proof of it goes in the same way as in §15 [FOn2]. Let us review the arguments in [FOn2], to glue the Kuranishi neighborhood constructed above. For each σ = ((Σ, z⃗), w) ∈ M_{k+1}(β) we have chosen a finite dimensional subspace E′_{σ} of Γ(Σ, Λ_{Σ}^{0,1} ⊗ w^{∗}T M) such that the equation

\overline{∂}w′ ≡ 0 mod E′_{σ}

is the equation defining the Kuranishi neighborhood U′_{σ} of σ. (In the last section, we denoted E_{((𝔱,ℓ),w,p⃗)} for E′_{σ}.) We may assume U′_{σ} to be a closed neighborhood of σ by shrinking it a bit. We take a finite number of σ’s, say σ_{1}, ⋯, σ_{N}, so that

(18.15)  ⋃_{i} U′_{σ_{i}} ⊃ M_{k+1}(β).

We put σ_{i} = ((Σ_{i}, z⃗_{(i)}), w_{i}). We may re-number σ_{i}’s so that if U′_{σ_{i}} ∩ U′_{σ_{j}} ≠ ∅ and i > j, topologically Σ_{j} is obtained by resolving some of the singular points of Σ_{i}.

Using the covering (18.15) we will modify E′_{σ} to construct a Kuranishi neighborhood of σ. For simplicity, we consider the case when (Σ, z⃗) is stable. (In the case when it is unstable we need additional argument. See §15 [FOn2] for it.) We collect all i’s such that σ ∈ U′_{σ_{i}} and put

E_{σ} = ⨁_{i:σ∈U′_{σ_{i}}} E′_{σ_{i}}.

Here we regard E′_{σ_{i}} as a subspace of Γ(Σ, Λ^{0,1}Σ ⊗ w^{∗}T M) in a way similar to the first hal of this section. A Kuranishi neighborhood U_{σ} is the set of isomorphism classes of pairs ((Σ′, z⃗′), w′) such that (Σ′, z⃗′) is close to (Σ, z⃗) in M_{m}^{b} (defined in §3) and w′ satisfies

\overline{∂}w′ ≡ 0 mod E_{σ}.

Using the fact that U′_{σ_{i}} is closed, we can choose U_{σ} so that if σ_{1} ∈ U_{σ_{2}} then E_{σ_{1}} ⊆ E_{σ_{2}}. (Here we use a map Γ(Σ_{1}, Λ^{0,1} ⊗ w_{1}^{∗}T M) → Γ(Σ_{2}, Λ^{0,1} ⊗ w_{2}^{∗}T M) (defined by the parallel transport) to compare E_{σ_{1}} and E_{σ_{2}}. (σ_{i} = ((Σ_{i}, z⃗_{i}), w_{i}).)) Therefore we may assume that if σ_{1} ∈ U_{σ_{2}}, then a neighborhood of σ_{1} in U_{σ_{1}} is contained in U_{σ_{2}}. We put this neighborhood U_{σ_{1},σ_{2}} which will satisfy the conditions (A2.1) in §A2.

§19. Construction of the fundamental chain.

Now we are in the position to use the Kuranishi structure we have produced in §§17,18, and to construct the fundamental chain which enables us to justify the constructions of previous chapters. As we already remarked, each irreducible component homeomorphic to D^{2} has either a marked point or a singular point in our situation and so does not have nontrivial automorphisms. So we need not to worry about negative multiple cover discs. However we still need to use multi-sections in general in place of single-valued sections because there may occur sphere-bubbles of negative multiple cover. Most of the argument is a direct application of general machinery developed in Chapter 1 [FOn2]. However we should be careful at one point, where we need to use something more than Chapter 1 [FOn2]. In fact the following statement is not true unless we perturb the Kuranishi map s_{k}(β_{0}), where we recall β_{0} = 0.

“There exists a family of multi-sections s_{k+1}(β) of Kuranishi structures on the space M_{k+1}^{main}(β) such that it is compatible with the isomorphism in Proposition 17.2.”

Let us explain the trouble that will appear in an attempt to prove this statement. Suppose that β_{1} ∈ π_{2}(M, L) is the element such that β_{1} ≠ 0, M_{1}^{main}(β_{1}) ≠ ∅ and that M_{1}^{main}(β) = ∅ for any β with 0 < ω(β) < ω(β_{1}). We first find s_{k+1}(β_{1}). Assuming, for simplicity, that M_{1}^{main}(β) = ∅ for any β with ω(β_{1}) ≤ ω(β) < 2ω(β_{1}), β_{1} ≠ β, the next step is to construct s_{k+1}(2β_{1}). We remark that

(19.1)  S_{2}M_{1}^{main}(2β_{1}) = M_{1}^{main}(β_{1})_{ev_{0}} ×_{ev_{1}} M_{3}^{main}(β_{0})_{ev_{2}} ×_{ev_{0}} M_{1}^{main}(β_{1}),

where β_{0} = 0. The difficulty in constructing s_{k+1}(2β_{1}) lies in that the fiber product of the right hand side of (19.1) is never transversal, even in the case when the moduli space M_{1}^{main}(β_{1}) is transversal. This is because the chain ev(M_{1}^{main}(β_{1})) is not transverse to itself, unless it has full dimension of L. We can not make it transversal by any choice of perturbations just defined on M_{1}^{main}(β_{1}) either. The trouble here is that we have to apply the same perturbation to the two copies of M_{1}^{main}(β_{1}) in the right hand side of (19.1), in order to make the construction compatible with the isomorphism in Proposition 17.2.

The trouble mentioned above is a rather essential point. It is related to the fact that we can not deform the singular chain complex so that the intersection pairing (the cup product) is everywhere defined in the chain level. Usual intersection pairing (in the chain level) makes sense only for the elements in a dense subset of singular chains. Especially the intersection pairing of a chain with itself is not well-defined in the chain level. This fact seems to have been well known to algebraic topologists.

However if we work in the category of A_{∞} algebras in place of differential graded algebras, we can perturb (a countably generated sub-complex of) the singular chain complex to another where products are everywhere well defined. This fact also seems to be known to algebraic topologists since the 70’s when rational homotopy theory was established. However we give a proof of this fact here because the structure of our proof is rather crucial in the construction of the fundamental chain in this section, which we use to extend the A_{∞} algebra by incorporating quantum contributions.

Let us first explain the idea briefly. Let C(L) be a countably generated subcomplex of \overline{S}(L) of the complex of currents realized by singular chains on L. (See §A1). We choose two diffeomorphisms φ_{0} : L → L and φ_{1} : L → L that are isotopic to the identity. We consider a countable set X_{0} of chains in L. It is easy to see that if φ_{i} are generic then φ_{0}(P) is transverse to φ_{1}(Q) for any chains P, Q ∈ X_{0}. (Note P may coincides with Q.) We put 𝔪_{2}(P, Q) = φ_{0}(P) ∩ φ_{1}(Q). Thus 𝔪_{2} is well defined for every pair of chains in X_{0}. This operator 𝔪_{2} is not associative in that

\begin{aligned}𝔪_{2}(𝔪_{2}(P, Q), R) & = φ_{0}(φ_{0}(P) ∩ φ_{1}(Q)) ∩ φ_{1}(R) \\ & ≠ φ_{0}(P) ∩ φ_{1}(φ_{0}(Q) ∩ φ_{1}(R)) = 𝔪_{2}(P, 𝔪_{2}(Q, R)). \end{aligned}

However it is easy to see that there exists a chain homotopy between (P, Q, R) ↦ 𝔪_{2}(𝔪_{2}(P, Q), R) and (P, Q, R) ↦ 𝔪_{2}(P, 𝔪_{2}(Q, R)). The operator 𝔪_{3} is defined to be this chain homotopy between them. We can continue this process and obtain an A_{∞} algebra.

Now we provide this construction more systematically. Let \overline{Gr}_{k+1} be the compactification of the moduli space of metric ribbon trees introduced in §18. Let Δ_{k} ⊂ L^{k} be the diagonal where Δ_{k} ≃ L and let N_{Δ_{k}}L^{k} be the normal bundle of Δ_{k} in L^{k}. We identify N_{Δ_{k}}L^{k} with a tubular neighborhood of Δ_{k} in L^{k}. We consider the bundle maps

(19.2)  𝔰^{k} : Δ_{k} × \overline{Gr}_{k+1} → N_{Δ_{k}}L^{k}.

(Here the spaces in (19.2) are regarded as bundles over Δ_{k}.) We assume that x ↦ 𝔰^{k}(x, T) is C^{1} close to the diagonal map x ↦ (x, ⋯, x) for each fixed T ∈ \overline{Gr}_{k+1}.

Note that \overline{Gr}_{3} is a point. Hence 𝔰^{2} is a map L → L^{2}. Since we assumed that the image of 𝔰^{2} is C^{1} close to the diagonal, we have 𝔰^{2}(p) = (φ_{0}(p), φ_{1}(p)), where φ_{0}, φ_{1} are diffeomorphisms. Let T ∈ \overline{Gr}_{k+1}, p ∈ L. We put (p_{1}, ⋯, p_{k}) = 𝔰^{k}(p, T) and set 𝔰_{i}^{k}(p, T) = p_{i}.

Let X be a countable set of currents realized by sigular chains on L such that if P ∈ X then ∂P ∈ X. Now impose the following assumptions on 𝔰^{k}:

(19.3.1) If P_{i} ∈ X then P_{1} × ⋯ × P_{k} is transverse to 𝔰^{k}. The fiber product (Δ_{k} × \overline{Gr}_{k+1})_{𝔰^{k}} ×_{L^{k}} (P_{1} × ⋯ P_{k}) has a triangulation. By π : Δ_{k} × \overline{Gr}_{k+1} × L^{k} → Δ_{k} = L it defines an (integral) current on L. We assume that it is an element of X. (19.3.2) Let k+1 = k_{1}+⋯+k_{m}+m′+1 and T = Comp_{k_{1},··· ,k_{m};m′;j_{1},··· ,j_{m}}(T_{1}, ⋯, T_{m} ; T′) ∈ Gr_{k+1}. Then we have :

𝔰^{k}(p, T) = (𝔰_{1}^{m+m′}(p, T′), ⋯, 𝔰_{j_{1}−1}^{m+m′}(p, T′),
𝔰_{1}^{k_{1}}(𝔰_{j_{1}}^{m+m′}(p, T′), T_{1}), ⋯, 𝔰_{k_{1}}^{k_{1}}(𝔰_{j_{1}}^{m+m′}(p, T′), T_{1}),
𝔰_{j_{1}+1}^{m+m′}(p, T′), ⋯, 𝔰_{j_{2}−1}^{m+m′}(p, T′),
𝔰_{1}^{k_{2}}(𝔰_{j_{2}}^{m+m′}(p, T′), T_{2}), ⋯,
· · · · · · · · · · · · · · · · · · · · · , 𝔰_{m+m′}^{m+m′}(p, T′).)

Figure 19.1

For each such system 𝔰^{k}, we define an A_{∞} algebra as follows. As an abelian group it is a free ℤ module over X. We write it as ℤX. The maps 𝔪 = {𝔪_{k}}_{k=1}^{∞} defining the A_{∞} algebra structure is given by:

(19.4.1) 𝔪_{1}(P) = (−1)^{n}∂P, (19.4.2) 𝔪_{k}(P_{1}, ⋯, P_{k}) = (−1)^{ϵ}((Δ_{k} × \overline{Gr}_{k+1})_{𝔰^{k}} ×_{L^{k}} (P_{1} × ⋯ P_{k}), π). Here π : N_{Δ_{k}}L^{k} → Δ_{k} = L is a projection and ϵ is given by

ϵ = (n + 1) ∑_{i=1}^{k−1} ∑_{j=1}^{i} deg P_{j},

where n = dim L and deg P_{j} = n − dim P_{j}. As for these signs in (19.4), recall and see Definition 13.18, Definition 13.17 and also Definition 25.1. We note that by identifying Δ_{k} ×\overline{Gr}_{k+1} with the moduli space of constant J holomorphic discs with k + 1 marked points, we here give the orientation on Δ_{k} × \overline{Gr}_{k+1}, as the orientation on the moduli space of constant maps which is given in Chapter 6. (We defined an orientation on M_{k+1}^{b,main,reg} in §3. By analyzing orientation on M_{k+1}^{b,main} and using the map \overline{Θ} : \overline{Gr}_{k+1} → M_{k+1}^{b,main} in §17, we can find an orientation on Δ_{k} × \overline{Gr}_{k+1}. This might be different from the orientation defined above. But one can see the difference exactly.) The product 𝔪_{k} in (19.4.2) is well defined on X by (19.3.1). The proof of the following lemma is immediate from (19.3.2).

Lemma 19.5. (ℤX , 𝔪) is an A_{∞} algebra with 𝔪_{0} = 0.

Now we have :

Theorem 19.6. For any countable set X_{0} of currents realized by singular chains, there exists a family of 𝔰_{k} and a countable set X ⊃ X_{0} satisfying (19.3) such that the homology group of (ℤX , 𝔪_{1}) coincides with the homology group of L.

Proof. We first remark that \overline{Gr}_{k+1} is a cell by Stasheff’s theorem [St]. Now the construction will be given by the induction on k. We first use diffeomorphisms φ_{0} and φ_{1} C^{1}-close to the identity so that φ_{0}(P) is transverse to φ_{1}(Q) for any P, Q ∈ X_{0}. This assumption is satisfied for φ_{i} in a Baire set of diffeomorphisms. We put 𝔰^{2}(x) = (φ_{0}^{−1}(x), φ_{1}^{−1}(x)). We add 𝔪_{2}(P, Q) = {x|φ_{0}^{−1}(x) ∈ P, φ_{1}^{−1}(x) ∈ Q} = φ_{0}(P) ∩ φ_{1}(Q) to X_{0} to get X_{0,1}. We then prove that φ_{0}(P) is transverse to φ_{1}(Q) for any P, Q ∈ X_{0,1}, by re-choosing φ_{0} and φ_{1} if necessary. This is again satisfied for φ_{i} in a Baire set, by re-choosing φ_{0} and φ_{1} if necessary. We then add 𝔪_{2}(P, Q) = φ_{0}(P)∩φ_{1}(Q) (P, Q ∈ X_{0,1}) to X_{0,1} and to obtain X_{0,2}. We continue this process and obtain X_{1} = ∪_{j}X_{0,j}. Since this is a countable process, the transversality is satisfied for φ_{i} in a Baire set. However in the above induction process, we need to re-choose φ_{0} and φ_{1} in every step and have to make sure that this re-choosing process converges and produces the limit diffeomorphisms that satisfy transversality for any pair of elements from X_{1}. Because of this, we will choose φ_{0} and φ_{1} all at once in the following way.

Let us first describe how an element in X_{1} = ∪_{j}X_{0,j} looks like. Each element from X_{0,j} can be associated to a binary tree with its depth j. If Γ has n legs, we denote the corresponding element from X_{0,j} by

𝔪_{Γ}^{φ_{0},φ_{1}}(P_{1}, ⋯, P_{n})

where P_{1}, ⋯, P_{n} ∈ X_{0}. If we denote by B_{n} the set of binary trees with n legs and B = ∪_{n=1}^{∞}B_{n}, then

X_{1} = ⋃_{n=1}^{∞} ⋃_{Γ∈B_{n}} ⋃_{(P_{1},··· ,P_{n})∈X_{0}}{𝔪_{Γ}^{φ_{0},φ_{1}}(P_{1}, ⋯ P_{n})}

and so X_{1} is a countable set. Now we would like to choose appropriate diffeomorphisms φ_{0}, φ_{1} so that

𝔪_{Γ}^{φ_{0},φ_{1}}(P_{1}, ⋯ P_{n}) ⋔ 𝔪_{Γ′}^{φ_{0},φ_{1}}(P′_{1}, ⋯ P′_{n})

for any choice of Γ, Γ′ ∈ B and (P_{1}, ⋯, P_{n}), (P′_{1}, ⋯, P′_{n}) ∈ X_{0}. It now follows from the standard transversality theorem that we can find a Baire set of diffeomorphisms (φ_{0}, φ_{1}) for which this is the case.

Next we consider 𝔰^{3}. We remark that 𝔰^{3}(p, T) is determined by (19.3.2) if T ∈ ∂\overline{Gr}_{4}. Transversality is satisfied there by induction hypothesis. Since fibers of the normal bundle N_{Δ_{3}}L^{3} → Δ_{3} are contractible, we can extend this map to Δ_{3} × \overline{Gr}_{4}. We thus obtained 𝔰^{3′}(p, T) which satisfies (19.3.2). We remark that transversality (19.3.1) is satisfied in a neighborhood of Δ_{3} × ∂\overline{Gr}_{4}. Therefore we can use usual transversality theorem to find that for a dense subset of perturbation 𝔰^{3}(p, T) of 𝔰^{3′}(p, T), the condition (19.3.1) is satisfied for P_{i} ∈ X_{1}. We add image of 𝔪_{3} to X_{1}. We repeat arguments similar to the one used above to obtain X_{1} and obtain X_{2} so that both 𝔪_{2} and 𝔪_{3} are well-defined in ℤX_{2}. Thus we obtain 𝔪_{3}. Repeating the similar process and taking X = ∪_{k=0}^{∞}X_{k}, we obtain 𝔪_{k} that are everywhere defined on X. Since this is again a countable process, we can choose 𝔰^{k+1}(p, T) all at once in a way similar to the above so that the transversality (19.3) holds. The proof of Theorem 19.6 is complete. □

We next show that the A_{∞} structure constructed in Theorem 19.6 is unique up to homotopy equivalence. We first recall that an A_{∞} algebra (C_{1}, 𝔪^{1}) with 𝔪_{0} = 0 is weakly homotopy equivalent to (C_{2}, 𝔪^{2}), if there exists an A_{∞} map 𝔣{𝔣_{k}}_{k≥1} from (C_{1}, 𝔪^{1}) to (C_{2}, 𝔪^{2}) which induces an isomorhpism :

(19.7)  𝔣_{1∗} : H(C_{1}; 𝔪_{1}^{1}) → H(C_{2}; 𝔪_{1}^{2}).

(Definition 15.5.) This definition is justified by the following theorem.

Theorem 19.8. Let (C_{1}, 𝔪^{1}), (C_{2}, 𝔪^{2}) be A_{∞} algebras with 𝔪_{0}^{1} = 𝔪_{0}^{2} = 0 and let 𝔣 = {𝔣_{k}}_{k≥1} be an A_{∞} homomorphism such that (19.7) is an isomorphism. Then there exists an A_{∞} homomorphism 𝔣′ = {𝔣′_{k}}_{k≥1} from (C_{2}, 𝔪^{2}) to (C_{1}, 𝔪^{1}) such that 𝔣′_{1} induces the inverse of (19.7).

Theorem 19.8 (which seems to be known) together with the sketch of the proof is in [K5] in the setting of L_{∞} algebra. We will give a proof of Theorem 19.8 in §A5 for completeness. (However our proof in §A5 is different from the one sketched in [K5].)

Note that Theorem 19.8 implies that the homotopy equivalence of A_{∞} algebra we defined above, is an equivalence relation. Now we state the following :

Theorem 19.9. The weak homotopy type of the A_{∞} algebra in Theorem 19.6 is unique, i.e., is independent of the choice of X.

We prove Theorem 19.9 in §A5. We now show the following :

Theorem 19.10. Let L ⊂ M be relatively spin. Then we can define a filtered A_{∞} algebra deforming the A_{∞} algebra constructed in Theorem 19.6.

Proof. To prove Theorem 19.10, we construct 𝔪_{k,β} by induction on the energy ω(β). So the first step is the case when ω(β) = 0. In that case, we only need to consider β = β_{0} = 0, since for β ≠ 0 with ω(β) = 0 the moduli space of pseudo-holomorphic discs M(β) is empty.

Set-theoretically the moduli space M_{k+1}(β_{0}) can be identified with the direct product M_{k+1}^{b,main} × L = \overline{Gr}_{k+1} × L. This moduli space is of correct dimension and in fact transversal. This means that for ((𝔱, ℓ), w) ∈ M_{k+1}(β_{0}) the linearization map D_{w}\overline{∂} is surjective. We remark that w is constant since its homology class is 0 and since it is pseudo-holomorphic. Let us put the constant map w ≡ p ∈ L. Then using the notations in §17,18, we can identify ((𝔱, ℓ), w) with ((𝔱, ℓ), w, (p, ⋯, p)) ∈ W_{k+1}^{1,p}(M, L). Hence the linearized operator is a linear map

(19.11)  D_{((𝔱,ℓ),w,(p,··· ,p))}\overline{∂} : T_{((𝔱,ℓ),w,(p,··· ,p))}^{small}W_{k+1}^{1,p}(M, L) → E_{((𝔱,ℓ),w,(p,··· ,p))}^{0,p}.

Since w is the constant map, both the induced bundle w^{∗}T M and the sub-bundle w^{∗}T L on the boundary are canonically trivialized. We can also make this trivialization satisfy that the boundary condition becomes the real boundary condition. Now the linearization D_{w}\overline{∂} becomes the standard (classical) Cauchy-Riemann operator with respect to the constant integrable complex structure on ℂ^{n} with the real boundary condition. It follows that the kernel of (19.11) consists of constant function with real values and so has n dimension. Therefore an index calculation implies that (19.11) is surjective. Hence one can choose E_{((𝔱,ℓ),w,(p,··· ,p))} = 0 for the construction of Kuranishi structure.

However then the evaluation map (ev_{0}, ⋯, ev_{k}) is not weakly submersive in general. So we still need to put nontrivial E_{((𝔱,ℓ),w,(p,··· ,p))} to enlarge the Kuranishi neighborhood. This is the process used in the proof of Lemma 17.3. But to proceed to the next step we need to carry out this more precisely.

We construct a homomorphism

K : T_{(p,··· ,p)}L^{k} → T_{((𝔱,ℓ),w,(p,··· ,p))}^{small}W_{k+1}^{1,p}(M, L),

as follows. Let

V⃗ = (V_{1}, ⋯, V_{k}) ∈ T_{(p,··· ,p)}L^{k}.

Let e_{i} be the ith exterior edge of 𝔱. Let χ be a cut-off function such that

χ(τ) = \{ \begin{matrix}1 & if τ < −2 \\ 0 & if τ > −1 \end{matrix}

We put

K(V⃗) = \{ \begin{matrix}χ(τ)V_{i} & on L_{e_{i}} i = 1, ⋯, k, \\ 0 & elesewhere. \end{matrix}

Here e_{0}, ⋯, e_{k} are the exterior edges. Regarding N_{Δ_{k}}L^{k} as a subbundle of T L^{k}|_{Δ_{k}}, we put

E_{((𝔱,ℓ),w,(p,··· ,p))} = (D_{((𝔱,ℓ),w,(p,··· ,p))}\overline{∂})(K(N_{Δ_{k},p}L^{k})).

Here N_{Δ_{k},p}L^{k} denotes the fiber of the normal bundle N_{Δ_{k}}L^{k} at (p, ⋯, p). We use this to construct our Kuranishi neighborhoods. Then the evaluation map (ev_{1}, ⋯, ev_{k}) : M_{k+1}(β_{0}) → L^{k} is weakly submersive.

The Kuranishi neighborhood just constructed is identified with a neighborhood of the zero section of N_{Δ_{k}}L^{k} times \overline{Gr}_{k+1}. The obstruction bundle over it (which is globally defined in our case) is just the pull-back of N_{Δ_{k}}L^{k}. We need to take an appropriate section s′_{k+1} of this obstruction bundle to perturb our moduli space M_{k+1}(β_{0}) so that it becomes transverse to the chains P_{i} on L.

Using the maps 𝔰^{k} we construct s′_{k+1} by defining

s′_{k+1}(v, T) = v − 𝔰^{k}(π(v), T).

Here v ∈ N_{Δ_{k}}L^{k}, π(v) ∈ Δ_{k} ≃ L. We remark that the Kuranishi map s_{k+1} before perturbation is given by s_{k+1}(v, T) = v. The moduli space perturbed by this section s′_{k+1} (namely the zero set of s′_{k+1}), together with evaluation map ev_{0}, is identified with the chain (𝔰^{k}(Δ_{k} × \overline{Gr}_{k+1}), π).

These perturbed moduli spaces for various k’s are compatible due to the condition (19.3.2). The transversality with P ∈ X is (19.3.1). The resulting 𝔪_{k,β_{0}} is equal to the one constructed by Theorem 19.10. Thus we have finished the first step of induction.

Once this first step is completed, the rest of the induction steps immediately follow from Chapter 1 [FOn2] by choosing (multi-)sections inductively on ω(β) and on k. We add countably many basis elements to the sub-complex C(L) if necessary, inductively in the same way as in the proof of Theorem 19.6. □

To show the construction more clearly, we go back to the example M_{1}^{main}(L; 2β_{1}) we mentioned at the beginning of this section. As we already explained, the first stratum S_{2}M_{1}^{main}(2β_{1}) (19.1) is perturbed to

(19.12)  𝔪_{2}(M_{1}^{main}(β_{1}), M_{1}^{main}(β_{1})).

We may assume that the right hand side of (19.12) is well-defined and is of expected dimension. Namely its dimension is

(19.13)  (n + μ(β_{1}) − 2) + (n + μ(β_{1}) − 2) − n = n + μ(2β_{1}) − 4.

(19.13) is the virtual dimension of M_{1}^{main}(2β_{1}) minus 2.

The next stratum we need to consider corresponds to the following figure

Figure 19.2

This component is, before perturbation,

(19.14)  S_{1}M_{1}^{main}(2β_{1}) = M_{1}^{main}(β_{1})_{ev_{0}} ×_{ev_{1}} M_{2}^{main}(β_{1}).

The point here is that the two spaces M_{1}^{main}(β_{1}) and M_{2}^{main}(β_{1}) in the right hand side are different. Hence we may choose generic perturbations so that they become transverse to each other. In fact, in a neighborhood of S_{2}M_{1}^{main}(2β_{1}) (inside S_{1}M_{1}^{main}(2β_{1})), the transversality is automatically satisfied if we take a perturbation that extends the one on S_{2}M_{1}^{main}(2β_{1}). So we can take a perturbation of S_{1}M_{1}^{main}(2β_{1}) to be of the expected dimension, (that is n+μ(2β_{1})−3 = the virtual dimension of M_{1}^{main}(2β_{1}) minus 1.) Once we can make these two strata transversal, then we can extend this perturbation to the whole M_{1}^{main}(2β_{1}). We can similarly proceed with M_{1}(kβ_{1}) for k ≥ 3 using 𝔪_{ℓ}, ℓ ≤ k.

§20 Homotopy unit.

The construction so far gives the fundamental chain (over ℚ) of the moduli spaces we have used. The fundamental chain thus defined has most of the properties we used in earlier chapters. However it lacks one of the properties which we used in an essential way in §7 and §14. Namely in the proof of Proposition 7.6 and in the verification of Conditions 13.11 and 14.3, we used the fact that the map

𝔣𝔬𝔯𝔤𝔢𝔱 : M_{k+1}^{main}(L; β) → M_{k}^{main}(L; β)

forgetting one marked point is well-defined and commutes with the evaluation map. The example of the stratum that we have explained at the end of the last section shows that this is not in fact the case after our perturbation: The evaluation map image of M_{2}^{main}(β_{1})_{ev_{1}} ×L must be transverse to and so can not coincide with that of M_{1}^{main}(β_{1}). In other words, the map M_{2}^{main}(β_{1}) → M_{1}^{main}(β_{1}) is not compatible with the evaluation map after taking perturbation.

Therefore e = [L] does not satisfy Condition 13.11 and is not a unit. However, we have another perturbation of M_{k+1}^{main}(β) that is compatible with the forgetful map. (But this perturbation is not compatible to the perturbation constructed in the earlier stage.) Using this second perturbation and the fact that “the A_{∞} operation 𝔪 = {𝔪_{k}}_{k≥0} is independent of perturbations up to homotopy”, we can prove that e = [L] is a homotopy unit (in the sense we will define below). Then we will prove that any A_{∞} algebra with a homotopy unit is homotopy equivalent to an A_{∞} algebra with a unit.

We would like to remark that we meet a similar trouble in [FOn2] when we prove that the Gromov-Witten invariant satisfies the Kontsevich-Manin axiom [KM]. Namely we can not choose a perturbation that is both compatible with the forgetful map and with the perturbations constructed in the earlier stage. However for the Gromov-Witten invariant, we can use the fact that the Gromov-Witten invariant is an invariant in the homology level and so it is enough to prove the compatibility in the homology level. In the current situation, the trouble is more serious since we have to perform the construction in the chain level.

Now let us carry out the plan mentioned above. We start with defining the notion of homotopy unit for which we need to develop some more homological algebra.

Let (C, 𝔪) be an A_{∞} algebra. We do not assume that it has a unit. We denote by B_{k}(B(C[1])) the tensor product of k copies of B(C[1])’s and put

B(B(C[1])) = ⨁_{k≥1} B_{k}(B(C[1])) = B(C[1]) ⊕ (B(C[1]) ⊗ B(C[1])) ⊕ · · · .

Let B̂(B̂(C[1])) be a completion of B(B(C[1])) with respect to the energy filtration.

To avoid confusion between the tensor product in B(C[1]) with the one in B(B(C[1])), we use \overline{⊗} for the later, and we use the bold face letter x etc. to denote elements of B(C[1]). Namely, for example, x_{1} ⊗x_{2} is an element of B_{2}(C[1]) and x_{1}\overline{⊗}x_{2} is an element of B_{2}(B(C[1])). If x_{i} = x_{i,1} ⊗ ⋯ ⊗ x_{i,ℓ_{i}} ∈ B_{ℓ_{i}}(C[1]) then x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k} ∈ B_{k}(B(C[1])) and x_{1} ⊗ ⋯ ⊗ x_{k} ∈ B_{ℓ_{1}+···+ℓ_{k}}(C[1]). (x_{1} ⊗ ⋯ ⊗ x_{k} = x_{1,1} ⊗ x_{1,2} ⊗ ⋯ ⊗ x_{1,ℓ_{1}} ⊗ x_{2,1} ⊗ ⋯ ⊗ x_{k,ℓ_{k}}.)

We consider operations

𝔥_{k} : B_{k}(B(C[1])) → C[1]

of degree 3 − 2k (k = 1, 2, 3, ⋯) such that

𝔥_{1} = 𝔪

and all 𝔥_{k} preserve the energy filtration. (If we count the degree of \overline{⊗} as deg′ \overline{⊗} = −2 the 𝔥_{k} will have degree +1 always.) We then define

𝔥̂_{k} : B(B(C[1])) → B(B(C[1])),  𝔥̂ : B(B(C[1])) → B(B(C[1]))

as follows: Let Δ : B(C[1]) → B(C[1])⊗B(C[1]) be the coproduct. Then for k ≥ 2, we put

𝔥̂_{k}(x_{1}\overline{⊗} ⋯ \overline{⊗}x_{n}) = ∑_{a,b,i}(−1)^{deg′ x_{1}+··· deg′ x_{i−1}+deg′ x′_{i,a}} x_{1}\overline{⊗} ⋯ \overline{⊗}x_{i−1}
\overline{⊗} (x′_{i,a} ⊗ 𝔥_{k}(x′′_{i,a}\overline{⊗}x_{i+1}\overline{⊗} ⋯ \overline{⊗}x_{i+k−2}\overline{⊗}x′_{i+k−1,b}) ⊗ x′′_{i+k−1,b})
\overline{⊗}x_{i+k}\overline{⊗} ⋯ \overline{⊗}x_{n},

where Δx_{i} = ∑_{a} x′_{i,a} ⊗ x′′_{i,a}. (Note 𝔥̂_{k} : B_{n}(B(C[1])) → B_{n−k+1}(B(C[1])), since ⊗ in the second line is not \overline{⊗}.) Here deg′ is the degree after shifted. Namely if x = x_{1} ⊗ ⋯ ⊗ x_{m} then

deg′ x = deg x_{1} + ⋯ + deg x_{m} − m.

For k = 1, we put

𝔥̂_{1}(x_{1}\overline{⊗} ⋯ \overline{⊗}x_{n}) = ∑_{a,i}(−1)^{deg′ x_{1}+··· deg′ x_{i−1}} x_{1}\overline{⊗} ⋯ \overline{⊗}x_{i−1}
\overline{⊗}d̂x_{i}\overline{⊗} x_{i+1}\overline{⊗} ⋯ \overline{⊗}x_{n} .

Here we recall

d̂x_{i} = ∑_{a}(−1)^{deg′ x{′ \atop i,a}}x′_{i,a} ⊗ 𝔪(x′′_{i,a}) ⊗ x′′′_{i,a}

when

ΔΔx_{i} = ∑ x′_{i,a} ⊗ x′′_{i,a} ⊗ x′′′_{i,a}.

We put 𝔥̂ = ∑ 𝔥̂_{k} and extend it to B̂(B̂(C[1])).

We also introduce a family of maps 𝔟_{k,i} : B_{k}(B(C[1])) → B_{k−1}(B(C[1])), (i = 1, ⋯, k − 1) by

𝔟_{k,i}(x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k}) = (−1)^{deg′ x_{1}+···+deg′ x_{i−1}}x_{1}\overline{⊗} ⋯ \overline{⊗}(x_{i} ⊗ e ⊗ x_{i+1})\overline{⊗} ⋯ \overline{⊗}x_{k}

for a given e ∈ C^{0} = C[1]^{−1}. (If we count deg′ \overline{⊗} = −2, then 𝔟_{k,i} is of degree +1.)

Definition 20.1. Let (C, 𝔪) be an A_{∞} algebra. We say an element e ∈ C^{0} = C[1]^{−1} is a homotopy unit if there exist 𝔥_{k} for k = 1, 2, ⋯ such that 𝔥_{1} = 𝔪 and that

(20.2.1)  ∑_{ℓ=1}^{k} 𝔥_{ℓ} ∘ 𝔥̂_{k−ℓ+1} − ∑_{i=1}^{k−1} 𝔥_{k−1} ∘ 𝔟_{k,i} + 𝔓_{k} = 0.

Here 𝔓_{k} = 0 if k ≠ 2. 𝔓_{2} = 0 on the complement of (B_{0}(C[1])\overline{⊗}B_{1}(C[1])) ⊕ (B_{1}(C[1])\overline{⊗}B_{0}(C[1])) ⊂ B_{2}(B(C[1])), and

(20.2.2)  −𝔓_{2}(x\overline{⊗}1) = 𝔓_{2}(1\overline{⊗}x) = x,

on (B_{0}(C[1])\overline{⊗}B_{1}(C[1])) ⊕ (B_{1}(C[1])\overline{⊗}B_{0}(C[1])) ⊂ B_{2}(B(C[1])).

Note that on B_{2}(B(C[1])), our formula (20.2.1) is

(20.3) \begin{aligned}0 = & ∑(−1)^{d_{i}}𝔪(x_{1}, ⋯, 𝔥_{2}((x_{i} ⊗ ⋯ ⊗ x_{m})\overline{⊗}(y_{1} ⊗ ⋯)), ⋯, y_{ℓ}) \\ & + ∑(−1)^{d_{i}}𝔥_{2}((x_{1} ⊗ ⋯ ⊗ 𝔪(x_{i}, ⋯) ⊗ ⋯ ⊗ x_{m})\overline{⊗}(y_{1} ⊗ ⋯ ⊗ y_{ℓ})) \\ & + ∑(−1)^{h_{i}}𝔥_{2}((x_{1} ⊗ ⋯)\overline{⊗}(⋯ ⊗ 𝔪(y_{i}, ⋯) ⊗ ⋯ ⊗ y_{ℓ})) \\ & − (−1)^{d_{m+1}}𝔪_{m+ℓ+1}(x_{1}, ⋯, x_{m}, e, y_{1}, ⋯, y_{ℓ}). \end{aligned}

where d_{i} = deg x_{1} +⋯+deg x_{i−1} −(i−1) and h_{i} = d_{m+1} +deg y_{1} +⋯+deg y_{i−1} − (i − 1) and for x\overline{⊗}y with x = x_{1} ⊗ ⋯ ⊗ x_{m} and y = y_{1} ⊗ ⋯ ⊗ y_{ℓ}, m + ℓ ≥ 2. (deg is the degree before shifted.) (In case m+ℓ = 1, 𝔓 ≠ 0 and we need to apply (20.2.2).) The formula in Definiton 20.1 looks rather complicated. It may be useful to see the proof of Proposition 20.8 below to understand the meaning of the formula. We also explain the geometric origin of the formula (20.2.1) after the proof of Proposition 20.8.

Proposition 20.4. Let 𝔪 be the operators we defined using the fundamental chains constructed in §19. Then e = [L] is a homotopy unit.

The proof of Proposition 20.4 will be given later in this section.

Let (C, 𝔪, e, 𝔥) be as in Definition 20.1. We define (C^{+}, 𝔪) as follows.

C^{+} = C ⊕ Λ_{0,nov}e^{+} ⊕ Λ_{0,nov}f,

where deg e^{+} = 0, deg f = −1. (Here deg is a degree before shifted.) We define

(20.5)  \begin{aligned}𝔪_{2}(e^{+}, x) = (−1)^{deg x}𝔪_{2}(x, e^{+}) & = x, \\ 𝔪_{k}(⋯, e^{+}, ⋯) & = 0,  k ≠ 2 \end{aligned}

for x ∈ C^{+}, and

(20.6)  𝔪(x_{1} ⊗ f ⊗ x_{2} ⊗ f ⊗ ⋯ ⊗ f ⊗ x_{k}) = 𝔥_{k}(x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k})

unless k = 2 and x_{1}, x_{2} ∈ Λ_{0,nov}, in which case we put

(20.7)  𝔪_{1}(f) = e^{+} − e + 𝔥_{2}(1\overline{⊗}1).

Proposition 20.8. If e is a homotopy unit then the following holds.

(20.8.1) (C^{+}, 𝔪) is an A_{∞} algebra and e^{+} is its unit. (Namely it satisfies Condition 13.11.) (20.8.2) If b ∈ C[1]^{0} = C^{1} is a bounding cochain of (C, 𝔪), then it is also a bounding cochain of (C^{+}, 𝔪). The natural inclusion (C, 𝔪^{b}) → (C^{+}, 𝔪^{b}) is a homotopy equivalence.

Proof. (20.8.1) follows from definition. Namely (20.2.1) and (20.2.2) will become the A_{∞} formulae. We will check it at the end of the proof. (20.5) then implies that e^{+} is its unit.

The first half of (20.8.2) is obvious. So, in view of Theorem 19.8, we have only to show that the natural inclusion (C, 𝔪_{1}^{b}) → (C^{+}, 𝔪_{1}^{b}) induces an isomorphism on cohomology, to prove the second half of (20.8.2). Clearly the natural inclusion is a cochain map and is injective. The quotient cochain complex is generated by [e^{+}] and [f]. (20.7) implies d[f] = [e^{+}]. Hence the quotient complex is acyclic.

Now we check the A_{∞} formula for (C^{+}, 𝔪). We compare the terms

(20.9.1)  (𝔪 ∘ d̂)(x_{1} ⊗ f ⊗ x_{2} ⊗ f ⊗ ⋯ ⊗ f ⊗ x_{k}).

and

(20.9.2)  ∑_{ℓ=1}^{k}(𝔥_{ℓ} ∘ 𝔥̂_{k−ℓ+1})(x_{1}\overline{⊗}x_{2}\overline{⊗} ⋯ \overline{⊗}x_{k})

and see that (20.9.1) is equal to

(20.10)

∑_{ℓ=1}^{k}(𝔥_{ℓ} ∘ 𝔥̂_{k−ℓ+1})(x_{1}\overline{⊗}x_{2}\overline{⊗} ⋯ \overline{⊗}x_{k})

+ ∑_{ℓ=1}^{k−1}(−1)^{ϵ_{ℓ}}𝔪(x_{1} ⊗ f ⊗ ⋯ ⊗ x_{ℓ}⊗

(𝔪_{1}(f) − 𝔥_{2}(1\overline{⊗}1)) ⊗ x_{ℓ+1} ⊗ ⋯ ⊗ f ⊗ x_{k}),

where

ϵ_{ℓ} = deg′ x_{1} + ⋯ + deg′ x_{ℓ−1}.

To be more specific, (20.9.1) has the term involving 𝔪_{1}(f) extra and (20.9.2) has the term involving 𝔥_{2} extra than the other, which appear in the second term of (20.10). By compensating them, (20.9.1) becomes (20.10).

Let us calculate the second term of (20.10). We first remark that

𝔪_{1}(f) − 𝔥_{2}(1\overline{⊗}1) = e^{+} − e

and

(20.11) (−1)^{ϵ_{ℓ}}𝔪(x_{1}⊗f⊗⋯⊗x_{ℓ}⊗e⊗x_{ℓ+1}⊗⋯⊗f⊗x_{k}) = 𝔥_{k−1}∘𝔟_{k,ℓ}(x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k}).

We next study

(20.12)  (−1)^{ϵ_{ℓ}}𝔪(x_{1} ⊗ f ⊗ ⋯ ⊗ x_{ℓ} ⊗ e^{+} ⊗ x_{ℓ+1} ⊗ ⋯ ⊗ f ⊗ x_{k}).

By (20.5), the formula (20.12) is 0 unless

x_{1} ⊗ f ⊗ x_{2} ⊗ f ⊗ ⋯ ⊗ f ⊗ x_{k} = f ⊗ f, x ⊗ f or f ⊗ x,

where x ∈ B_{1}C[1] = C[1]. In the first case where x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k} = 1\overline{⊗}1\overline{⊗}1, we have :

∑_{ℓ=1}^{k−1}(−1)^{ϵ_{ℓ}}𝔪(x_{1} ⊗ f ⊗ ⋯ ⊗ x_{ℓ} ⊗ e^{+} ⊗ x_{ℓ+1} ⊗ ⋯ ⊗ f ⊗ x_{k})
= 𝔪_{2}(f ⊗ e^{+}) + 𝔪_{2}(e^{+} ⊗ f) = (−1)^{deg f}f + f = 0.

In the second case, where x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k} = x\overline{⊗}1, we have :

∑_{ℓ=1}^{k−1}(−1)^{ϵ_{ℓ}}𝔪(x_{1} ⊗ f ⊗ ⋯ ⊗ x_{ℓ} ⊗ e^{+} ⊗ x_{ℓ+1} ⊗ ⋯ ⊗ f ⊗ x_{k})
= (−1)^{deg x+1}𝔪_{2}(x ⊗ e^{+}) = −x = 𝔓_{2}(x\overline{⊗}1).

In the third case, where x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k} = 1\overline{⊗}x, we have :

∑_{ℓ=1}^{k−1}(−1)^{ϵ_{ℓ}}𝔪(x_{1} ⊗ f ⊗ ⋯ ⊗ x_{ℓ} ⊗ e^{+} ⊗ x_{ℓ+1} ⊗ ⋯ ⊗ f ⊗ x_{k})
= 𝔪_{2}(e^{+} ⊗ x) = x = 𝔓_{2}(1\overline{⊗}x).

Thus we verified

(20.13) ∑_{ℓ=1}^{k−1}(−1)^{ϵ_{ℓ}}𝔪(x_{1} ⊗f⊗⋯⊗x_{ℓ} ⊗e^{+} ⊗x_{ℓ+1} ⊗⋯⊗f⊗x_{k}) = 𝔓_{k}(x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k}).

(20.11) and (20.13) together with (20.2.1) imply that (20.10) is zero. The proof of Proposition 20.8 is now complete. □

We now explain the geometric background of Proposition 20.8. Let (X, x_{0}) be a pointed space. We assume that there exists a product structure · : X × X → X such that its restrictions to {x_{0}} × X and X × {x_{0}} are homotopic to identity. Let H_{1} : {x_{0}} × X × [0, 1] → X, H_{2} : X × {x_{0}} × [0, 1] → X be the homotopy. Namely

H_{1}(x_{0}, x, 1) = H_{2}(x, x_{0}, 1) = x,
H_{1}(x_{0}, x, 0) = x_{0} · x,
H_{2}(x, x_{0}, 0) = x · x_{0}.

In other words, x_{0} is a homotopy unit. We consider the disjoint union X ∪ [0, 1] and identify x_{0} and 0. Let X_{+} be the space obtained in this way. We extend · to · : X_{+} × X_{+} → X_{+} as follows : If s, s′ ∈ [0, 1] and x, x′ ∈ X then

\begin{aligned}s · x′ & = H_{1}(x_{0}, x′, s), \\ x · s′ & = H_{2}(x, x_{0}, s′). \end{aligned}

It follows that 1 · x = x · 1 = x. Namely 1 is a unit. The construction of C^{+} and the proof of Proposition 20.8 is an algebraic version of this construction.

Proof of Proposition 20.4. Let M_{k+1}(β) be the moduli space of pseudo-holomorphic discs with k+1 marked points on the boundary and of homotopy class β. We regard it as a space with Kuranishi structure.

For a⃗ = (a_{1}, ⋯, a_{|a⃗|}) ∈ {1, ⋯, k}^{|a⃗|} with a_{1} < ⋯ < a_{|a⃗|}, let

𝔣𝔬𝔯𝔤𝔢𝔱a⃗ : M_{k+1}(β) → M_{k+1−|a⃗|}(β)

be the map forgetting the a_{1} ⋯, a_{|a⃗|}-th marked points. We put 𝔣𝔬𝔯𝔤𝔢𝔱 = 𝔣𝔬𝔯𝔤𝔢𝔱_{(1,··· ,k+1)}. By construction, the Kuranishi structure is compatible with forgetful maps in the following sense.

Let x ∈ M_{k+1}(β), U_{x} = V_{x}/Γ_{x} be its Kuranishi neighborhood, E_{x} → U_{x} be its obstruction (orbi)bundle and s_{x} be its Kuranishi map. We denote \overline{x} = 𝔣𝔬𝔯𝔤𝔢𝔱a⃗(x) and define U\overline{x}, E\overline{x}, s\overline{x} in a similar way.

Note that the Kuranishi maps here are those before perturbation. Hence s_{x}^{−1}(0) is the neighborhood of x in M_{k+1}(β), (which may be of wrong dimension.) (We remark that the evaluation maps ev_{x} : U_{x} → L are of maximal rank by construction.) We also have

(20.14)  U_{x} ≃ U\overline{x} × ℝ^{|a⃗|},  E_{x} ≃ E\overline{x}.

Suppose we are given a perturbation (multi-section) s_{ϵ,\overline{x}} of s\overline{x} : U\overline{x} → e\overline{x} which is transverse to 0. The trouble we have been discussing so far is caused by the fact that we can not choose s_{x,a⃗}^{1} : U_{x} → E_{x} defined by

(20.15)  s_{x,a⃗}^{1}(y, r⃗) := s_{ϵ,\overline{x}}(y)

if we want the perturbed moduli spaces to become transverse to one another with respect to the evaluation maps. However, the perturbation s_{x,a⃗}^{1} as it is defined by (20.15) is transversal for the moduli space itself.

Let s_{x}^{0} be the multi-section we have chosen to define 𝔪_{k+1} in §19. The idea of the proof of Proposition 20.4 is to use a homotopy between these two multisections. Since there are several marked points to deal with, we use the homotopy parameterized by [0, 1]^{k}. Namely we use the following Lemma 20.16. For given a⃗, let a^{c} = (a_{1}^{c}, ⋯, a_{k−|a⃗|}^{c}) such that a_{i}^{c} < a_{i+1}^{c} and a⃗ ∪ a^{c} = {1, ⋯, k}.

Lemma 20.16. There exists a family of multi-sections s(t_{1}, ⋯, t_{k}) of M_{k+1}(β) parameterized by (t_{1}, ⋯, t_{k}) ∈ [0, 1]^{k} with the following properties.

(20.17.1) s(0, ⋯, 0) = s_{x}^{0}. (20.17.2) If t_{a_{i}} = 1 for i = 1, ⋯, |a⃗|, then s(t_{1}, ⋯, t_{k}) is induced from

s(t_{a_{1}^{c}}, ⋯, t_{a_{k−|a⃗|}^{c}}) by 𝔣𝔬𝔯𝔤𝔢𝔱a⃗.

Proof. The proof of Lemma 20.16 is by induction on k. If k = 1 we put

s_{x}^{t} = (1 − t)s_{x}^{0} + ts_{x,(1)}^{1}.

(See [FOn2] Definition 3.4 for the definition of sum of two multi-sections.) It is easy to see that s_{x}^{t} can be glued together and has required properties.

We assume that we have constructed s(t_{1}, ⋯, t_{j}) for j ≤ k − 1. Then requiring (20.17.1) and (20.17.2), s(t_{1}, ⋯, t_{k}) are determined for (t_{1}, ⋯, t_{k}) such that

∃i : t_{i} = 1 or ∀i : t_{i} = 0.

By using the induction hypothesis, we can check that they are consistent where two or more t_{i} are zero. Hence we can extend it to (t_{1}, ⋯, t_{k}) ∈ [0, 1]^{k} by using the partition of unity. □

We remark that the family of multisections thus constructed is compatible with the gluing map describing the boundaries. Since we need this compatibility later let us state it precisely here.

Let 1 ≤ ℓ < m ≤ k and β_{1} + β_{2} = β. We define a map

gl_{m,ℓ,k,β_{1},β_{2}} : M_{k−m+ℓ+2}(β_{1}) × M_{m−ℓ}(β_{2}) → M_{k+1}(β)

which is a homeomorphism onto the boundary component of M_{k+1}(β) described by the figure below.

Figure 20.1

Moreover this map is compatible with the Kuranishi structure in the following way. Let 𝔵_{1} ∈ M_{k−m+ℓ+2}(β_{1}), 𝔵_{2} ∈ M_{m−ℓ}(β_{2}), and 𝔵 = gl_{m,ℓ,k,β_{1},β_{2}}(𝔵_{1}, 𝔵_{2}). Then there exist Kuranishi neighborhoods (U_{1}, E_{1}, s_{1}) and (U_{2}, E_{2}, s_{2}),(U, E, s) of 𝔵_{1}, 𝔵_{2} and 𝔵 respectively and a smooth map gl : U_{1} × U_{2} → U and a bundle isomorphism I : E_{1} ⊕ E_{2} → gl^{∗}E such that I ∘ (s_{1}, s_{2}) = s ∘ gl.

Now let us consider t⃗ = (t_{1}, ⋯, t_{k}). We put t⃗_{(1)} = (t_{1}, ⋯, t_{ℓ−1}, 0, t_{m+1}, ⋯, t_{k}), t⃗_{(2)} = (t_{ℓ}, ⋯, t_{m}). Now the compatibility required is

(20.17.3)  s(t⃗)(gl(𝔶_{1}, 𝔶_{2})) = I(s(t⃗_{(1)})(𝔶_{1}), s(t⃗_{(2)})(𝔶_{2})).

(20.17.3) can be checked easily from construction.

We may also assume that the family of multi-sections s(t_{1}, ⋯, t_{k}) satisfies various transversality properties we need, by perturbing it if necessary.

Now, we are ready to define 𝔥_{k+1}. We start with 𝔥_{2,β_{0}} which will be the first step of the inductive construction of 𝔥_{k}. Note the moduli space we use to define 𝔥_{2,β_{0}} is M_{3}(L; β_{0})_{(1)} and M_{3}(L; β_{0})_{(2)}. We recall that M_{3}(L; β_{0}) = L as a set. However the evaluation map is not the identity but is

(ev_{0}, ev_{1}, ev_{2})(x) = (x, φ_{0}(x), φ_{1}(x)),

where φ_{0}, φ_{1} : L → L are as in the proof of Theorem 19.6. Let φ_{i}^{t} : L → L a small isotopy such that φ_{i}^{1}(x) = x for each i, φ_{i}^{0}(x) = φ_{i}(x). We define M_{3}(L; β_{0})_{(1)} = L × [0, 1] together with the evaluation map

(20.18.1)  (ev_{0}, ev_{1}, ev_{2})(x, t) = (x, φ_{0}^{t}(x), φ_{1}(x)).

Similarly M_{3}(L; β_{0})_{(2)} = L × [0, 1] together with

(20.18.2)  (ev_{0}, ev_{1}, ev_{2})(x, t) = (x, φ_{0}(x), φ_{1}^{t}(x)).

We then define

\begin{aligned}𝔥_{2,β_{0}}(x\overline{⊗}1) & = (M_{3}(L; β_{0})_{(1)} × (x × L), ev_{0}), \\ 𝔥_{2,β_{0}}(1\overline{⊗}x) & = (M_{3}(L; β_{0})_{(2)} × (L × x), ev_{0}). \end{aligned}

Remark 20.19. The reader may want to put M_{3}(L; β_{0})_{(1,2)} = L × [0, 1]^{2} with

(ev_{0}, ev_{1}, ev_{2})(x, t, s) = (x, φ_{0}^{t}(x), φ_{1}^{s}(x))

and

𝔥_{3,β_{0}}(1\overline{⊗}1\overline{⊗}1) = M_{3}(L; β_{0})_{(1,2)} × (L × L).

However the right hand side actually vanishes as a current since the fiber product is of degenerate one. So we actually do not need it and can put

𝔥_{3,β_{0}}(1\overline{⊗}1\overline{⊗}1) = 0.

Now we consider the case where either k ≥ 3 or k = 2, β ≠ 0. We will do this by induction on k and on the energy of β.

Let x_{i} ∈ B_{j_{i}}C[1], x_{i} = x_{i,1} ⊗ ⋯ ⊗ x_{i,j_{i}} and put j_{1} + ⋯ + j_{k+1} = m, k ≥ 0. We put b⃗ = (j_{1} + 1, j_{1} + j_{2} + 2, ⋯, j_{1} + ⋯ + j_{k} + k), a⃗ = {1, ⋯, k + m} − b⃗. We consider elements (((Σ, z⃗), φ), t_{1}, ⋯, t_{m+k}) with the following properties.

(20.20.1) ((Σ, z⃗), φ) is in a Kuranishi neighborhood of M_{k+1}(L; β). (20.20.2) t_{a_{ℓ}} = 0, ℓ = 1, 2, ⋯, |a⃗|. (20.20.3) (s(t_{1}, ⋯, t_{m+k}))((Σ, z⃗), φ) = 0.

We denote by M_{k+m+1}(L; β)b⃗ the set of all such (((Σ, z⃗), φ), t_{1}, ⋯, t_{m+k}). Let ev : M_{k+m+1}(β)b⃗ → L^{k+m+1} be the evaluation map. We may choose the multisection s(t_{1}, ⋯, t_{m+k}) so that (M_{k+m+1}(β)b⃗, ev) is a ℚ-chain of dimension k+m+ 1 + k + μ(β) + dim L. We define

(20.21)  \begin{matrix}𝔥_{k+1,β}(x_{1}, & ⋯, x_{k+1}) \\ = & (M_{k+m+1}(β)b⃗ ×_{L^{k+m}} (x_{1,1} × ⋯ x_{1,j_{1}} × L \\ × x_{2,1} × ⋯ × x_{2,j_{2}} × L × ⋯ × x_{k+1,j_{k+1}}), ev_{0}). \end{matrix}

Then we set 𝔥_{k+1} = ∑_{β} 𝔥_{k+1,β} ⊗ [β].

Remark 20.22. Actually we need to put a sign in the definition of (20.21) in a way similar to other cases which will be discussed in Chapter 6. In this preprint version, we do not discuss this point in detail. Details of the argument for choosing the correct sign in the proof of Proposition 20.4 will be included in the final (published) version of this book. (All other sign problems will be discussed in detail in Chapter 6 except this case.)

We have finished the definition of 𝔥_{k}. The formula 𝔥_{1} = 𝔪 immediately follows from (20.17.1).

We now verify properties (20.2.1), (20.2.2). Consider the boundary of the chain 𝔥_{k+1,β}(x_{1}, ⋯, x_{k+1}). The boundary of M_{k+m+1}(L; β)b⃗ is described by the following 3 cases.

Case 1 t_{b_{i}} goes to 1. Case 2 t_{b_{i}} goes to 0. Case 3 The domain splits into two discs.

The boundary appearing in Case 1 is actually 0 as a cochain (current) in case 𝔓 = 0. This is because the multi-section is compatible with the forgetful map of b_{i}-th marked point if t_{b_{i}} = 1, and we take fiber product with L using the b_{i} th evaluation map in the definition of 𝔥_{k+1,β}(x_{1}, ⋯, x_{k+1}) and hence we can apply the argument of the proof of Proposition 7.6.

In case when 𝔓 ≠ 0, x_{1}\overline{⊗} ⋯ \overline{⊗}x_{k+1} = x\overline{⊗}1 or 1\overline{⊗}x. Hence (20.20) and φ_{i}^{0} = id implies that Case 1 gives −x = 𝔓_{2}(x\overline{⊗}1) or x = 𝔓_{2}(1\overline{⊗}x).

Case 2 gives −𝔥_{k} ∘ 𝔟_{k+1,i} by definition. (See (20.18).)

In a way similar to the argument of Chapter 2, we can show that Case 3 together with (−1)^{n}∂(𝔥_{k+1,β}(x_{1}, ⋯, x_{k+1})), after summed over β ∈ π_{2}(M, L), gives ∑_{ℓ=1}^{k+1} 𝔥_{ℓ} ∘ 𝔥̂_{k−ℓ+2}. (Recall that ∂ = (−1)^{n}𝔪_{1,β_{0}}.) (We use (20.17.3) here.)

We have thus verified (20.2). The proof of Proposition 20.4 is complete. □

We next show that the construction so far justifies the proof of Theorem 10.6 in §10. In the proof of Theorem 10.6, we used the fact that Floer cohomology HF (L, L) is nontrivial if the Maslov index is zero. We will prove that, for any bounding cochain b of (C(L), 𝔪), we still have HF ((L, b), (L, b)) ≠ 0 even if e = [L] is not a unit but a homotopy unit.

We first need some remarks. One is that in Theorem 10.6, we do not assume L is oriented. (Then we work over ℤ_{2} coefficient.) The discussion above in this section works in this situation. (Once the problem constructing fundamental chain over ℤ_{2} is understood as in section A3, it becomes easier to work over ℤ_{2} coefficient, since we do not need to worry about sign.) We introduce a formal parameter \overline{e} whose degree is 1 in place of e to deal with the case when L is unoriented.

The second remark is as follows. We consider

x = ∑ x_{i}T^{λ_{i}} \overline{e}^{n_{i}}.

We denote by gdeg x_{i} the geometric degree of x_{i} which is nothing but a degree as a cochain. (The degree of x_{i}T^{λ_{i}} \overline{e}^{n_{i}} is n_{i} + gdeg x_{i} and is different from its geometric degree.) Let b = ∑ b_{i}T^{λ_{i}} \overline{e}^{n_{i}} be a bounding chain. As a bounding cochain we considered such b with condition

(20.23)  gdeg b_{i} + n_{i} = 1

in Chapter 4. In Chapters 2 and 3, we considered b = ∑ b_{i}T^{λ_{i}} \overline{e}^{n_{i}} such that there exists β_{i} ∈ π_{2}(M; L) with the following properties in addition.

(20.24.1) gdeg b_{i} = 1 − μ_{L}(β_{i}), (20.24.2) n_{i} = μ_{L}(β_{i})

(20.24.3) β_{i} is contained in the monoid generated by β such that M(β) is nonempty.

This difference may look rather technical. However the authors do not know whether the existence of b under additional assumption (20.24) is equivalent to the existence of b without it in our geometric situation.

We consider the following condition :

(20.25) For the bounding cochain we consider only those which satisfy (20.24.1) for some β_{i} ∈ π_{2}(M, L).

Requiring (20.25) is a bit weaker condition than requiring (20.24). The unobstructedness assumption in Theorem 10.6 and in Corollary 10.8 are to be understood as the existence of bounding cochain b satisfying (20.25). This assumption follows from vanishing of the sequence of obstruction classes defined in Chapter 2, since in that case we obtain b satisfying the stronger condition (20.24). In particular Theorem G follows from Theorem 10.8 if we take the unobstructedness assumption in this sense.

Now we go back to the proof of Theorem 10.6. Let b be a bounding cochain satisfying Condition (20.25). By Proposition 20.8, b is also a bounding cochain of (C(L)^{+}, 𝔪). Then e^{+} is a δ_{b} cocycle since it is a unit. Since e^{+} = e−𝔥_{2}(1\overline{⊗}1)+𝔪_{1}(f), it follows

δ_{b}(f) = e^{+} − e + 𝔥_{2}(e^{b}\overline{⊗}e^{b}).

Hence we have

δ_{b}(e − 𝔥_{2}(e^{b}\overline{⊗}e^{b})) = 0.

We next prove that e − 𝔥_{2}(e^{b}\overline{⊗}e^{b}) is not a δ_{b}-coboundary. We need to modify the proof of Theorem 10.6 mentioned in §10 a bit here, since 𝔥_{2} is of negative degree. Let π_{1} : C(L; Λ_{nov}) → C^{0}(L; ℚ) and π_{2} : C^{0}(L; ℚ) → H^{0}(L; ℚ) ≃ ℚ be the projections and π : C(L; Λ_{nov}) → ℚ be the composition of them. (π(e) = 1 by definition.) We put b = ∑ b_{i}T^{λ_{i}} \overline{e}^{n_{i}}.

Lemma 20.26. If Im μ_{L} ∩ {1, ⋯, n − 1} = ∅, and if b is a bounding cochain satisfying (20.25), then gdeg b_{i} = 1.

Proof. We remark that the assumption implies Im μ_{L}∩{−1, ⋯, 1−n} = ∅. Suppose the contrary that gdeg b_{i} = 1 − μ_{L}(β_{i}) ≠ 1. Then by assumption, we have 1 − gdeg b_{i} ∉ {−n + 1, ⋯, n − 1}. Since the degree of current on L is between 0 and n, this is a contradiction. □

Lemma 20.27. If 2 ∉ Im μ_{L} and if b is a bounding cochain satisfying (20.25), then we have π(𝔥_{2}(e^{b}\overline{⊗}e^{b})) = 0.

Proof. Let us study

(20.28)  𝔥_{2,β}((b_{i_{1}} ⊗ ⋯ ⊗ b_{i_{k}})\overline{⊗}(b_{j_{1}} ⊗ ⋯ ⊗ b_{j_{ℓ}})).

Since gdeg b_{i} = 1 it follows that the geometric degree of (20.28) is 2 − μ_{L}(β). By the assumption that Σ ∉ Im μ_{L}, we have 2 − μ_{L}(β) ≠ 0 and so the projection π(𝔥_{2}(e^{b}\overline{⊗}e^{b})) = 0. □

Now we are in the position to complete the proof of Theorem 10.6. We remark the assumption of Theorem 10.6 implies that the assumptions 2 ∉ Im μ_{L}, Im μ_{L} ∩ {1, ⋯, n − 1} = ∅ of Lemmas 20.26 and 20.27 are satisfied.

We can then use Lemma 20.27 in a similar way to the proof of Theorem 10.6 in §10 to show that e − 𝔥_{2}(e^{b}\overline{⊗}e^{b}) remains to be nonzero in the E_{∞} term of the spectral sequence calculating HF ((L, b), (L, b)). Therefore HF ((L, b), (L, b)) ≠ 0 as required. We have thus completed the proof of Theorem 10.6. □

Remark 20.29. We remark that if b ≠ b′ then HF ((L, b), (L, b′)) may be zero even in the case when Maslov index is zero. In fact, we find that

δ_{b,b′}(e^{+}) = 𝔪(e^{b}\overline{⊗}e^{+}\overline{⊗}e^{b′}) = b − b′.

Hence e^{+} is not a δ_{b,b′} cocycle if b ≠ b′. More explicitly, let us consider the two parallel (nontrivial) circles S_{1}^{1}, S_{2}^{1} in T^{2}. They do not intersect each other. Hence the Floer cohomology HF ((S_{1}^{1}, 0), (S_{2}^{1}, 0)) vanishes. We can find an appropriate bounding cochain b_{2,1} (i.e. a cocycle in this case since the obstruction vanishes) such that we have an isomorphism HF ((S_{1}^{1}, b_{2,1}), (L, b)) ≃ HF ((S_{2}^{1}, 0), (L, b)). (The cohomology class of b_{2,1} is determined by the area of the annulus between S_{1}^{1} and S_{2}^{1}.) Thus in case when b−b′ is obtained from a codimension one cycle, HF ((L, b), (L, b′)) is related to the Novikov homology [N].

We next justify the proof of Proposition 7.1 in §7. Using the terminology so far, we can re-state the proposition as follows.

Proposition 20.30. Let b ∈ C^{+}[1]^{0} = C^{+,1}. We assume 𝔪(e^{b}) = ce^{+}, where c is an element of the universal Novikov ring. Then δ_{b} ∘ δ_{b} = 0.

Proof. We have d̂d̂(e^{b}xe^{b}) = 0. Therefore

d̂(ce^{b}e^{+}e^{b}xe^{b}) + (−1)^{deg x+1}d̂(ce^{b}xe^{b}e^{+}e^{b}) + d̂(e^{b}δ_{b}(x)e^{b}) = 0.

Using the fact that e^{+} is a unit the sum of the first two terms are zero. Hence δ_{b} ∘ δ_{b} = 0 as required. □

We remark that e^{+} − e + higher = 𝔪_{1}(f), where higher means higher order term with respect to the energy. Hence it is easy to see that under the assumption of Proposition 7.1 we can find b ∈ C^{+}[1]^{0} = C^{+,1} such that 𝔪(e^{b}) = ce^{+}. Therefore Proposition 20.30 implies Proposition 7.1. □

We next turn to the proof of Theorem 14.5, that is the verification of Condition 14.3. For this purpose, we need to define the homotopy unit version of Condition 14.3, to prove it for C(L), and to use it to verify Condition 14.3 for C(L)^{+}. The construction is similar to what we already discussed so we will explain it only briefly.

Let (C_{1}, 𝔪^{1}), (C_{2}, 𝔪^{2}) be A_{∞} algebras with homotopy units e_{1}, e_{2} respectively. We define (C_{1}^{+}, 𝔪^{1}), (C_{2}^{+}, 𝔪^{2}) as above. Let D be a graded free filtered left Λ_{nov} module. We assume that there exists 𝔫 which defines a left (C_{1}^{+}, 𝔪^{1}) and right (C_{2}^{+}, 𝔪^{2}) A_{∞} bimodule structure on D. We do not assume Condition 14.3. Let 𝔥_{k}^{1} : B_{k}(B(C_{1}[1])) → B(C_{1}[1]), 𝔥_{k}^{2} : B_{k}(B(C_{2}[1])) → B(C_{2}[1]) be the operators that appeared in Definition 20.1.

We consider operators

𝔤_{k,ℓ} : B_{k}(B(C_{1}[1])) ˜⊗D ˜⊗B_{ℓ}(B(C_{2}[1])) → D.

Here we use ˜⊗ to distinguish it from ⊗ and \overline{⊗}. Using it and 𝔥_{ℓ}^{1}, 𝔥_{k}^{2} in a similar way to the definition of 𝔥̂_{k}, we define

𝔤̂_{k,ℓ}^{m,n} : B_{m}(B(C_{1}[1])) ˜⊗D ˜⊗B_{n}(B(C_{2}[1]))
→ B_{m−k+1}(B(C_{1}[1])) ˜⊗D ˜⊗B_{n−ℓ+1}(B(C_{2}[1])).

(Here m + n ≥ 0.) We put 𝔤̂_{k,ℓ} = ⊕_{m,n}𝔤̂_{k,ℓ}^{m,n} and 𝔤̂ = ∑ 𝔤̂_{k,ℓ}.

We assume 𝔤_{0,0} = 𝔫 . Hence 𝔤̂_{0,0} = d̂, where d̂ is defined by (14.2).

We next define

𝔟_{m,i}^{1} : B_{m}(B(C_{1}[1])) ˜⊗D ˜⊗B_{ℓ}(B(C_{2}[1])) → B_{m−1}(B(C_{1}[1])) ˜⊗D ˜⊗B_{ℓ}(B(C_{2}[1]))

by 𝔟_{m,i}^{1} = 𝔟_{m,i} ˜⊗id ˜⊗id and define

𝔟_{ℓ,i}^{2} : B_{m}(B(C_{1}[1])) ˜⊗D ˜⊗B_{ℓ}(B(C_{2}[1])) → B_{m}(B(C_{1}[1])) ˜⊗D ˜⊗B_{ℓ−1}(B(C_{2}[1]))

by

𝔟_{ℓ,i}^{2}(𝔵_{1} ˜⊗v ˜⊗𝔵_{2}) = (−1)^{deg′ 𝔵_{1}+deg v+1}𝔵_{1} ˜⊗v ˜⊗𝔟_{ℓ,i}(𝔵_{2}).

Definition 20.31. We say that {e_{1}, e_{2}} acts as a homotopy unit on D, if there exists 𝔤_{k,ℓ} such that

(20.32.1) ∑_{a,b} 𝔤_{a,b} ∘ 𝔤̂_{k−a+1,ℓ−b+1} − ∑_{i} 𝔤_{k−1,ℓ} ∘ 𝔟_{k,i}^{1} − ∑_{j} 𝔤_{k,ℓ−1} ∘ 𝔟_{ℓ,j}^{2} + 𝔓_{k,ℓ} = 0

Here 𝔓_{k,ℓ} = 0 if (k, ℓ) ≠ (0, 2), (2, 0).

𝔓_{2,0} = 0 also holds on the complement of

(B_{0}(C_{1}[1])\overline{⊗}B_{0}(C_{1}[1])) ˜⊗D ˜⊗B_{0}(B(C_{2}[1])))

in B_{2}(C_{1}[1]) ˜⊗D ˜⊗B_{0}(C_{1}[1])) and

(20.32.2)  𝔓_{2,0}((1\overline{⊗}1) ˜⊗v ˜⊗1) = v.

𝔓_{0,2} = 0 also holds on the complement of

B_{0}(B(C_{1}[1]) ˜⊗D ˜⊗(B_{0}(C_{1}[1])\overline{⊗}B_{0}(C_{1}[1]))

in B_{0}(C_{1}[1]) ˜⊗D ˜⊗B_{2}(C_{1}[1])) and

(20.32.3)  𝔓_{0,2}(1 ˜⊗v ˜⊗(1\overline{⊗}1)) = −v.

Let L_{1}, L_{2} be Lagrangian submanifolds intersecting transversely. We put

C(L_{1}, L_{2}; Λ_{nov}) = ⨁_{p∈L_{1}∩L_{2}} Λ_{nov}[p].

Then, there exists 𝔫 defining A_{∞} bimodule structure, as we already proved in §14 and this chapter.

We know from Proposition 20.4 that [L_{1}], [L_{2}] are homotopy units of A_{∞} algebras (C(L_{1}), 𝔪^{1}), (C(L_{2}), 𝔪^{2}) respectively.

Proposition 20.33. {[L_{1}], [L_{2}]} acts as a homotopy unit on C(L_{1}, L_{2}; Λ_{nov}).

The proof is quite similar to that of Proposition 20.4 and is omitted.

Proposition 20.34. Let (C_{1}, 𝔪^{1}), (C_{2}, 𝔪^{2}) be A_{∞} algebras with homotopy units e_{1}, e_{2} and D be a graded free filtered left Λ_{nov} module. We assume that there exists 𝔫 defining A_{∞} bimodule structure on D. Moreover we assume that {e_{1}, e_{2}} acts as a homotopy unit on D.

Then 𝔫 can be extended to C_{1}^{+}, and C_{2}^{+} so that {e_{1}^{+}, e_{2}^{+}} acts as a unit.

The proof is quite similar to the proof of Proposition 20.8 and is omitted. Propositions 20.33 and 20.34 justify the construction of §14.

Remark 20.35. We remark that the analog of Proposition 20.30 does not hold in this situation where L_{1} ≠ L_{2}. Namely let b_{1} ∈ C^{1}[1], b_{2} ∈ C^{2}[1] be elements of degree 0 (after shifted). We assume 𝔪_{1}(e^{b_{1}}) = ce_{1}, 𝔪_{2}(e^{b_{2}}) = c′e_{2}. But δ_{b_{1},b_{2}} is not a coboundary operator in general. (Namely δ_{b_{1},b_{2}} ∘ δ_{b_{1},b_{2}} ≠ 0.)

To see this, we assume that e_{i} are not only homotopy units but also units, for simplicity. Let

π : B(C_{1}[1]) ˜⊗D ˜⊗B(C_{2}[1]) → B_{0}(C_{1}[1]) ˜⊗D ˜⊗B_{0}(C_{2}[1]) ≃ D

be the projection. Then we have

0 = π(δ_{b_{1},b_{2}} ∘ δ_{b_{1},b_{2}}(v)) = π(d̂(e^{b_{1}} d̂(e^{b_{1}}ve^{b_{2}})e^{b_{1}})) = (c − c′)v + δ_{b_{1},b_{2}} ∘ δ_{b_{1},b_{2}}(v).

Thus the condition for Floer cohomology HF (L_{1}, L_{2}) to be well-defined is more restrictive than the one for HF (L, L) to be well-defined. This fact was already pointed out by the second named author in [Oh2,5] in the case of monotone Lagrangian submanifolds.

Chapter 6. Orientation.

§21. Orientation on the moduli space of unmarked pseudo-holomorphic discs.

21.1. The case of holomorphic discs.

Let (M, ω) be a symplectic manifold and L a Lagrangian submanifold. We pick an almost complex structure J compatible with ω. Then L is a totally real submanifold, namely J(T L) ∩ T L is the zero section of T M|_{L}. In this section, we consider an orientation on the moduli space of pseudo-holomorphic discs, w : (D^{2}, ∂D^{2}) → (M, L) with totally real boundary condition. First of all, we should note that the moduli space of pseudo-holomorphic discs is not always orientable. This is also observed by de Silva [Sil] independently. In the case of pseudo-holomorphic curves without boundary, it is well known that it has a canonical orientation. In this chapter, we put an assumption on the second Stiefel-Whitney class w_{2}(T L) in order to consider the orientation problem. Our result is the following:

Theorem 21.1. The moduli space of pseudo-holomorphic discs is orientable, if L ⊂ (M, ω) is relatively spin Lagrangian submanifold. Furthermore the choice of relative spin structure on L determines an orientation on M(L; β) canonically for all β ∈ π_{2}(P, L).

Remark 21.2. In particular, if L is a spin manifold, the assumption holds. A spin structure of L gives an orientation on the moduli space of pseudo-holomorphic discs. To give an orientation only under the relative spinness as in the theorem, we need to use the notion of relative spin structure which we introduce in this paper (see Definition 21.5).

Proof: It suffices to show that the index of the linearized operator is oriented. The linearized operator D\overline{∂} for the pseudo-holomorphic curve equation is the first order elliptic differential operator with the same symbol as the Dolbeault operator:

D\overline{∂}_{w} : W^{1,p}(D^{2}, ∂D^{2}; w^{∗}T M, ℓ^{∗}T L) → L^{p}(D^{2}; w^{∗}T M).

Here ℓ is the restriction of w to the boundary ∂D^{2} and we take p as p > 2. Since the zero-th order term does not affect the index problem, we assume that the operator is the Dolbeault operator \overline{∂}_{w}

\overline{∂}_{(w^{∗}T M,l^{∗}T L)} : W^{1,p}(D^{2}, ∂D^{2}; w^{∗}T M, ℓ^{∗}T L) → L^{p}(D^{2}; w^{∗}T M).

Firstly, we study a pointwise orientation of the index bundle.

Proposition 21.3. Let E be a complex vector bundle over the 2-disc D^{2} and F a totally real subbundle of E|_{∂D^{2}} over ∂D^{2}. We denote by \overline{∂}_{(E,F)} the Dolbeault operator on D^{2} with coefficient (E, F),

\overline{∂}_{(E,F)} : W^{1,p}(D^{2}, ∂D^{2}; E, F) → L^{p}(D^{2}; E).

Assume F is trivial. Take a trivialization of F over ∂D^{2}. Then the trivialization gives an orientation of the virtual vector space Ker \overline{∂}_{(E,F)} − Coker \overline{∂}_{(E,F)}.

Remark 21.4. In our situation, we apply the proposition by putting E = w^{∗}T M and F = ℓ^{∗}T L which is trivial because L is orientable.

Proof of Proposition 21.3: Suppose that the operators are surjective. (Otherwise we consider a quotient of the range by a finite dimensional complex subspace, which carries the complex orientation.) By deforming the Hermitian connection, we may assume that the totally real subbundle F is parallel (especially, trivially flat) and the connection is “product” in a collar neighborhood of ∂D^{2}. Let C be a concentric circle in the collar neighborhood of ∂D^{2}. If we pinch C to a point, we have the union of a 2-disc D^{2} and a 2-sphere ℂP^{1} with the center O ∈ D^{2} and S ∈ ℂP^{1} identified. By the parallel translation along radials, the trivial vector bundle F extends up to C and its complexification gives a trivialization of E|_{C}. Thus the bundle E descends to D^{2} ∪ ℂP^{1}. We also denote this vector bundle by E.

We compare the indices of the following two operators:

\overline{∂}_{(E,F)} : W^{1,p}(D^{2}, ∂D^{2}; E, F) → L^{p}(D^{2}; E ⊗ T_{0,1}^{∗}D^{2})

and

\overline{∂} : \{(ξ_{0}, ξ_{1}) ∈ W^{1,p}(D^{2}, ∂D^{2}; E, F) × W^{1,p}(ℂP^{1}, E) | ξ_{0}(O) = ξ_{1}(S)\}
→ L^{p}(D^{2}, E ⊗ T_{0,1}^{∗}D^{2}) × L^{p}(ℂP^{1}, E ⊗ T_{0,1}^{∗}ℂP^{1}).

We follow the argument in Appendix A [MS]. Choose small coordinate neighborhoods D_{δ}^{2}(O) and D_{δ}^{2}(S) of O and S. Here D_{δ}^{2} is the disc of radius δ. Fix a positive real number r. We glue D^{2} and ℂP^{1} around O and S by identifying z ∈ D_{δ}^{2}(O) and w ∈ D_{δ}^{2}(S) whenever zw = 1/r. We denote the resulting Riemann surface by X_{r}, which is biholomorphic to the unit disc. Suppose that the operator on D^{2} ∪ ℂP^{1} is also surjective. (Otherwise we consider a quotient of the range by a finite dimensional complex subspace, which is spanned by sections supported away from O and S.) Suppose that χ_{r} is a sequence of solutions \overline{∂}_{(E,F)}χ_{r} = 0 on X_{r}. Then we may choose a subsequence of χ_{r} which converges to a solution (ξ_{0}, ξ_{1}) on D^{2} ∪ ℂP^{1}. Note that if the L^{p}-norm of χ_{r} is 1 independent of r, then (ξ_{0}, ξ_{1}) ≠ (0, 0). Because the L^{p}-norm on the neck region D_{δ}^{2}(O) ∪ D_{δ}^{2}(S) ⊂ X_{r} is uniformly dominated by the L^{p}-norm on D_{δ}^{2}(O) − D_{δ/2}^{2}(O) and D_{δ}^{2}(S) − D_{δ/2}^{2}(S), it is not possible that the L^{p}-norm concentrated in the neck C.

On the other hand, we glue solutions ξ_{0} and ξ_{1} to a solution χ_{r} for a sufficiently large r. The argument is same as Appendix A [MS]. Hence we have an isomorphism (canonical up to homotopy) of the indices.

Since the real vector bundle F is trivialized, the kernel of this operator is the kernel of the homomorphism:

(ξ_{0}, ξ_{1}) ∈ Hol(D^{2}, ∂D^{2}; ℂ^{n}, ℝ^{n}) × Hol(ℂP^{1}; E) ↦ ξ_{0}(O) − ξ_{1}(S) ∈ ℂ^{n} ≃ E_{S}.

Note that the orientation of F ≃ ℝ^{n} gives the orientation on Hol(D^{2}, ∂D^{2}; ℂ^{n}, ℝ^{n}) ≃ ℝ^{n} via the evaluation map. Since Hol(ℂP^{1}; E) and ℂ^{n} have canonical complex orientations and we assumed the Dolbeault operator is surjective, the kernel inherits the orientation. This proves Proposition 21.3. □

Now let us fix a map w_{0} and an orientation on

Ker \overline{∂}_{(w_{0}^{∗}T M,l_{0}^{∗}T L)} − Coker \overline{∂}_{(w_{0}^{∗}T M,l_{0}^{∗}T L)}

as in Proposition 21.3. For any given element w in the path component of the moduli space containing w_{0}, we consider a path w_{t} : (D^{2}, ∂D^{2}) → (M, L) for t ∈ [0, 1] starting from w_{0} and ending at w. By Proposition 21.3 and since [0,1] is contractible, we can provide an orientation of the virtual vector space

Ker \overline{∂}_{(w_{t}^{∗}T M,l_{t}^{∗}T L)} − Coker \overline{∂}_{(w_{t}^{∗}T M,l_{t}^{∗}T L)}

continuously over t ∈ [0, 1], by taking a trivialization of ℓ_{t}^{∗}T L over S^{1} × [0, 1] where ℓ_{t} = w_{t}|_{∂D^{2}}. This will define an orientation for w.

In order to prove Theorem 21.1, we need to prove that this assignment is independent of the choice of path {w_{t}}_{t∈[0,1]}. Here enters the relative spinness of on L ⊂ M. By the definition of relative spinness below, there exists a cohomology class st ∈ H^{2}(M; ℤ_{2}) such that st|_{L} = w_{2}(T L). Given a triangulation of M, there exists an oriented real vector bundle V on the 3-skeleton M^{(3)} of M with w_{2}(V) = st|_{M^{(3)}}. (Here we first triangulate L and then extend it to M.) We choose a spin structure on T L ⊕ V on M^{(3)} ∩ L, which is spin by the choice of V.

Let w_{t} : (D^{2}, ∂D^{2}) → (M, L) be a loop in the space of pseudo-holomorphic discs and define the map

Φ : D^{2} × S^{1} → M; Φ(z, t) = w_{t}(z).

By the simplicial approximation theorem, Φ can be homotoped to a map to the 3-skeleton M^{(3)} whose boundary is mapped into M^{(2)} ∩ L = L^{(2)}. We pull back V to a vector bundle over D^{2} × S^{1},which we denote by Φ^{∗}V. The spin structure chosen on T L ⊕ V |_{L^{(2)}} above gives a stable trivialization on M^{(2)} ∩ L which induces a (stable) trivialization on T L⊕Φ^{∗}V |_{Φ(∂D^{2}×S^{1})}. On the other hand, Φ^{∗}V is defined on D^{2} × S^{1} ≃ S^{1} which is also oriented and so trivial. Furthermore the orientation on Φ^{∗}V also induces a unique homotopy class of trivialization on D^{2} × {t} (which is independent of the choice of the spin structure on T L ⊕ V |_{M^{(3)}∩L}). Combining these, we have a stable trivialization of (Φ|_{∂D^{2}×{t}})^{∗}T L which is continuous over t ∈ S^{1}.

This implies that for any w, ℓ^{∗}T L with ℓ = w|_{∂D^{2}} can be given a consistent stable trivialization. This completes the proof of Theorem 21.1 □

From the above proof, the choice of orientation on L, st ∈ H^{2}(M, ℤ_{2}) and the vector bundle V on M^{(3)} and the spin structure T L ⊕ V |_{M^{(3)}∩L} canonically determine the orientations on M(L; β) for all β. Note that existence of V on M^{(3)} is automatic under the assumption that st|_{L} = w_{2}(T L).

Definition 21.5. The choice of an orientation of L, st ∈ H^{2}(M; ℤ_{2}) and a spin structure on T L ⊕ V |_{L^{(2)}} is called a relative spin structure of the Lagrangian submanifold L ⊂ M. A pair of Lagrangian submanifolds L, L′ is called relative spin, if the class st satisfies st|_{L} = w_{2}(T L) and st|_{L′} = w_{2}(T L′) simultaneously. A relative spin structure of the pair L, L′ is the choice of orientations of L and L′, st ∈ H^{2}(M; ℤ_{2}) and spin structures on T L ⊕ V |_{L^{(2)}} and T L′ ⊕ V |_{L^{′(2)}}.

Remark 21.6. If we change the homotopy class of a stable trivialization of ℓ^{∗}T L, the orientation on the index bundle changes. Take the trivial bundle (D^{2} × [0, 1]) × ℂ^{n}. Identifying D^{2} × {0} and D^{2} × {1}, we get D^{2} × S^{1}. We lift this identification to the vector bundle such that its restriction to ∂D^{2} × {i} preserves the real part ∂D^{2} × ℝ^{n} and identify them by homotopically non-trivial loop γ : ∂D^{2} → SO(n). Such identification exists, because the image of the loop by SO(n) → U(n) is null-homotopic and extends to a continuous map D^{2} → U(n). Then Proposition 21.7 below implies that the index bundle is not orientable . This means that the change of a stable trivialization by γ reverses the orientation on the moduli space of holomorphic discs.

21.2. Examples of non-orientable family index

Another description of the resulting vector bundle E (for n = 2) over D^{2} × S^{1} is the following. Take a trivial vector bundle (D^{2} × [0, 1]) × ℍ, where ℍ is the quaternions considered as a left ℂ-module. Identify the fibers over D^{2} × {0} and D^{2} × {1} by multiplication of i ∈ ℍ^{∗}, that is, (z, 0, ζ) ∼ (z, 1, ζi) for z ∈ D^{2} and ζ ∈ ℍ. For (z, t) ∈ ∂D^{2} × [0, 1], the subspace

(ℝ1 ⊕ ℝj)\{t(\frac{z + \overline{z}}{2} + \frac{z − \overline{z}}{2i} j) + (1 − t)k\}

gives a totally real subspace in ℍ, hence we have a totally real subbundle over ∂D^{2} × [0, 1] which descends to a totally real subbundle F over ∂D^{2} × S^{1}.

Proposition 21.7. The family index of Dolbeault operators twisted by (E, F) is a non-orientable virtual vector bundle over S^{1}.

Proof: By using the Fourier expansion, we calculate the kernels and cokernels of Dolbeault operators explicitly. We take a basis ⟨1+k, 1−k⟩ of the left ℂ-module ℍ.

By regarding ℍ as ℂ⊕ℂj, we write the basis as column vectors ⟨( \begin{matrix}1 \\ i \end{matrix} ) , ( \begin{matrix}1 \\ −i \end{matrix} )⟩. Let

f be a holomorphic map from D^{2} to ℂ^{2} = ℍ. Then we have the Fourier expansion of f such that

f(z) = ∑_{n=0}^{∞} z^{n}\{α_{n} ( \begin{matrix}1 \\ i \end{matrix} ) + β_{n} ( \begin{matrix}1 \\ −i \end{matrix} ) \}

for z ∈ D^{2} and α_{n}, β_{n} ∈ ℂ. Now suppose that the image of f|_{∂D^{2}} is in the totally real subspace

(ℝ1 ⊕ ℝj)\{t(\frac{z + \overline{z}}{2} + \frac{z − \overline{z}}{2i} j) + (1 − t)k\}

of ℍ. This condition is equivalent to

f(z)\{t(\frac{z + \overline{z}}{2} + \frac{z − \overline{z}}{2i} j) + (1 − t)k\}^{−1} ∈ ℝ1 ⊕ ℝj for |z| = 1.

By using a homotopy, we can replace the condition by

(21.8)  f(z)\{t(\frac{z + \overline{z}}{2} − \frac{z − \overline{z}}{2i} j) − (1 − t)k\} ∈ ℝ1 ⊕ ℝj  for |z| = 1.

When we describe the multiplication from the right by t(\frac{z+\overline{z}}{2} − \frac{z−\overline{z}}{2i} j) − (1 − t)k as a 2 × 2 complex matrix acting on column vectors of ℍ = ℂ ⊕ ℂj from the left, we find that the multiplication can be described by

t\frac{z + \overline{z}}{2} ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) − t\frac{z − \overline{z}}{2i} ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) − (1 − t) ( \begin{matrix}0 & i \\ i & 0 \end{matrix} )

=z \frac{t}{2} ( \begin{matrix}1 & −i \\ i & 1 \end{matrix} ) + \overline{z} \frac{t}{2} ( \begin{matrix}1 & i \\ −i & 1 \end{matrix} ) − (1 − t) ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) .

It follows that we have

f(z)\{t(\frac{z + \overline{z}}{2} − \frac{z − \overline{z}}{2i} j) − (1 − t)k\}

= ∑_{n=0}^{∞} z^{n}\{z \frac{t}{2}α_{n} ( \begin{matrix}1 & −i \\ i & 1 \end{matrix} ) ( \begin{matrix}1 \\ i \end{matrix} ) + \overline{z} \frac{t}{2}α_{n} ( \begin{matrix}1 & i \\ −i & 1 \end{matrix} ) ( \begin{matrix}1 \\ i \end{matrix} ) − (1 − t)α_{n} ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) ( \begin{matrix}1 \\ i \end{matrix} )

+ z \frac{t}{2}β_{n} ( \begin{matrix}1 & −i \\ i & 1 \end{matrix} ) ( \begin{matrix}1 \\ −i \end{matrix} ) + \overline{z} \frac{t}{2}β_{n} ( \begin{matrix}1 & i \\ −i & 1 \end{matrix} ) ( \begin{matrix}1 \\ −i \end{matrix} ) − (1 − t)β_{n} ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) ( \begin{matrix}1 \\ −i \end{matrix} )\}.

Note that \overline{z} = z^{−1} for |z| = 1. So we can put it as ∑_{n=−1}^{∞} z^{n}a⃗_{n} with a⃗_{n} ∈ ℂ ⊕ ℂj. In

this expression, we have

a⃗_{−1} = tβ_{0} ( \begin{matrix}1 \\ −i \end{matrix} ) ,
a⃗_{0} = −(1 − t)β_{0} ( \begin{matrix}1 \\ i \end{matrix} ) + \{(1 − t)α_{0} + tβ_{1}\} ( \begin{matrix}1 \\ −i \end{matrix} ) ,
a⃗_{n} = \{tα_{n−1} − (1 − t)β_{n}\} ( \begin{matrix}1 \\ i \end{matrix} ) + \{(1 − t)α_{n} + tβ_{n+1}\} ( \begin{matrix}1 \\ −i \end{matrix} ) for n ≥ 1.

Since ℝ1 ⊕ ℝj is a totally real subspace in ℍ, the condition (21.8) implies that \overline{a⃗_{n}} = a⃗_{−n} for any n. These conditions yield that (21.9.1) \overline{a⃗_{−1}} = a⃗_{1}

(21.9.2)  a⃗_{0} ∈ ℝ^{2}
(21.9.3)  a⃗_{n} = 0 for n ≥ 2.

Now let us consider the case t ≠ 0, 1. From (21.9.1) and (21.9.3) we have α_{n} = 0 (n ≥ 1) and β_{n} = 0 (n ≥ 2). The condition (21.9.2) is equivalent to the equality −(1 − t)\overline{β_{0}} = (1 − t)α_{0} + tβ_{1}. Combining it with (21.9.1), we easily find that α_{0} and β_{1} are determined by β_{0}. (Explicitly we have α_{0} = \frac{2t−1}{2t^{2}−2t+1} \overline{β_{0}} and β_{1} = −\frac{1−t}{t}(\overline{β_{0}} + α_{0})). Hence our f is determined by only β_{0} ∈ ℂ. Therefore we find that the real dimension of the kernel of the Dolbualt operator for t ≠ 0, 1 is equal to 2. On the other hand, we can see that the Fredholm index of the operator is 2. So we can find that the operator is surjective.

Next, we describe the kernels for the cases t = 0, 1. In both cases, we can see that α_{n} = 0 (n ≥ 1) and β_{n} = 0 (n ≥ 2) by using (21.9.1) and (21.9.3). Moreover, by using (21.9.2), we find that β_{1} = 0 in both cases and

α_{0} = \{ \begin{matrix}−\overline{β_{0}} & for t = 0, \\ \overline{β_{0}} & for t = 1. \end{matrix}

Thus β_{0} determines the kernels of the operators in both cases.

With these explicit descriptions of the kernels understood, we can show the non-orientability of our index bundle. For t ∈ [0, 1], the kernel is isomorphic to {β_{0} ∈ ℂ}. So we have the (complex) orientation. What we want to show is that the identification map between the kernels on t = 0 and t = 1 is an orientation

reversing map. When t = 0, the kernel is given by V_{0} = \{−\overline{β_{0}} ( \begin{matrix}1 \\ i \end{matrix} ) + β_{0} ( \begin{matrix}1 \\ −i \end{matrix} )\}.

When t = 1, the kernel is given by V_{1} = \{\overline{β_{0}} ( \begin{matrix}1 \\ i \end{matrix} ) + β_{0} ( \begin{matrix}1 \\ −i \end{matrix} )\}. Recall that we

identify V_{0} with V_{1} by the right multiplication of i. As before, this multiplication can be written by a 2 × 2 complex matrix acting on ℂ ⊕ ℂj from the left as

( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) .

Thus V_{0} is identified with

V_{1} · i = \{\overline{β_{0}} ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) ( \begin{matrix}1 \\ i \end{matrix} ) + β_{0} ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) ( \begin{matrix}1 \\ −i \end{matrix} )\} = \{\overline{β_{0}} ( \begin{matrix}i \\ 1 \end{matrix} ) + β_{0} ( \begin{matrix}i \\ −1 \end{matrix} )\}.

We find that this identification map is an orientation reversing map. Because when

we put β_{0} = 1 and i, we have bases ⟨( \begin{matrix}0 \\ −2i \end{matrix} ) , ( \begin{matrix}2i \\ 0 \end{matrix} )⟩ and ⟨( \begin{matrix}2i \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ −2i \end{matrix} )⟩ of the

real vector spaces V_{0} and V_{1} ·i respectively. Clearly, the orientations on these spaces are opposite. Therefore we have finished the proof of Proposition 21.7. □

Note that the totally real subbundle F above is orientable, but not spin.

21.3. The case of connecting orbits in the Floer theory.

Let L_{i}, i = 0, 1, be a pair of relatively spin Lagrangian submanifolds, (see Definition 21.5), intersecting transversely. A connecting orbit in the Floer complex for L_{0}, L_{1} is a pseudo-holomorphic map u : ℝ × [0, 1] → M, which maps ℝ × {i} to L_{i}, u(−∞, ∗) = p_{j} and u(+∞, ∗) = p_{k}. Consider the pull back of T M by the projection M × [0, 1] → M. For each p_{j} ∈ L_{0} ∩ L_{1}, we choose a path of oriented Lagrangian subspaces Λ(p_{j}) : [0, 1] → Λ(T_{p_{j}}M) so that

Λ(p_{j}, 0) = T_{p_{j}}L_{0}, Λ(p_{j}, 1) = T_{p_{j}}L_{1}

and a trivialization (relative to fixed trivialization at end points) of the bundle

{ \atop Λ(p_{j})} = ⋃_{t∈[0,1]}\{t\} × Λ(p_{j}, t) → [0, 1].

Then we have a Lagrangian subbundle

L → L_{0} × {0} ∪ (L_{0} ∩ L_{1}) × [0, 1] ∪ L_{1} × {1}.

Let st ∈ H^{2}(M; ℤ_{2}) such that w_{2}(T L_{i}) = st|_{L_{i}}, for i = 0, 1. As in Subsection 21.1 above, we choose an oriented vector bundle V on the 3-skeleton of M with w_{2}(V) = st. Note that the spin structures over L_{0} and L_{1} and the trivializations of { \atop Λ(p_{j})} gives a spin structure over L. For each p_{j}, we consider the half discs D_{±}^{2} and glue the set of z ∈ D^{2} with positive (resp. negative) real part with the half infinite cylinder [0, ∞) × [0, 1] (resp. (−∞, 0] × [0, 1]). We denote them by

Θ_{−} = D_{−}^{2}♯([0, ∞) × [0, 1])
Θ_{+} = ((−∞, 0] × [0, 1])♯D_{+}^{2}.

Take the Dolbeault operator

\overline{∂}_{(±,Λ(p_{j}))} : W_{Λ(p_{j})}^{1,p}(Θ_{±}) → L^{p}(Θ_{±})

on the space

W_{Λ(p_{j})}^{1,p}(Θ_{±}) := \{ζ ∈ W^{1,p}(Θ_{±}, ℂ^{n}) | ζ(z) ∈ Λ(p_{j}, t) for z = e^{π\sqrt{−1}±(t− \frac{1}{2} )},
ζ(τ, 1) ≡ T_{p_{j}}L_{1}, ζ(τ, 0) ≡ T_{p_{j}}L_{0}}

through a trivialization.

Note that the Maslov-Viterbo index of the intersection point p ∈ L_{0} ∩ L_{1} with a path Λ(p_{j}) of Lagrangian subspaces in T_{p}M is given by the index of \overline{∂}_{(−,Λ(p_{j}))}.

By the gluing argument for a sufficiently large constant R > 0, we have

(21.10)

Index \overline{∂}_{(−,Λ(p_{j}))} + Index D\overline{∂}_{u} + Index \overline{∂}_{(+,Λ(p_{k}))}
= Index \overline{∂}_{(u;Λ(p_{j}),Λ(p_{k}))},

where D\overline{∂}_{u} is the linearization operator for the pseudo-holomorphic curve equation at a connecting orbit joining p_{j} and p_{k}, and \overline{∂}_{(u;Λ(p_{j}),Λ(p_{k}))} is the Dolbeault operator defined on

Θ_{−}♯_{R}(ℝ × [0, 1])♯_{R}Θ_{+} ≃ D^{2}

with Lagrangian boundary condition coming from Λ(p_{j}), T_{u(s,0)}L_{0}, T_{u(s,1)}L_{1} and Λ(p_{k}) in an obvious way. The virtual vector space Index \overline{∂}_{(u,Λ(p_{j}),Λ(p_{k}))} is oriented as a virtual vector space, from the choice of a spin structure of L ⊕ V, in a similar way as in Theorem 21.1. Once we choose an orientation on Index \overline{∂}_{(±,Λ(p_{j}))}, we get an orientation of index D\overline{∂}_{u} from (21.10). Hence we get the following:

Theorem 21.11. Suppose that a pair of Lagrangian submanifolds L_{0}, L_{1} are relatively spin. Then for any p, q ∈ L_{0} ∩ L_{1}, the moduli space M(p, q) of connecting orbits in Lagrangian intersection Floer homology is orientable. Furthermore, choose an orientation on Index \overline{∂}_{(±,Λ(p_{j}))}. Then a choice of relative spin structures for the pair (L_{0}, L_{1}) determine the orientation on M(p, q).

Remark 21.12. (1) In Theorem 21.11, we do not have to choose orientations on both Index \overline{∂}_{(+,Λ(p_{j}))} and Index \overline{∂}_{(−,Λ(p_{j}))}. It is enough to choose an orientation on either Index \overline{∂}_{(+,Λ(p_{j}))} or Index \overline{∂}_{(−,Λ(p_{j}))}.

(2) The orientation determined by the relative spin structure (Theorem 21.1 and 21.11) satisfies the coherent condition [FH], i.e., the compatibility under gluing process, automatically. See Proposition 23.2 and Lemma 23.4 below for compatibility of orientation on moduli spaces of holomorphic discs. The compatibility for the moduli space of connecting orbits is also proven based on Lemma 23.4. For the procedure of giving orientation in Theorem 21.11, we follow the argument from §21 [FOn2] (see also [FH] for argument on the coherent orientation problem).

(3) In [Oh7], the second named author previously studied the orientation problem for the Floer homology of Lagrangian intersection for the case where L_{0} = ϕ(o_{N}) and L_{1} = o_{N} on M = T^{∗}N (or more generally L_{1} = ν^{∗}S the conormal bundle of S ⊂ N). He proved that in this case there exists a coherent orientation on the Floer complex for any compact manifold N irrespecitve of orientability of N. In this case, the condition on w_{2} is automatically satisfied and there are no holomorphic discs with boundary on L_{0} nor L_{1}.

§22. Conventions and Preliminaries.

In this section, we will fix some basic conventions and prepare some formulae concerning orientation. At the beginning, we assume that spaces are smooth (so manifolds), and mappings are submersions. Later we incorporate Kuranishi structure and deal with the space with Kuranishi structure and the fiber product of smooth mappings in the sense of Kuranishi structure. In this section, we denote dimensions of spaces X or Y by small letters x or y respectively. We will denote by ∗ a generic point without specifying it from now on. Convention 22.1.

(1) Let X be an oriented smooth manifold with boundary ∂X. Then we define an orientation on ∂X so that

(22.1.1)  T_{∗}X ≃ ℝ_{out} × T_{∗}(∂X)

is an isomorphism of oriented vector spaces. Here ℝ_{out} is ℝ oriented by an outer normal vector.

(2) Let G be a Lie group given an orientation. When G acts on an oriented manifold X smoothly and freely, then we define an orientation of the quotient space X/G so that

(22.1.2)  T_{∗}X ≃ T_{∗}(X/G) × Lie G

is an isomorphism of oriented vector spaces. Here Lie G is the Lie algebra of G.

(3) (Fiber product orientation). Let X_{i} (i = 1, 2) and Y be oriented smooth manifolds and f_{i} : X_{i} → Y two submersions. We denote the fibre product of f_{i}’s by X_{1}_{f_{1}} ×_{f_{2}} X_{2} or X_{1} ×_{Y} X_{2}. Take a point q ∈ f_{1}(X_{1}) ∩ f_{2}(X_{2}). Let ⟨u_{1}, ⋯, u_{y}⟩ be an oriented basis of T_{q}Y. By transversality, we have ⟨v_{X_{i},1}, ⋯, v_{X_{i},y}⟩ ∈ T_{p_{i}}X_{i} for p_{i} ∈ f_{i}^{−1}(q) such that (df_{i})_{p_{i}}(v_{X_{i},k}) = u_{k} for k = 1, ⋯, y. We can choose a basis ⟨η_{X_{i},1}, ⋯, η_{X_{i},x_{i}−y}⟩ ∈ Ker(df_{i})_{p_{i}} so that

⟨η_{X_{1},1}, ⋯, η_{X_{1},x_{1}−y}, v_{X_{1},1}, ⋯, v_{X_{1},y}⟩ ∈ T_{p_{1}}X_{1} ≃ Ker(df_{1})_{p_{1}} × Im(df_{1})_{p_{1}}

is the given oriented basis of T_{p_{1}}X_{1} and

⟨v_{X_{2},1}, ⋯, v_{X_{2},y}, η_{X_{2},1}, ⋯, η_{X_{2},x_{2}−y}⟩ ∈ T_{p_{2}}X_{2} ≃ Im(df_{2})_{p_{2}} × Ker(df_{2})_{p_{2}}

is the given oriented basis of T_{p_{2}}X_{2} respectively. Then we define an orientation on the fibre product X_{1} ×_{Y} X_{2} so that

⟨η_{X_{1},1}, ⋯, η_{X_{1},x_{1}−y}, u_{1}, ⋯, u_{y}, η_{X_{2},1}, ⋯, η_{X_{2},x_{2}−y}⟩

is an oriented basis at T_{[p_{1},p_{2}]}(X_{1} ×_{Y} X_{2}). Here we identify T_{q}Y with T_{[p_{1},p_{2}]}{(s_{1} × s_{2})Y }, where s_{i} is a local section of f_{i} : X_{i} → Y.

In other words, we identify T_{[p_{1},p_{2}]}(X_{1} ×_{Y} X_{2}) and Ker(df_{1})_{p_{1}} ×T_{q}Y ×Ker(df_{2})_{p_{2}}, as vector spaces and define an orientation on T_{[p_{1},p_{2}]}(X_{1} ×_{Y} X_{2}) so that

T_{[p_{1},p_{2}]}(X_{1} ×_{Y} X_{2}) ≃ Ker(df_{1})_{p_{1}} × T_{q}Y × Ker(df_{2})_{p_{2}}

is an isomorphism of oriented vector spaces. Here we define the orientation on Ker(df_{i})_{p_{i}} such that

(22.1.3)  T_{p_{1}}X_{1} ≃ Ker(df_{1})_{p_{1}} × Im(df_{1})_{p_{1}}
(22.1.4)  T_{p_{2}}X_{2} ≃ Im(df_{2})_{p_{2}} × Ker(df_{2})_{p_{2}}

are isomorphisms as oriented vector spaces respectively. Here Im(df_{i})_{p_{i}} is isomorphic to T_{q}Y as oriented vector spaces. (When (df_{i})_{p_{i}} (i = 1, 2) is bijective, we define the orientation on the one-point set Ker(df_{i})_{p_{i}} by ϵ_{i}, where ϵ_{i} = +1 if (df_{i})_{p_{i}} is an orientation preserving isomorphism, and ϵ_{i} = −1 if (df_{i})_{p_{i}} is an orientation reserving isomorphism.) In this sense, when we consider orientations of fibre products, we will hereafter write oriented isomorphisms (22.1.3)-(22.1.4) such as

(22.1.5)  X_{1} = X_{1}^{∘} × Y
(22.1.6)  X_{2} = Y ×^{∘}X_{2}.

Then we will write the fibre product X_{1} ×_{Y} X_{2} as X_{1}^{∘} × Y ×^{∘}X_{2}. (Of course, these equalities make sense as equalities at tangent space level.)

(4) (Fibre product of Kuranishi structure.) Let (s; E → U) be a local chart of a space with Kuranishi structure. Here s is a section of the vector bundle E over U. By definition of the orientation of the Kuranishi structure (see Definition 5.8 in [FOn2]), we have an orientation on det E_{∗} ⊗ det T_{∗}U. As long as a finite group acts on E → U preserving orientation of det E ⊗ det T U, the group action does not effect on orientation problem. So we may assume here the action is trivial. Then we have the exact sequence

(22.2)  0 ⟶ T_{∗}s^{−1}(0) ⟶ T_{∗}U ⟶^{Ds(∗)} E_{∗} ⟶ 0

where Ds(∗) : T_{∗}U → E_{∗} is the covariant derivative of s : U → E at ∗ with respect to a connection. Note that this derivative does not depend on the choice of connection as long as ∗ ∈ s^{−1}(0). We arrange the basis of T_{∗}U so that the basis of E_{∗} comes first and then that of T_{∗}s^{−1}(0) next. We define the orientation on s^{−1}(0) by

E_{∗} × T_{∗}s^{−1}(0) = T_{∗}U.

(Note that since we have a given orientation on det T_{∗}U ⊗det E_{∗}, the above equality is enough for determing the orientation on T_{∗}s^{−1}(0)). Let (s_{1}; E_{1} → U_{1}) and (s_{2}; E_{2} → U_{2}) be local charts of spaces X_{1} and X_{2} with Kuranishi structure. Then we can take (s_{1} ⊕ s_{2}; E_{1} ⊕ E_{2} → U_{1} × U_{2}), which gives a Kuranishi structure on the product X_{1} × X_{2}. Note that the orientation on (s_{1} ⊕ s_{2})^{−1}(0) defined by (s_{1} ⊕ s_{2}; E_{1} ⊕ E_{2} → U_{1} × U_{2}) is different from the product orientation on X_{1} × X_{2} = s_{1}^{−1}(0) × s_{2}^{−1}(0). The difference is given by

X_{1} × X_{2} = (−1)^{rankE_{2} dim X_{1}}(s_{1} ⊕ s_{2})^{−1}(0)
= (−1)^{rankE_{2}(dim U_{1}−rankE_{1})}(s_{1} ⊕ s_{2})^{−1}(0).

As for the general case of fibre products, we define the orientation on the fibre product with Kuranishi structures as follows. Let f_{1} : U_{1} → B and f_{2} : U_{2} → B be submersions. Take the fiber product of them, i.e., U_{1} ×_{B} U_{2} = U_{1}_{f_{1}} ×_{f_{2}} U_{2}. We have the orientation on U_{1} ×_{B} U_{2} defined by Convention 10.1 (3). We restrict the bundle E_{1} ⊕E_{2} on U_{1} ×U_{2} to U_{1} ×_{B} U_{2}. The orientation on E_{1} ⊕E_{2} is also induced. Then (s_{1} ⊕ s_{2}; E_{1} ⊕ E_{2} → U_{1} ×_{B} U_{2}) defines an orientation on (s_{1} ⊕ s_{2})^{−1}(0) by

(E_{1} ⊕ E_{2})_{∗} × T_{∗}(s_{1} ⊕ s_{2})^{−1}(0) = T_{∗}(U_{1} ×_{B} U_{2}).

Then we define the fibre product X_{1} ×_{B} X_{2} by

X_{1} ×_{B} X_{2} = (−1)^{rankE_{2}(dim X_{1}−dim B)}(s_{1} ⊕ s_{2})^{−1}(0)
= (−1)^{rankE_{2}(dim U_{1}−rankE_{1}−dim B)}(s_{1} ⊕ s_{2})^{−1}(0).

As consequences of these conventions, we have the following fundamental formulae which will be used later.

Lemma 22.3. (1) Assume X_{1} and X_{2} have boundaries and the boundary of Y is empty. For X_{1} → Y and X_{2} → Y, we have

∂(X_{1} ×_{Y} X_{2}) = ∂X_{1} ×_{Y} X_{2} ⨆(−1)^{x_{1}+y}(X_{1} ×_{Y} ∂X_{2}).

(2) (Associativity.) For X_{1} → Y_{1}, X_{2} → Y_{1} × Y_{2} and X_{3} → Y_{2}, we have

(X_{1} ×_{Y_{1}} X_{2}) ×_{Y_{2}} X_{3} = X_{1} ×_{Y_{1}} (X_{2} ×_{Y_{2}} X_{3}).

(3) (Iteration formula.) For X_{1} → Y_{1} × Y_{2}, X_{2} → Y_{1} and X_{3} → Y_{2}, we have

X_{1} ×_{Y_{1}×Y_{2}} (X_{2} × X_{3}) = (−1)^{y_{2}(y_{1}+x_{2})}(X_{1} ×_{Y_{1}} X_{2}) ×_{Y_{2}} X_{3}.

(4) Let f_{i} : X_{i} → X′_{i} (i = 1, 2) be an ϵ(f_{i})-oriented isomorphism and g : Y → Y′ an ϵ(g)-oriented isomorphism. Here ϵ(f_{i}) and ϵ(g) are ±1 and the +1 (resp.−1)- oriented isomorphism means the orientation preserving (resp. reversing) isomorphism. Then for X_{i} → Y and X′_{i} → Y′, the induced map

f_{1} ×_{g} f_{2} : X_{1} ×_{Y} X_{2} ⟶ X′_{1} ×_{Y′} X′_{2}

is an ϵ(f_{1})ϵ(f_{2})ϵ(g)-oriented isomorphism.

Proof: All claims are trivial except signs. So we only check signs.

(1) We put X_{1} = X_{1}^{∘} × Y and X_{2} = Y ×^{∘}X_{2} as in (22.1.5)-(22.1.6). Moreover we put ∂X_{1} = (∂X_{1})^{∘} × Y and ∂X_{2} = Y × ^{∘}(∂X_{2}). From the convention of the boundary orientation (22.1.1), we have

ℝ_{X_{1}} × (∂X_{1})^{∘} × Y = X_{1}^{∘} × Y
ℝ_{X_{2}} × (Y ×^{∘}(∂X_{2})) = Y ×^{∘}X_{2},

where ℝ_{X_{i}} is oriented by the outward normal vector. Thus we have

ℝ_{X_{1}} × (∂X_{1})^{∘} = X_{1}^{∘}
(−1)^{y}(ℝ_{X_{2}} ×^{∘}(∂X_{2})) = ^{∘}X_{2}.

Hence we have

ℝ_{X_{1}×_{Y} X_{2}} × ∂(X_{1} ×_{Y} X_{2}) = X_{1} ×_{Y} X_{2} = X_{1}^{∘} × Y ×^{∘}X_{2}
= ℝ_{X_{1}} × (∂X_{1})^{∘} × Y ×^{∘}X_{2}
⨆ X_{1}^{∘} × Y × (−1)^{y}(ℝ_{X_{2}} ×^{∘}(∂X_{2}))
= ℝ_{X_{1}} × (∂X_{1} ×_{Y} X_{2}) ⨆(−1)^{x_{1}+y}ℝ_{X_{2}} × (X_{1} ×_{Y} ∂X_{2})),

which proves (1).

(2) Put

X_{1} = X_{1}^{∘} × Y_{1}
X_{2} = Y_{1} ×^{∘}X_{2} = X_{2}^{∘} × Y_{2} = Y_{1} ×^{∘}X_{2}^{∘} × Y_{2}
X_{3} = Y_{2} ×^{∘}X_{3}.

Then we have ^{∘}X_{2} = ^{∘}X_{2}^{∘} × Y_{2} and X_{2}^{∘} = Y_{1} ×^{∘}X_{2}^{∘}. Since

X_{1} ×_{Y_{1}} X_{2} = X_{1}^{∘} × Y_{1} ×^{∘}X_{2} = X_{1}^{∘} × Y_{1} ×^{∘}X_{2}^{∘} × Y_{2},

we have

(X_{1} ×_{Y_{1}} X_{2}) ×_{Y_{2}} X_{3} = X_{1}^{∘} × Y_{1} ×^{∘}X_{2}^{∘} × Y_{2} ×^{∘}X_{3}.

On the other hand, since

X_{2} ×_{Y_{2}} X_{3} = X_{2}^{∘} × Y_{2} ×^{∘}X_{3} = Y_{1} ×^{∘}X_{2}^{∘} × Y_{2} ×^{∘}X_{3},

th right hand side in (2) can be written as

X_{1} ×_{Y_{1}} (X_{2} ×_{Y_{2}} X_{3}) = X_{1}^{∘} × Y_{1} ×^{∘}X_{2}^{∘} × Y_{2} ×^{∘}X_{3},

which implies the associativity.

(3) We put

X_{1} = X_{1}^{∘} × Y_{1} × Y_{2},  X_{2} = Y_{1} ×^{∘}X_{2},  X_{3} = Y_{2} ×^{∘}X_{3}.

Then since we have X_{2} × X_{3} = (−1)^{y_{2}(x_{2}−y_{1})}Y_{1} × Y_{2} × ^{∘}X_{2} × ^{∘}X_{3}, the left hand side is written as

(22.4)  X_{1} ×_{Y_{1}×Y_{2}} (X_{2} × X_{3}) = (−1)^{y_{2}(x_{2}−y_{1})}X_{1}^{∘} × Y_{1} × Y_{2} ×^{∘}X_{2} ×^{∘}X_{3}.

On the other hand, since we have

X_{1} ×_{Y_{1}} X_{2} = (−1)^{y_{1}y_{2}}X_{1}^{∘} × Y_{2} × Y_{1} ×^{∘}X_{2}
=(−1)^{y_{1}y_{2}}(−1)^{y_{2}x_{2}}X_{1}^{∘} × Y_{1} ×^{∘}X_{2} × Y_{2},

the right hand side is written as

(X_{1} ×_{Y_{1}} X_{2}) ×_{Y_{2}} X_{3}
=(−1)^{y_{2}(y_{1}+x_{2})}X_{1}^{∘} × Y_{1} ×^{∘}X_{2} × Y_{2} ×^{∘}X_{3}
(22.5)  =(−1)^{y_{2}(y_{1}+x_{2})}(−1)^{y_{2}(x_{2}−y_{1})}X_{1}^{∘} × Y_{1} × Y_{2} ×^{∘}X_{2} ×^{∘}X_{3}.

Comparing (22.4) and (22.5), we get the formula.

(4) We put X_{1} = X_{1}^{∘} × Y and X_{2} = Y ×^{∘}X_{2}. Then we can write as

X_{1} ×_{Y} X_{2} = X_{1}^{∘} × Y ×^{∘}X_{2}.

Similarly we have X′_{1}×_{Y′} X′_{2} = X_{1}^{′∘}×Y′×^{∘}X′_{2}. Since Y′ = ϵ(g)Y and X′_{i} = ϵ(f_{i})X_{i}, we have X_{1}^{′∘} = ϵ(f_{1})ϵ(g)X_{1}^{∘} and ^{∘}X′_{2} = ϵ(f_{2})ϵ(g)^{∘}X_{2}. Thus we get (4).

The same conclusions hold for spaces with Kuranishi structures. We note that when, for example, dim X^{∘} is negative, it indicates the obstruction bundle which is a virtual space of negative dimension. □

Remark 22.6. By using our expression of a fibre product, we can write the fibre product in a simple manner as following. Let f_{k} : X → Y_{k} and g_{k} : X_{k} → Y_{k} (k = 1, …, ℓ). For k = 1, …, ℓ, we put

X =: X^{∘} × ∏_{k=1}^{ℓ} Y_{k}, X_{k} =: Y_{k} ×^{∘}X_{k}.

Then the orientation on the fibre product is given by

(22.7)  X_{(f_{1},... ,f_{ℓ})} ×_{g_{1}×···×g_{ℓ}} (∏_{k=1}^{ℓ} X_{k}) = X^{∘} × ∏_{k=1}^{ℓ}(Y_{k} ×^{∘}X_{k}).

Because, we put

∏_{k=1}^{ℓ} X_{k} =: (∏_{k=1}^{ℓ} Y_{k}) ×^{∘}(∏_{k=1}^{ℓ} X_{k}).

Then by definition of the orientation on the fibre product, we have

X_{(f_{1},... ,f_{ℓ})} ×_{g_{1}×···×g_{ℓ}} (∏_{k=1}^{ℓ} X_{k}) = X^{∘} × (∏_{k=1}^{ℓ} Y_{k}) ×^{∘}(∏_{k=1}^{ℓ} X_{k}).

But it is easy to see that

∘(∏_{k=1}^{ℓ} X_{k}) = (−1)^{δ_{1}} ∏_{k=1}^{ℓ}^{∘}X_{k}, where δ_{1} = ∑_{k=2}^{ℓ} y_{k} ∑_{j=1}^{k−1}(x_{j} − y_{j}).

Thus we have

X_{(f_{1},... ,f_{ℓ})} ×_{g_{1}×···×g_{ℓ}} (∏_{k=1}^{ℓ} X_{k}) = (−1)^{δ_{1}}X^{∘} × (∏_{k=1}^{ℓ} Y_{k}) × (∏_{k=1}^{ℓ}^{∘}X_{k})

= (−1)^{δ_{1}+δ_{2}}X^{∘} × ∏_{k=1}^{ℓ}(Y_{k} ×^{∘}X_{k}).

But it is clear that δ_{1} = δ_{2}, so we find the formula (22.7).

§23. Orientation on the singular strata on the moduli space.

In this section, we describe the orientation on the moduli space of reducible curves of 2 components as that of the boundary of the moduli space of irreducible curves.

Firstly, we fix the orientation on P SL(2; ℝ) as follows. Note that the orientation on ∂D^{2} is the counter-clock-wise orientation. We pick three distinct marked points z_{0}, z_{1} and z_{2} on ∂D^{2}, which are in cyclic order with respect to the counter-clock-wise orientation. Then we embed P SL(2; ℝ) in ∂D^{2}×∂D^{2}×∂D^{2} by g ↦ (g·z_{0}, g·z_{1}.g·z_{2}) and define the orientation on P SL(2; ℝ) as an open subset of ∂D^{2} × ∂D^{2} × ∂D^{2}. If we fix the first two marked points z_{0} and z_{1}, the common stabilizer of z_{0} and z_{1} is ℝ, whose orientation is given as follows. The orbit t ↦ t · p, where p ≠ z_{0}, z_{1}, converges to z_{0} as t tends to +∞ and converges to z_{1} as t tends to −∞.

Let M̃(C) = M̃(L; C) be the space of holomorphic maps from the unit disc representing the homotopy class C ∈ π_{2}(M, L). (For simplicity of notations, we will often omit L.) Recall that under the assumptions in §21 we have an orientation on M̃(C). We write M̃_{m+1}(C) = M̃(C) × (∂D^{2})^{m−1}. Note that this can be regarded as the space of holomorphic maps from the unit disc representing the homotopy class C with m + 1 marked points on the boundary of the disc, two points of which are fixed. (Strictly speaking, the marked points are distinct. So M̃_{m+1}(C) is an open subset of M̃(C) × (∂D^{2})^{m−1}. But when we consider the orientation problem, we often write it as M̃_{m+1}(C) = M̃(C) × (∂D^{2})^{m−1} for simplicity.) We assign the point −1 ∈ ∂D^{2} to the first marked point z_{0} and +1 to the second one z_{1}. So dim M̃_{m+1}(C) = n + μ(C) + m − 1. We denote by M_{m+1}(C) the quotient space of M̃_{m+1}(C) by the biholomorphic automorphism group of D^{2} fixing {±1}. In later arguments, we denote this automorphism group by ℝ_{C}. (Under our definition of the orientation on P SL(2; ℝ) above understood, orbits of this ℝ-action converge to −1 (resp. +1) as t ∈ ℝ tends to +∞ (resp. −∞).) Orientation on these spaces are defined as the product of oriented spaces and the quotient by the automorphism group (see Convention (22.1.2) in §22). So when we consider orientations, we shall simply write as

(23.1)  M̃_{m+1}(C) = M_{m+1}(C) × ℝ_{C},

and dim M_{m+1}(C) = n + μ(C) + m − 2 ≡ n + m (mod 2). Recall the Maslov index is even since we assume that Lagrangian submanifolds are oriented.

Proposition 23.2. We denote by ev_{j}^{C} the evaluation map from M_{m+1}(C) at the (j + 1)-th marked point z_{j} (j = 0, …, m). We have an isomorphism

∂M_{m+1}(A + B) = (−1)^{(m_{1}−1)(m_{2}−1)+(n+m_{1})}M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{m_{2}+1}(B)

as oriented spaces with Kuranishi structures. Here m = m_{1}+m_{2}−1 and n = dim L.

Remark 23.3. Here the order of marked points on the boundary of the unit disc is specified as follows. After gluing holomorphic discs, the second marked point z_{1,A} of the A-disc is glued with the first marked point z_{0,B} of the B-disc and marked points disappear after smoothing. After gluing, the first marked point z_{0,A+B} of the (A + B)-disc becomes the first marked point z_{0,A} of the A-disc. The rest of the marked points of B-disc are numbered from 1 to m_{2} and then the rest of the marked points of A-disc are numbered from m_{2} + 1 to m = m_{1} + m_{2} − 1. Namely, z_{j,A+B} = z_{j,B} (j = 1, …, m_{2}) and z_{m_{2}+j,A+B} = z_{j+1,A} (j = 1, …, m_{1} − 1). We write this convention as

∂D_{2,A+B}^{2} × ⋯ × ∂D_{m,A+B}^{2} = ∂D_{2,B}^{2} × ⋯ × ∂D_{m_{2},B}^{2} × ∂D_{2,A}^{2} × ⋯ × ∂D_{m_{1},A}^{2},

or simply

(∂D^{2})^{m−1} = (∂D^{2})^{m_{2}−1} × (∂D^{2})^{m_{1}−1}.

Figure 23.1

Proof of Proposition 23.2: Firstly we state the following. Lemma 23.4. The gluing map

M̃(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M̃(B) → M̃(A + B)

is orientation preserving in the sense of Kuranishi structure. We will simply write

M̃(A + B) = M̃(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M̃(B),

when we consider the orientation problem.

The proof of Lemma 23.4 is given at the end of this section. Using this lemma, the proof of Proposition 23.2 goes as follows.

M̃_{m+1}(A + B) = M̃(A + B) × (∂D^{2})^{m−1}

=(M̃(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M̃(B)) × (∂D^{2})^{m−1}
=(−1)^{(m_{1}−1)(m_{2}−1)}(M̃(A) × (∂D^{2})^{m_{1}−1})_{ev_{1}^{A}} ×_{ev_{0}^{B}} (M̃(B) × (∂D^{2})^{m_{2}−1})
=(−1)^{(m_{1}−1)(m_{2}−1)}(M_{m_{1}+1}(A) × ℝ_{A})_{ev_{1}^{A}} ×_{ev_{0}^{B}} (M_{m_{2}+1}(B) × ℝ_{B})  by (23.1)
=(−1)^{(m_{1}−1)(m_{2}−1)+(n+m_{1})}(ℝ_{A} × M_{m_{1}+1}(A))_{ev_{1}^{A}} ×_{ev_{0}^{B}} (M_{m_{2}+1}(B) × ℝ_{B})
=(−1)^{(m_{1}−1)(m_{2}−1)+(n+m_{1})}ℝ_{A} × (M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{m_{2}+1}(B)) × ℝ_{B}
=(−1)^{(m_{1}−1)(m_{2}−1)+(n+m_{1})}ℝ_{out} × (M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{m_{2}+1}(B)) × ℝ_{A+B}.

Here we explain the reasons for some equalities. At the equality in the third line, we put

M̃(A) = M̃(A)^{∘} × L,  M̃(B) = L ×^{∘}M̃(B).

Then since

M̃(A) × (∂D^{2})^{m_{1}−1} = (−1)^{(m_{1}−1)n}M̃(A)^{∘} × (∂D^{2})^{m_{1}−1} × L,

we have

(M̃(A) × (∂D^{2})^{m_{1}−1})_{ev_{1}^{A}} ×_{ev_{0}^{B}} (M̃(B) × (∂D^{2})^{m_{2}−1})

=(−1)^{(m_{1}−1)n}M̃(A)^{∘} × (∂D^{2})^{m_{1}−1} × L ×^{∘}M̃(B) × (∂D^{2})^{m_{2}−1}
=(−1)^{(m_{1}−1)n+(m_{1}−1)(n+m_{2}−1)}M̃(A)^{∘} × L ×^{∘}M̃(B) × (∂D^{2})^{m_{2}−1} × (∂D^{2})^{m_{1}−1}
=(−1)^{(m_{1}−1)(m_{2}−1)}(M̃(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M̃(B)) × (∂D^{2})^{m−1}.

In the last equality, ℝ_{out} and ℝ_{A+B} are anti-diagonal and diagonal oriented submanifolds in ℝ_{A}×ℝ_{B}, i.e., ℝ_{out} is generated by (1, −1) ∈ ℝ_{A}×ℝ_{B} and ℝ_{A+B} is generated by (1, 1). So the orientation on ℝ_{A} ×ℝ_{B} coincides with that on ℝ_{out} ×ℝ_{A+B}. The ℝ_{out} can be regarded as the space of the gluing parameter of holomorphic discs and the ℝ_{A+B} can be regarded as an ℝ-action on M̃_{m+1}(A + B). (See Figure 23.2 below.) From our definition of orientations on the boundary and the quotient, we get Proposition 23.2. □

\begin{matrix}(1, −1) & (1, 1) \\ ℝ_{out}, gluing & ℝ_{A+B} \end{matrix}

Figure 23.2

Proof of Lemma 23.4: Let u ∈ M̃(A) and v ∈ M̃(B) are holomorphic discs with u(1) = v(−1). For a fixed sufficiently large R > 0, we glue u and v in 1/Rneighborhoods of −1 and 1, respectively, using a partition of unity and Picard’s iteration method. Denoty by u♯_{R}v ∈ M̃(A + B) the glued holomorphic disc. A relative spin structure on L gives a stable trivialization of (u|_{∂D^{2}})^{∗}T L, (u|_{∂D^{2}})^{∗}T L and ((u♯_{R}v|_{∂D^{2}})^{∗}T L, which are compatible under the gluing. Hence Lemma 23.4 reduces to the following linearized problem.

Let (E_{1}, F_{1}) and (E_{2}, F_{2}) be complex vector bundles over the unit disc with totally real subbundles on the boundary. Then we glue them together near points 1 and −1 on the boundaries of two copies of unit discs and get a complex vector bundle E = E_{1}♯E_{2} and a totally real subbundle F = F_{1}♯F_{2}. We should compare orientation of indices of Dolbeault operators \overline{∂}_{(E_{1},F_{1})}, \overline{∂}_{(E_{2},F_{2})}, and \overline{∂}_{(E,F)}. Lemma 23.5. The linear gluing map induces an orientation preserving isomorphism, as virtual vector spaces, between Ind \overline{∂}_{(E,F)} and Ind \overline{∂}_{(E_{1},F_{1})} ×_{F_{1}|_{\{1\}}=F_{2}|_{\{−1\}}} Ind \overline{∂}_{(E_{2},F_{2})}, where the fiber product of the indices is the formal difference of

Ker \overline{∂}_{(E_{1},F_{1})} ×_{F_{1}|_{\{1\}}=F_{2}|_{\{−1\}}} Ker \overline{∂}_{(E_{2},F_{2})}

and

Coker \overline{∂}_{(E_{1},F_{1})} ⊕ Coker \overline{∂}_{(E_{2},F_{2})}.

Proof. If some operators above are not surjective, we may take ”obstruction bundles”, i.e., finite dimensional complex subspace in the range which is transversal to the image of the operator. Note that complex vector spaces have natural orientation via complex structures and of even real dimension. Therefore the procedure of quotient by obstruction bundles does not effect orientation problem in Lemma 23.5. We may assume that these operators are surjective and the index is nothing but the kernel.

The linear gluing argument gives an isomorphism

Ker \overline{∂}_{(E,F)} ∼ Ker \overline{∂}_{(E_{1},F_{1}) ev_{1}} ×_{ev_{−1}} Ker \overline{∂}_{(E_{2},F_{2})}.

We check that this isomorphism is orientation preserving.

First of all, we consider the case that (E_{1}, F_{1}) and (E_{2}, F_{2}) are both trivial, i.e., F_{1} and F_{2} extends to a totally real subbundle of E_{1} and E_{2} over the disc, respectively. In this case, the conclusion is obvious.

In general, we use the (stable) trivialization of the totally real subbundle over the boundary to push down the bundle to the one-point union of the disc and the Riemann sphere (see Proposition 21.3). Note that the points which are identified are interior points of the disc and the Riemann sphere. Therefore the fiber product is taken over a fiber of a complex vector bundle, which has complex orientation.

In order to check that orientations on Ker \overline{∂}_{(E,F)} and Ker \overline{∂}_{(E_{1},F_{1}) ev_{1}} ×_{ev_{−1}} Ker \overline{∂}_{(E_{2},F_{2})} are compatible, it is enough to show that the linear gluing gives an orientation preserving isomorphism between the real index of the Dolbeault operator on the genus 0 Riemann surface with two double points and the real index of the Dolbeault operator on the glued Riemann sphere with the glued vector bundle. Since they carries a natural orientation coming from the complex structure, it is obvious that they are compatible. □

§24 The obstruction chain is a cycle.—Proof of Theorem 4.9.

For β_{k} ∈ G_{++}(L), we have the chain B(L; β_{k}) of L,(see Section 4). To simplify notations, we will write as B_{k} = B(L; β_{k}). In this section, we will complete proving that our obstruction ℚ-chain o_{k}(L) is a ℚ-cycle. First of all, we recall the definition of o_{k}(L) (Definition 4.8):

\begin{aligned}o_{k}(L) & := o(L; β_{k}) \\ & = ∑_{ℓ=0,1,2,... ; i_{1},... ,i_{ℓ}<k ; β_{k}−∑{ℓ \atop j=1} β_{i_{j}} ∈G_{++}(L)} \frac{1}{ℓ!}(M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{ℓ}}), ev_{0}). \end{aligned}

To complete proving that o_{k}(L) is a ℚ-cycle, we give a precise definition of the orientation on M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{ℓ}}) in this section. Recall that M_{1}(β_{k}; β_{i_{1}}, …, β_{i_{ℓ}}) is defined, as a space, by the fiber product

M_{ℓ+1}(β_{k} − β_{i_{1}} − ⋯ − β_{i_{ℓ}})_{(ev_{1},... ,ev_{ℓ})} × (B_{i_{1}} × ⋯ × B_{i_{ℓ}}).

To simplify notations, we use a new notation M_{1}(C; B_{1}, ⋯, B_{ℓ}) defined, as a space, by the fibre product

M_{ℓ+1}(C)_{(ev_{1},... ,ev_{ℓ})} × (B_{1} × ⋯ × B_{ℓ})

for C ∈ π_{2}(M, L). By considering the orientation on M_{ℓ+1}(C) defined as in Section 23 and the orientations on B_{j} (j = 1, …, ℓ), we define an orientation by the fibre product orientation in the way defined in §22. But we assign another orientation on M_{1}(C; B_{1}, ⋯, B_{ℓ}) not as the fibre product orientation, but as the following.

Definition 24.1. We define M_{1}(C; B_{1}, …, B_{ℓ}) as the oriented space given by

M_{1}(C; B_{1}, ⋯, B_{ℓ}) = (−1)^{(n+1)ℓ(ℓ−1)/2}M_{ℓ+1}(C)_{(ev_{1},... ,ev_{ℓ})} × (B_{1} × ⋯ × B_{ℓ}).

Here n = dim L and the orientation on M_{ℓ+1}(C)_{(ev_{1},... ,ev_{ℓ})} ×(B_{1} ×⋯×B_{ℓ}) is given by the fibre product orientation. Remark 24.2. (1) We shall define orientations in the more general cases where we replace B_{∗}’s with general chains in L. See the next section.

(2) We can see that the ℓ-th symmetric group S_{ℓ} acts on M_{1}(C; B_{1}, ⋯, B_{ℓ}) in a way of orientation preserving as follows. Let σ = (i, i + 1) be the transposition element in S_{ℓ} and consider the following diagram;

M_{ℓ+1}(C) ⟶^{(ev_{1},... ,ev_{i},ev_{i+1},... ,ev_{ℓ})} L × ⋯ × L ← B_{1} × ⋯ × B_{i} × B_{i+1} × ⋯ × B_{ℓ}
σ ↓ (−1)  (−1)^{n} ↓  ↓ (−1)^{n−1}
M_{ℓ+1}(C) ⟶^{(ev_{1},... ,ev_{i},ev_{i+1},... ,ev_{ℓ})} L × ⋯ × L ← B_{1} × ⋯ × B_{i+1} × B_{i} × ⋯ × B_{ℓ}.

Therefore the action of the ℓ-th symmetric group preserves the orientation on M_{1}(C; B_{1}, …, B_{ℓ}). In the more general situation, we shall discuss this problem in the next section. (See Lemma 25.3).

(3) In this chapter, we exclusively reserve the notation M_{k+1}(C; B_{1}, ⋯, B_{ℓ}) as the oriented space instead of M_{k+1+ℓ}(C+β_{1}+⋯+β_{ℓ}; β_{1}, β_{2}, ⋯, β_{ℓ}) which was used in the previous chapters as an abstract space. Here the first k +1 marked points are free and the last ℓ ones are mapped to B_{∗} correspondingly. Using this convention, we will show that the chain o_{k}(L) is a ℚ-cycle (Theorem 4.9). In this notation, we note that the chain o_{k}(L) is defined by using union of M_{1}(β_{k}−β_{i_{1}} −⋯−β_{i_{ℓ}}; B_{i_{1}}, …, B_{i_{ℓ}}) in Definition 24.1. That is, by using our new notation, we can rewrite the definition of o_{k}(L) as follows.

\begin{aligned}o_{k}(L) & = o(L; β_{k}) \\ & = ∑_{ℓ=0,1,2,... ; i_{1},... ,i_{ℓ}<k ; β_{k}−∑{ℓ \atop j=1} β_{i_{j}} ∈G_{++}(L)} \frac{1}{ℓ!}(M_{1}(β_{k} − β_{i_{1}} − … − β_{i_{ℓ}}; B_{i_{1}}, …, B_{i_{ℓ}}), ev_{0}). \end{aligned}

To prove that the obstruction chain o_{k}(L) defines a cycle, we should show that its codimension 1 components of the boundary cancel one another. There are two types of codimension 1 boundary components. One is the component corresponding to bubbling off of holomorphic discs, and the other is the component consisting of M_{1}(C − β_{i_{1}} − ⋯ − β_{i_{ℓ}}; B_{1}, …, ∂B_{j}, …, B_{ℓ}) for some j. The latter ,in the case e.g., j = 1, is also considered as the fiber product of M_{2}(C−β_{i_{1}}−⋯−β_{i_{ℓ}}; B_{2}, …, B_{ℓ}) and M_{1}(β_{i_{1}}), where the fiber product is taken with respect to the evaluation map at the second marked point on the first factor and the evaluation map at the first marked point on the second factor. This can be also thought of component corresponding to bubbling of holomorphic discs in M_{1}(β_{i_{1}}). We shall compare their orientations and show that they are opposite.

Note that from Remark 24.2 (2) or Lemma 25.3 below for the case P_{j} = B_{j}, we find that the action of the ℓ-th symmetric group on M_{1}(C; B_{1}, …, B_{ℓ}) is orientation preserving. (Recall that

deg B_{j} = n − dim B_{j} = n − (n + μ(β_{j}) − 1) ≡ 1 ( mod 2 ),

where μ(β_{j}) is the Maslov index which is even for the oriented Lagrangian submanifold L.) Therefore it is enough to consider the case that bubbling off or ∂B occurs at the second marked point z_{1} (i.e., j = 1 case).

Firstly let us consider the orientation on the first type of codimension 1 boundary components, namely, those corresponding to bubbling off of holomorphic discs. We put m = m_{1} + m_{2} − 1. Then we have

M_{1}(A + B; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} )

=(−1)^{ϵ_{1}}M_{m+1}(A + B)_{(ev_{1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}} × ∏_{k=2}^{m_{1}} B_{i_{k}})

=(−1)^{ϵ_{2}}(M_{m+1}(A + B)_{(ev_{1}^{A+B},... ,ev_{m_{2}}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}}))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}})

=(−1)^{ϵ_{3}}ℝ_{out} × ((M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{m_{2}+1}(B))_{(ev_{1}^{A+B},... ,ev_{m_{2}}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}}))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}}),

where

ϵ_{1} = \frac{(n + 1)m(m − 1)}{2},
ϵ_{2} = ϵ_{1} + nm_{2}(m_{1} − 1),
ϵ_{3} = ϵ_{2} + (m_{1} − 1)(m_{2} − 1) + n + m_{1}.

Here we used iteration formula (3) of Lemma 22.3 in the second equality, and Proposition 23.2 in the third equality. (Recall dim B_{k} = n + μ(β_{k}) − 1 ≡ n − 1 (mod 2).) Using the associativity property (2) of Lemma 22.3, we have

((M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{m_{2}+1}(B))_{(ev_{1}^{A+B},... ,ev_{m_{2}}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}}))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}})

=(M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} (M_{m_{2}+1}(B)_{(ev_{1}^{B},... ,ev_{m_{2}}^{B})} × (∏_{k=1}^{m_{2}} B_{j_{k}})))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}})

=(−1)^{(n+1)m_{2}(m_{2}−1)/2}(M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}}).

Hence ∂M_{1}(A + B; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} ) contains (24.3)

(−1)^{ϵ_{4}}(M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))_{(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B})} × (∏_{k=2}^{m_{1}} B_{i_{k}}),

where

ϵ_{4} = ϵ_{3} + \frac{(n + 1)m_{2}(m_{2} − 1)}{2}

=\frac{(n + 1)m(m − 1)}{2} + nm_{2}(m_{1} − 1) + (m_{1} − 1)(m_{2} − 1)

+ n + m_{1} + \frac{(n + 1)m_{2}(m_{2} − 1)}{2}.

Next, let us consider the orientation on the second type of codimension 1 boundary components. Since we assume that all lower obstruction classes vanish, i.e., they bound chains, we have a chain B such that (−1)^{n−1}ev_{0}(M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} )) is one of main component of ∂B. Using the iteration formula (3) in Lemma 22.3, we have

∂M_{1}(A; B, B_{i_{2}}, …, B_{i_{m_{1}}} )
=(−1)^{δ_{1}}∂(M_{m_{1}+1}(A)_{(ev_{1}^{A},... ,ev_{m_{1}}^{A})} × (B × ∏_{k=2}^{m_{1}} B_{i_{k}}))

=(−1)^{δ_{2}}∂((M_{m_{1}+1}(A)_{ev_{1}^{A}} × B)_{(ev_{2}^{A},... ,ev_{m_{1}}^{A})} × (∏_{k=2}^{m_{1}} B_{i_{k}})),

where

δ_{1} = \frac{(n + 1)m_{1}(m_{1} − 1)}{2},
δ_{2} = δ_{1} + n(m_{1} − 1).

By the formula (1) in Lemma 22.3, this contains, as one of the main strata,

(−1)^{δ_{2}}∂(M_{m_{1}+1}(A)_{ev_{1}^{A}} × B)_{(ev_{2}^{A},... ,ev_{m_{1}}^{A})} × (∏_{k=2}^{m_{1}} B_{i_{k}}).

Since the formula (1) again tells

∂(M_{m_{1}+1}(A)_{ev_{1}^{A}} × B) = ∂M_{m_{1}+1}(A)_{ev_{1}^{A}} × B ⨆(−1)^{n+m_{1}+n}M_{m_{1}+1}(A)_{ev_{1}^{A}} × (∂B),

OBSTRUCTION TO FLOER HOMOLOGY 211

we see that ∂M_{1}(A; B, B_{i_{2}}, …, B_{i_{m_{1}}} ) contains

(−1)^{δ_{2}+m_{1}}(M_{m_{1}+1}(A)_{ev_{1}^{A}} × (∂B))_{(ev_{2}^{A},... ,ev_{m_{1}}^{A})} × (∏_{k=2}^{m_{1}} B_{i_{k}})

=(−1)^{δ_{2}+m_{1}}(M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} (−1)^{n−1}M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}})

=(−1)^{δ_{3}}(M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))_{(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B})} × (∏_{k=2}^{m_{1}} B_{i_{k}})

(24.4)

where

δ_{3} = δ_{2} + m_{1} + n − 1
=\frac{(n + 1)m_{1}(m_{1} − 1)}{2} + n(m_{1} − 1) + m_{1} + n − 1.

Then comparing with (24.3) and (24.4), it is easy to check that the orientations on

(M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))_{(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}−1}^{A+B})} × (∏_{k=2}^{m_{1}} B_{i_{k}})

as the boundary component of M_{1}(A + B; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} ) and the boundary component of M_{1}(A; B, B_{i_{2}}, …, B_{i_{m_{1}}} ) are opposite, because

ϵ_{4} + δ_{3}

=\frac{(n + 1)m(m − 1)}{2} + nm_{2}(m_{1} − 1) + (m_{1} − 1)(m_{2} − 1) + n + m_{1}

+ \frac{(n + 1)m_{2}(m_{2} − 1)}{2} + \frac{(n + 1)m_{1}(m_{1} − 1)}{2} + n(m_{1} − 1) + m_{1} + n − 1

≡1  (mod 2).

This implies that our obstruction ℚ-chain o_{k}(L) actually defines a ℚ-cycle. □

§25. Orientation of the moduli spaces defining boundary operators. 25.1. Definition of the orientation on M_{1}(β; P_{1}, …, P_{ℓ}).

Let L be a relatively spin Lagrangian submanifold of (M, ω). The result in Section 21 in Chapter 6 shows that the moduli space M_{ℓ+1}(β) of pseudo-holomorphic maps from the unit disc representing the homotopy class β ∈ π_{2}(M, L) with (ℓ + 1)- marked points on the boundary is orientable. When we consider an orientation, it is convenient that we fix two marked points. Because the group of biholomorphic automorphisms of the disc fixing two points is ℝ, it can be interpreted as gluing parameters of two holomorphic discs. (See Section 23). In general, we can choose the fixed marked points arbitrarily. For example, if we choose z_{i} and z_{j} as the fixed points, then we can define an orientation on M_{ℓ+1}(β) by the following equalities;

M̃_{ℓ+1}(β) := M̃(β) × ∂D_{0}^{2} × ⋯ × ˇ ∂D_{i}^{2} × ⋯ × ˇ ∂D_{j}^{2} × ⋯ × ∂D_{ℓ}^{2}
M_{ℓ+1}(β) := M̃_{ℓ+1}(β)/ℝ.

If we change the order of marked points by an element σ of (ℓ + 1)-th symmetric group S_{ℓ+1}, then it might define different connected component. The change of orientation is given by the sign of σ. (Note that, even if we fix some two marked points to define an orientation, we can compare orientations on a moduli space and a moduli space after changing the order of any two points.) Hereafter we fix the two fixed points as the first marked point z_{0} and the second one z_{1}. This convention is the same as in Section 23. (Recall our definition of the orientation on P SL(2; ℝ) in Section 23. We also note that in the case when we consider the moduli space with only one marked point, we add the second marked point arbitrarily and we follow the convention explained above. After that, we consider the map of forgetting the second marked point.)

Now let [P_{k}, f_{k}] be a chain in L with dim P_{k} = p_{k}. We put deg P_{k} = n − p_{k}, which is the degree as a cochain. We define the oriented space M_{1}(β; P_{1}, …, P_{ℓ}) as the following; Definition 25.1.

M_{1}(β; P_{1}, …, P_{ℓ}) := (−1)^{ϵ_{1}}M_{ℓ+1}(β)_{(ev_{1},... ,ev_{ℓ})} ×_{f_{1}×···×f_{ℓ}} (∏_{k=1}^{ℓ} P_{k}),

where

ϵ_{1} = (n + 1) ∑_{k=1}^{ℓ−1} ∑_{j=1}^{k} deg P_{j}.

Remark 25.2. (1) If we use the iteration formula in Lemma 22.3 (3) or the formula (22.7) in Remark 22.6, we can rewrite the right hand side as

M_{1}(β; P_{1}, …, P_{ℓ})

=(−1)^{∑_{k=1}^{ℓ−1} ∑_{j=1}^{k} deg P_{j}}(⋯ ((M_{ℓ+1}(β)_{ev_{1}} ×_{f_{1}} P_{1})_{ev_{2}} ×_{f_{2}} P_{2}) × ⋯)_{ev_{ℓ}} ×_{f_{ℓ}} P_{ℓ}.

(2). If all P_{k} are bounding chains B(L; β_{k}) with β_{k} ∈ π_{2}(M, L), then we have deg B(L; β_{k}) ≡ 1 (mod 2). In this case, ϵ_{1} is given by

\frac{(n + 1)ℓ(ℓ − 1)}{2},

which is nothing but the power of the sign factor in Definition 24.1.

When we change the ordering of marked points, we can find the difference of orientations by the following key Lemma, which plays a fundamental role in our argument later. Lemma 25.3. Let σ be the transposition element (i, i + 1) in the ℓ-th symmetric group S_{ℓ}. (i = 1, …, ℓ−1). Then the action of σ on M_{1}(β; P_{1}, …, P_{i}, P_{i+1}, …, P_{ℓ}) by changing the order of marked points is described by the following.

σ(M_{1}(β; P_{1}, …, P_{i}, P_{i+1}, …, P_{ℓ}))
=(−1)^{(deg P_{i}+1)(deg P_{i+1}+1)}M_{1}(β; P_{1}, …, P_{i+1}, P_{i}, …, P_{ℓ}).

Proof: By definition we have

(25.4.1)M_{1}(β; P_{1}, …, P_{i}, P_{i+1}, …, P_{ℓ})

= (−1)^{ϵ_{1}}M_{ℓ+1}(β)_{(ev_{1},... ,ev_{i},ev_{i+1},... ,ev_{ℓ})} ×_{f_{1}×···f_{i}×f_{i+1}···×f_{ℓ}} (∏_{k=1}^{ℓ} P_{k}),

(25.4.2)M_{1}(β; P_{1}, …, P_{i+1}, P_{i}, …, P_{ℓ})

= (−1)^{ϵ_{2}}M_{ℓ+1}(β)_{(ev_{1},... ,ev_{i},ev_{i+1},... ,ev_{ℓ})}×_{f_{1}×···f_{i+1}×f_{i}···×f_{ℓ}}

(∏_{k=1}^{i−1} P_{k} × P_{i+1} × P_{i} × ∏_{k=i+2}^{ℓ} P_{k}),

where ϵ_{1} is just the same as in Definition 25.1 and ϵ_{2} is given by

ϵ_{2} = ϵ_{1} − (n + 1)(deg P_{i} − deg P_{i+1}).

To compute the difference between the fibre product orientations on the right hand sides above, we recall Lemma 22.3 (4). The σ acts on M_{ℓ+1}(β) by changing the order of marked points. Thus the σ induces an (−1)-oriented isomorphism on M_{ℓ+1}(β). Clearly we have

∏_{k=1}^{ℓ} P_{k} = (−1)^{p_{i}p_{i+1}} ∏_{k=1}^{i−1} P_{k} × P_{i+1} × P_{i} × ∏_{k=i+2}^{ℓ} P_{k}.

Here p_{i} = dim P_{i} and p_{i+1} = dim P_{i+1}. Moreover, as for the orientation of base spaces of the fibre products, we have

∏_{k=1}^{ℓ} L_{k} = (−1)^{n} ∏_{k=1}^{i−1} L_{k} × L_{i+1} × L_{i} × ∏_{k=i+2}^{ℓ} L_{k}.

Here L_{k} is a copy of L. Therefore by Lemma 22.3 (4), we can find that (25.5)

\begin{matrix}M_{ℓ+1}(β)_{(ev_{1},... ,ev_{i},ev_{i+1},... ,ev_{ℓ})} ×_{f_{1}×···f_{i}×f_{i+1}···×f_{ℓ}} (∏_{k=1}^{ℓ} P_{k}) = (−1)^{1+p_{i}p_{i+1}+n} \\ M_{ℓ+1}(β)_{(ev_{1},... ,ev_{i},ev_{i+1},... ,ev_{ℓ})} ×_{f_{1}×···f_{i+1}×f_{i}···×f_{ℓ}} (∏_{k=1}^{i−1} P_{k} × P_{i+1} × P_{i} × ∏_{k=i+2}^{ℓ} P_{k}) \end{matrix}

By combining (25.4) and (25.5), we can see that

ϵ_{1} + ϵ_{2} + 1 + p_{i}p_{i+1} + n ≡ (deg p_{i} + 1)(deg p_{i+1} + 1) (mod 2),

which proves Lemma 25.3. □

25.2. Orientation on the moduli space of connecting orbits.

When all our obstruction classes vanish, we modify the definition of coboundary operator in usual Lagrangian intersection Floer cohomology theory to get our modified Floer cohomology. In this section, we define the precise orientation on the moduli space of connecting orbits to define our modified coboundary operator. Suppose that a pair of Lagrangian submanifolds L_{0}, L_{1} of (M, ω) is relatively spin. (See Definition in Definition 21.5). Now let [ℓ_{p}, w], [ℓ_{q}, w′] ∈ Cr(L_{0}, L_{1}), where we include the data Λ(p) (resp. Λ(q)) and its trivialization in w (resp. w′). (See Section 2.2 for the definition of Cr(L_{0}, L_{1}), Section 21.3 for Λ(p).) We recall M̃([ℓ_{p}, w], [ℓ_{q}, w′]) = M̃(L_{0}, L_{1}; [ℓ_{p}, w], [ℓ_{q}, w′]) is the space of pseudo-holomorphic maps u from the infinite cylinder ℝ × [0, 1] ⊂ ℂ to M such that

u(−∞, t) = p, u(+∞, t) = q, u(ℝ × {i}) ⊂ L_{i}(i = 0, 1) and w#u ∼ w.′

(For simplicity of notations, we often omit L_{0} and L_{1} in M̃(L_{0}, L_{1}; [ℓ_{p}, w], [ℓ_{q}, w′]) etc.) From now on, we identify the infinite cylinder ℝ × [0, 1] with the unit disc D^{2} ⊂ ℂ so that the ends {−∞}×[0, 1] and {+∞}×[0, 1] correspond to z_{0} = −1 and z_{1} = 1 respectively. Then under the assumption L_{i} ⊂ M are relatively spin, we have the orientation on M̃([ℓ_{p}, w], [ℓ_{q}, w′]) by Theorem 21.11 in Section 21 in Chapter 6. We denote by M̃_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) the space of connecting orbits u with ℓ + m marked points on ∂D^{2} − {±1} such that u(−1) = p and u(1) = q. Here z_{0} and z_{1} are fixed marked points, and the ℓ marked points are on ∂_{−}D^{2} and the m marked points are on ∂_{+}D^{2}. (We denote by ∂_{−}D^{2} (resp. ∂_{+}D^{2}) {z ∈ ∂D^{2} | Im z < 0} (resp. {z ∈ ∂D^{2} | Im z > 0}. The space M̃_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) is an open subset of M̃([ℓ_{p}, w], [ℓ_{q}, w′]) × (∂D^{2})^{ℓ+m}. (The marked points are distinct.) So, under this convention, the orientation on M̃_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) is given by

M̃_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) ⊂ M̃([ℓ_{p}, w], [ℓ_{q}, w′]) × (∂D_{2}^{2} × ⋯ × ∂D_{ℓ+m+1}^{2}),

where ∂D_{i}^{2} is a parameter space of the marked point z_{i} (i = 2, …, ℓ + m + 1). We denote by M_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) the quotient space of M̃_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) by the biholomorphic automorphism group of D^{2} fixing the two marked points, z_{0} and z_{1}, which is isomorphic to ℝ. (See Definition 5.1.) The orientation on M_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) is defined by the quotient orientation as in Section 22. Like as in Section 23, we simply write as

M̃_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) = M_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) × ℝ.

Now suppose all our obstruction classes of L_{0} and L_{1} vanish. Let β_{0,i_{1}}, …, β_{0,i_{ℓ}} ∈ G_{++}(L_{0}), β_{1,j_{1}}, …, β_{1,j_{m}} ∈ G_{++}(L_{1}). We choose oriented chains B(L_{0}; β_{0,i_{k}}) in L_{0} and B(L_{1}; β_{1,j_{k}}) in L_{1} so that ∂B(L_{0}; β_{0,i_{k}}) = (−1)^{n−1}o(L_{0}; β_{0,i_{k}}) (k = 1, …, ℓ) and ∂B(L_{1}; β_{1,j_{k}}) = (−1)^{n−1}o(L_{1}; β_{1,j_{k}}) (k = 1, …, m). (The orientations of o(L_{0}; β_{0,i_{k}}) and o(L_{1}; β_{1,j_{k}}) are given by the rule in Section 24, see Definition 24.1.) Now we define

Definition 25.6. (1)

M(L_{0}, L_{1}; [ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{ℓ}}; β_{1,j_{1}}, …, β_{1,j_{m}})
= M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{ℓ}}; β_{1,j_{1}}, …, β_{1,j_{m}})
:= (−1)^{ϵ_{1}}M_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′])_{(ev_{2},... ,ev_{ℓ+1},ev_{ℓ+2},... ,ev_{ℓ+m+1})}×

(∏_{k=1}^{ℓ} B(L_{0}; β_{0,i_{k}}) × ∏_{k=1}^{m} B(L_{1}; β_{1,j_{k}})),

where

ϵ_{1} = (n + 1)(∑_{k=2}^{ℓ+m} ∑_{j=2}^{k} 1) = (n + 1)( ∑_{k=1}^{ℓ+m−1} ∑_{j=1}^{k} 1) = \frac{(n + 1)(ℓ + m − 1)(ℓ + m)}{2},

and u ∈ M_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′]) satisfies (∑_{k=1}^{ℓ} β_{0,i_{k}} + ∑_{k=1}^{m} β_{1,j_{k}}) · (w#u) ∼ w′ instead of w#u ∼ w′.

(2) More generally, for P_{0,i_{k}} ∈ C^{∗}(L_{0}, ℚ) (k = 1, …, ℓ) and P_{1,j_{k}} ∈ C^{∗}(L_{1}, ℚ) (k = 1, …, m), we define

M(L_{0}, L_{1}; [ℓ_{p}, w], [ℓ_{q}, w′] : P_{0,i_{1}}, …, P_{0,i_{ℓ}}; P_{1,j_{1}}, …, P_{1,j_{m}})
= M([ℓ_{p}, w], [ℓ_{q}, w′] : P_{0,i_{1}}, …, P_{0,i_{ℓ}}; P_{1,j_{1}}, …, P_{1,j_{m}})
:= (−1)^{ϵ_{2}}M_{ℓ,m}([ℓ_{p}, w], [ℓ_{q}, w′])_{(ev_{2},... ,ev_{ℓ+1},ev_{ℓ+2},... ,ev_{ℓ+m+1})}×

(∏_{k=1}^{ℓ} P_{0,i_{k}} × ∏_{k=1}^{m} P_{1,j_{k}}),

where

ϵ_{2} = (n + 1)(∑_{k=1}^{ℓ} ∑_{j=1}^{k} deg P_{0,i_{j}} + ∑_{k=1}^{m−1} ∑_{j′=1}^{k}((∑_{h=1}^{ℓ} deg P_{0,i_{h}}) + deg P_{1,j_{j′}} )).

Remark 25.7. (1) When we consider the space of connecting orbits as a space of pseudo-holomorphic maps from the unit disc, this sign is nothing but one determined by the rule in Definition 25.1. Note that we do not treat the first two marked points z_{0},_{1} as other marked points. This is because we treat them in a different way in order to define the orientation on the space M̃([ℓ_{p}, w], [ℓ_{q}, w′]). Namely, we glue half discs with these ends and after gluing, marked points disappear.

(2) In the case when L_{0} and L_{1} intersect cleanly, we refer more detail to Subsection 25.3.

Using these oriented spaces, we define δ_{β_{0,i_{1}} ,... ,β_{0,i_{ℓ}};β_{1,j_{1}} ,... ,β_{1,jm}} by

\begin{matrix}⟨δ_{β_{0,i_{1}} ,... ,β_{0,i_{ℓ}};β_{1,j_{1}} ,... ,β_{1,jm}} [ℓ_{p}, w], & [ℓ_{q}, w′]⟩ \\ := #M([ℓ_{p}, w], [ℓ_{q}, w′]) : & β_{0,i_{1}}, …, β_{0,i_{ℓ}}; β_{1,j_{1}}, …, β_{1,j_{m}}). \end{matrix}

as in Definition 5.4. Then we can show Proposition 25.8. We define our modified coboundary operator δ by

δ := ∑_{ℓ,m} ∑_{β_{0,i_{1}} ,... ,β_{0,i_{ℓ}};β_{1,j_{1}} ,... ,β_{1,jm}} \frac{1}{ℓ!m!}δ_{β_{0,i_{1}} ,... ,β_{0,i_{ℓ}};β_{1,j_{1}} ,... ,β_{1,jm}} .

Then it satisfies δ ∘ δ = 0. Proof: The proof goes in a similar way as in the one of the fact that our obstruction ℚ-chains are in fact ℚ-cycles (Section 24). We have two kinds of ends in M([ℓ_{p}, w], [ℓ_{q}, w′] : β_{0,i_{1}}, …, β_{0,i_{ℓ}}; β_{1,j_{1}}, …, β_{1,j_{m}}) except the end of splitting into two “stable connecting orbits”, which corresponds to δ ∘ δ. One type of ends correspond to bubbling off of holomorphic discs and the other type of ends correspond to the convergence of the values (in B) of one of evaluation maps ev_{k} to a point in ∂B.

In Section 24, we have proved that the cancellation actually holds when these phenomena happen at the second marked point. In our case, we do not consider the evaluation map at the second marked point z_{1} = 1 and the sign ϵ_{1} does not contain any contribution from the second marked point (Remark 25.7). So the exchange of the second marked point and the third marked point simply makes the orientation on the space of holomorphic maps from D^{2} with ℓ + m + 2 marked points (here we also consider ±1 as marked points) opposite. This effects equally for all components, so the cancellation for the second marked point implies the cancellation for the third marked point in this case. Note that deg B ≡ 1 mod 2. Lemma 25.3 implies that the action of symmetry group on M(β; B_{1}, …, B_{ℓ}) is orientation preserving. Hence the cancellation also holds for other marked points. □

25.3. The Bott-Morse case I.

Next we will define the precise orientation on the moduli spaces used when we define the coboundary operator in the Bott-Morse situation (Section 6). Firstly, we consider the case in which two Lagrangian submanifolds coincide. For the case of general clean intersection, the intersection of two Lagrangian submanifolds may not be orientable and we have to deal with orientation bundles of spaces with Kuranishi structures and chains with values in certain local systems. So we discuss it in section 25.6.

Let L be a relatively spin Lagrangian submanifold of (M, ω). Now suppose all the obstruction classes of L vanish. We have bounding chains B(L; β_{k}) with β_{k} ∈ G_{++}(L). Let [P, f] be a chain of L. To simplify notations, we put B_{k} = B(β_{k}) = B(L; β_{k}). According to Definition 25.1, we have Definition 25.9.

M_{1}(β − ∑_{k=1}^{ℓ} β_{i_{k}} − ∑_{k=1}^{m} β_{j_{k}}; β_{i_{1}}, …, β_{i_{ℓ}}, β_{j_{1}}, …, β_{j_{m}}; [P, f])

:=M_{1}(β − ∑_{k=1}^{ℓ} β_{i_{k}} − ∑_{k=1}^{m} β_{j_{k}}; P, B(β_{i_{1}}), …, B(β_{i_{ℓ}}), B(β_{j_{1}}), …, B(β_{j_{m}}))

=(−1)^{ϵ_{1}}(M_{ℓ+m+2}(β − ∑_{k=1}^{ℓ} β_{i_{k}} − ∑_{k=1}^{m} β_{j_{k}})

(ev_{1},... ,ev_{ℓ+1},ev_{ℓ+2},... ,ev_{ℓ+m+1}) ×_{f×...} (P × ∏_{k=1}^{ℓ} B(β_{i_{k}}) × ∏_{k=1}^{m} B(β_{j_{k}}))),

where

ϵ_{1} ≡ (n + 1)((deg P)(ℓ + m) + ∑_{k=1}^{ℓ+m−1} ∑_{j=1}^{k} 1) = \frac{(n + 1)(2 deg P + ℓ + m − 1)(ℓ + m)}{2}.

By using these spaces, we can define

δ_{β:β_{i_{1}} ,... ,β_{i_{ℓ}};β_{j_{1}} ,... ,β_{jm}} ([P, f])
:=(M_{1}(β − ∑_{k=1}^{ℓ} β_{i_{k}} − ∑_{k=1}^{m} β_{j_{k}}; β_{i_{1}}, …, β_{i_{ℓ}}, β_{j_{1}}, …, β_{j_{m}}; [P, f]), ev_{0}),

and our coboundary operator is defined by

δ([P, f] ⊗ α) := ∑_{β;β_{i_{1}} ,... ,β_{i_{ℓ}};β_{j_{1}} ,... ,β_{jm}} \frac{1}{ℓ!m!}δ_{β:β_{i_{1}} ,... ,β_{i_{ℓ}};β_{j_{1}} ,... ,β_{jm}} ([P, f]) ⊗ (α + β)

as in (6.17). Then we can prove the following; Proposition 25.10. The operator δ above satisfies δ ∘ δ = 0. Proof: The proof seems similar to that in Section 24. From 25.13 (2) below implies that

(−1)M_{1}(C; ∂P, B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} )
⊂∂M_{1}(C; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} , P).

Since ∂M_{1}(C; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} , P) may have other components corresponding to bubbling and the case that a marked point is mapped to ∂B. So we investigate these components. Suppose that C = A + B. Thanks to Lemma 25.3, we note that

M_{1}(A + B; P, B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} )
=M_{1}(A + B; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} , P),

because deg B + 1 is even. Here we put B_{k} = B(β_{k}) = B(L; β_{k}). So it is enough to prove the cancellation between orientations of codimension one boundaries corresponding to the case where bubbling happens at the second marked point and and the case that the image of the evaluation map at the second marked point goes to ∂B, by changing the ordering of marked points if necessary. We may assume P is assigned at the last marked point. Now let us consider the orientation on the boundary corresponding to bubbling off of holomorphic discs.

There are two kinds of bubbling off. One is the bubbling off of a holomorphic disc from connecting orbits, which will cancel with the ends corresponding to ∂B mentioned above. The other is the splitting of connecting orbits, which will cancel with a part of ∂M_{1}(C; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} , P). Note that ∂ above corresponds to the classical coboundary operator and we modified it by multiplying with (−1)^{n} in δ.

Firstly, we consider the first case. We put m = m_{1} + m_{2}. Then we have

M_{1}(A + B; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} , P)

=(−1)^{ϵ_{1}}M_{m+1}(A + B)_{(ev_{1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}} × ∏_{k=2}^{m_{1}} B_{i_{k}} × P)

=(−1)^{ϵ_{2}}(M_{m+1}(A + B)_{(ev_{1}^{A+B},... ,ev_{m_{2}}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}}))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)

=(−1)^{ϵ_{3}}ℝ_{out} × ((M_{m_{1}+2}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{m_{2}+1}(B))_{(ev_{1}^{A+B},... ,ev_{m_{2}}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}}))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}} × P),

where

ϵ_{1} = \frac{(n + 1)(m_{1} + m_{2})(m_{1} + m_{2} − 1)}{2},
ϵ_{2} = ϵ_{1} + nm_{1}m_{2},
ϵ_{3} = ϵ_{2} + m_{1}(m_{2} − 1) + n + m_{1} + 1.

Here we used the iteration formula (3) of Lemma 22.3 in the second equality, and Proposition 22.2 in the third equality. (Recall dim B_{k} = n + μ(β_{k}) − 1 ≡ n − 1 (mod 2).) Using the associativity property (2) of Lemma 22.3, we have

((M_{m_{1}+2}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{m_{2}+1}(B))_{(ev_{1}^{A+B},... ,ev_{m_{2}}^{A+B})} × (∏_{k=1}^{m_{2}} B_{j_{k}}))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)

= (M_{m_{1}+2}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} (M_{m_{2}+1}(B)_{(ev_{1}^{B},... ,ev_{m_{2}}^{B})} × (∏_{k=1}^{m_{2}} B_{j_{k}})))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)

= (−1)^{(n+1)m_{2}(m_{2}−1)/2}(M_{m_{1}+1}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}} × P).

Hence ∂M_{1}(A + B; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} , P) contains (25.11)

(−1)^{ϵ_{4}}(M_{m_{1}+2}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))_{(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B})}×(∏_{k=2}^{m_{1}} B_{i_{k}} ×P)

where

ϵ_{4} = ϵ_{3} + \frac{(n + 1)m_{2}(m_{2} − 1)}{2}

=\frac{(n + 1)(m_{1} + m_{2})(m_{1} + m_{2} − 1)}{2} + nm_{1}m_{2} + m_{1}(m_{2} − 1)

+ n + m_{1} + 1 + \frac{(n + 1)m_{2}(m_{2} − 1)}{2}.

Next, let us consider the orientation on the second type of codimension 1 boundary components. Since we assume that all the obstruction classes vanish, i.e., they bound chains, we have a chain B such that (−1)^{n−1}ev_{0}(M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} )) is one of main component of ∂B. Using the iteration formula (3) in Lemma 22.3, we have

∂M_{1}(A; B, B_{i_{2}}, …, B_{i_{m_{1}}} , P)
=(−1)^{δ_{1}}∂(M_{m_{1}+2}(A)_{(ev_{1}^{A},... ,ev_{m_{1}+1}^{A})} × (B × ∏_{k=2}^{m_{1}} B_{i_{k}} × P))

=(−1)^{δ_{2}}∂((M_{m_{1}+2}(A)_{ev_{1}^{A}} × B)_{(ev_{2}^{A},... ,ev_{m_{1}+1}^{A})} × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)),

where

δ_{1} = \frac{(n + 1)(m_{1} + 1)m_{1}}{2} ,
δ_{2} = δ_{1} + nm_{1}.

By the formula (1) in Lemma 22.3, this contains, as one of main strata,

(−1)^{δ_{2}}∂(M_{m_{1}+2}(A)_{ev_{1}^{A}} × B)_{(ev_{2}^{A},... ,ev_{m_{1}+1}^{A})} × (∏_{k=2}^{m_{1}} B_{i_{k}} × P).

Since the formula (1) again tells

∂(M_{m_{1}+2}(A)_{ev_{1}^{A}} × B)
=∂M_{m_{1}+2}(A)_{ev_{1}^{A}} × B ⨆(−1)^{n+m_{1}+1+n}M_{m_{1}+2}(A)_{ev_{1}^{A}} × (∂B),

we see that ∂M_{1}(A; B, B_{i_{2}}, …, B_{i_{m_{1}}} ) contains

(−1)^{δ_{2}+m_{1}+1}(M_{m_{1}+2}(A)_{ev_{1}^{A}} × (∂B))_{(ev_{2}^{A},... ,ev_{m_{1}+1}^{A})} × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)

=(−1)^{δ_{2}+m_{1}+1}(M_{m_{1}+2}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} (−1)^{n−1}M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)

=(−1)^{δ_{3}}(M_{m_{1}+2}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))

(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B}) × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)

(25.12)

where

δ_{3} = δ_{2} + m_{1} + 1 + n − 1
=\frac{(n + 1)(m_{1} + 1)m_{1}}{2} + nm_{1} + m_{1} + n.

Then comparing with (25.11) and (25.12), it is easy to check that the orientations on

(M_{m_{1}+2}(A)_{ev_{1}^{A}} ×_{ev_{0}^{B}} M_{1}(B; B_{j_{1}}, …, B_{j_{m_{2}}} ))_{(ev_{m_{2}+1}^{A+B},... ,ev_{m_{2}+m_{1}}^{A+B})} × (∏_{k=2}^{m_{1}} B_{i_{k}} × P)

as the boundary component of M_{1}(A + B; B_{j_{1}}, …, B_{j_{m_{2}}} , B_{i_{2}}, …, B_{i_{m_{1}}} , P) and the boundary component of M_{1}(A; B, B_{i_{2}}, …, B_{i_{m_{1}}} , P) are opposite, because

ϵ_{4} + δ_{3}

=\frac{(n + 1)(m_{1} + m_{2})(m_{1} + m_{2} − 1)}{2} + nm_{1}m_{2} + m_{1}(m_{2} − 1)

+ n + m_{1} + 1 + \frac{(n + 1)m_{2}(m_{2} − 1)}{2} + \frac{(n + 1)(m_{1} + 1)m_{1}}{2} + nm_{1} + m_{1} + n

≡1  (mod 2).

Next we consider the case of splitting of connecting orbits. Since the bounding chains B are treated in the same way, we consider the case that no B appears. Then the cancellation follows from Proposition 23.2 with m_{1} = 1, m_{2} = 0. Then the sign in Proposition 23.2 becomes (−1)^{n+1} which cancels with the factor (−1)^{n} which we put the classical coboundary operator in δ.

Hence we have finished the proof of Proposition 25.10. □

25.4. A_{∞}-algebra case.

We recall that, when we study the structure of the A_{∞}-algebra associated to a Lagrangian submanifold L, we use the main component of the moduli space. (See §3 for the definition of the main component.) So the ordering of the marked points on the boundary of the disc is cyclic. Let [P_{i}, f_{i}] ∈ C^{g_{i}}(L, Λ_{0,nov}). We define the orientation of the main component by

M_{1}^{main}(β; P_{1}, …, P_{k}) := (−1)^{ϵ_{1}}M_{k+1}^{main}(L; β)_{(ev_{1},... ,ev_{k})} ×_{f_{1}×···×f_{k}} (∏_{i=1}^{k} P_{i}),

where

ϵ_{1} = (n + 1) ∑_{j=1}^{k−1} ∑_{i=1}^{j} deg P_{i}.

This is the same rule as in Definition 25.1. We will show and check signs in the proof of Theorem 13.22 that (C(L; Λ_{0,nov}), 𝔪) is a filtered A_{∞}-algebra. We have two points where to check signs. The first one is in the proof of d̂∘ d̂ = 0, and the second one is in the comparison of our orientation on (M_{1}^{main}(L, 0; P_{1}, P_{2}), ev_{0}) and the geometric orientation on P_{1} ∩ P_{2}. We will check the first part in this subsection and the second one in the next subsection. Firstly we can show the following; Proposition 25.13. (1) For β = β_{1} + β_{2}, we have

M_{1}^{main}(β_{1}; P_{1}, …, P_{i−1}, M_{1}^{main}(β_{2}; P_{i}, …, P_{i+k_{2}−1}), P_{i+k_{2}}, …, P_{k})
⊂ (−1)^{ϵ_{2}}∂M_{1}^{main}(β; P_{1}, …, P_{i−1}, P_{i}, …, P_{i+k_{2}−1}, P_{i+k_{2}}, …, P_{k}),

where

ϵ_{2} = n + 1 + ∑_{j=1}^{i−1}(deg P_{j} + 1).

(2) We have

M_{1}^{main}(β; P_{1}, …, ∂P_{i}, …, P_{k}) ⊂ (−1)^{ϵ_{3}}∂M_{1}^{main}(β; P_{1}, …, P_{i}, …, P_{k}),

where

ϵ_{3} = 1 + ∑_{j=1}^{i−1}(deg P_{j} + 1).

Proof: (1) Step 1. Since we consider the main component, we can not change the order of marked points except by cyclic permutations. But we can compare orientations by using the non-main components intermediately. From Proposition 13.16, we note that

deg M_{1}^{main}(β_{2}; P_{i}, …, P_{i+k_{2}−1}) ≡ ∑_{j=i}^{i+k_{2}−1}(deg P_{j} + 1) ( mod 2 ).

Then by using Lemma 25.3 repeatedly, we find

M_{1}^{main}(β_{1}; P_{1}, …, M_{1}^{main}(β_{2}; P_{i}, …, P_{i+k_{2}−1}), P_{i+k_{2}}, …, P_{k})
= (−1)^{δ_{1}}M_{1}(β_{1}; M_{1}^{main}(β_{2}; P_{i}, …, P_{i+k_{2}−1}), P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k}),

where

δ_{1} = ( ∑_{j=i}^{i+k_{2}−1}(deg P_{j} + 1) + 1)(∑_{j=1}^{i−1}(deg P_{j} + 1)).

Step 2. Next, we compare with the orientation on

M_{1}(β_{1}; M_{1}^{main}(β_{2}; P_{i}, …, P_{i+k_{2}−1}), P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k})

and that on

∂M_{1}(β; P_{i}, …, P_{i+k_{2}−1}, P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k}).

This can be done by a similar, but slightly generalized, argument in the proof of Proposition 23.2 as follows. By Definition 25.1, we have

\begin{matrix}M_{1}(β; P_{i}, …, P_{i+k_{2}−1}, P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k}) \\ = & (−1)^{γ_{1}}M_{k+1}(β)_{(ev_{1},... ,ev_{k})} × ( ∏_{j=i}^{i+k_{2}−1} P_{j} × ∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j}), \end{matrix}

where

γ_{1} = (n + 1)( ∑_{j=i}^{i+k_{2}−1} ∑_{ℓ=i}^{j} deg P_{ℓ} + ∑_{j=1}^{i−1}( ∑_{m=i}^{i+k_{2}−1} deg P_{m} + ∑_{ℓ=1}^{j} deg P_{ℓ})

+ ∑_{j=i+k_{2}}^{k−1}( ∑_{m=i}^{i+k_{2}−1} deg P_{m} + ∑_{m=1}^{i−1} deg P_{m} + ∑_{ℓ=i+k_{2}}^{j} deg P_{ℓ})).

Using the iteration formula Lemma 22.3 (3), we have

\begin{matrix}M_{k+1}(β)_{(ev_{1},... ,ev_{k})} × ( ∏_{j=i}^{i+k_{2}−1} P_{j} × ∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j}) \\ = & (−1)^{γ_{2}}(M_{k+1}(β)_{(ev_{1},... ,ev_{k_{2}} )} × ( ∏_{j=i}^{i+k_{2}−1} P_{j}))_{(ev_{k_{2}+1},... ,ev_{k})} × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j}), \end{matrix}

where

γ_{2} = (k − k_{2})n(k_{2}n + ∑_{j=i}^{i+k_{2}−1}(n − deg P_{j})) ≡ (k − k_{2})n ∑_{j=i}^{i+k_{2}−1} deg P_{j}.

When we glue two holomorphic discs with marked points, our convention (see Remark 23.3) of the ordering of marked points and of the boundary orientation (and Proposition 23.2) show that

∂(M_{k+1}(β)_{(ev_{1},... ,ev_{k_{2}} )} × ( ∏_{j=i}^{i+k_{2}−1} P_{j}))_{(ev_{k_{2}+1},... ,ev_{k})} × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j})

⊃(−1)^{γ_{3}}((M_{k−k_{2}+2}(β_{1})_{ev_{1}^{β_{1}}} ×_{ev_{0}^{β_{2}}} M_{k_{2}+1}(β_{2}))_{(ev_{1}^{β},... ,ev_{k_{2}}^{β})} × ( ∏_{j=i}^{i+k_{2}−1} P_{j}))

(ev_{k_{2}+1}^{β},... ,ev_{k}^{β}) × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j}),

where

γ_{3} = (k − k_{2})(k_{2} − 1) + n + k − k_{2} + 1 ≡ n + 1 + kk_{2} + k_{2} (mod 2).

By the associativity property Lemma 22.3 (2), we have

((M_{k−k_{2}+2}(β_{1})_{ev_{1}^{β_{1}}} ×_{ev_{0}^{β_{2}}} M_{k_{2}+1}(β_{2}))_{(ev_{1}^{β},... ,ev_{k_{2}}^{β})} × ( ∏_{j=i}^{i+k_{2}−1} P_{j}))

(ev_{k_{2}+1}^{β},... ,ev_{k}^{β}) × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j})

=(M_{k−k_{2}+2}(β_{1})_{ev_{1}^{β_{1}}} ×_{ev_{0}^{β_{2}}} (M_{k_{2}+1}(β_{2})_{(ev_{1}^{β_{2}} ,... ,ev_{k_{2}}^{β_{2}} )} × ( ∏_{j=i}^{i+k_{2}−1} P_{j})))

(ev_{k_{2}+1}^{β},... ,ev_{k}^{β}) × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j})

=(−1)^{γ_{4}}(M_{k−k_{2}+2}(β_{1})_{ev_{1}^{β_{1}}} ×_{ev_{0}^{β_{2}}} M_{1}(β_{2}; P_{i}, …, P_{i+k_{2}−1}))

(ev_{k_{2}+1}^{β},... ,ev_{k}^{β}) × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j}),

OBSTRUCTION TO FLOER HOMOLOGY 225

where

γ_{4} = (n + 1) ∑_{j=i}^{i+k_{2}−2} ∑_{ℓ=i}^{j} deg P_{ℓ}.

Again by using the iteration formula, we find that

(M_{k−k_{2}+2}(β_{1})_{ev_{1}^{β_{1}}} ×_{ev_{0}^{β_{2}}} M_{1}(β_{2}; P_{i}, …, P_{i+k_{2}−1}))

(ev_{k_{2}+1}^{β},... ,ev_{k}^{β}) × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j})

=(−1)^{γ_{5}}M_{k−k_{2}+2}(β_{1})_{(ev_{1}^{β_{1}} ,... ,ev_{k−k_{2}+1}^{β_{1}})}×

(M_{1}(β_{2}; P_{i}, …, P_{i+k_{2}−1})_{(ev_{k_{2}+1}^{β},... ,ev_{k}^{β})} × (∏_{j=1}^{i−1} P_{j} × ∏_{j=i+k_{2}}^{k} P_{j}))

=(−1)^{γ_{5}+γ_{6}}M_{1}(β; M_{1}(β_{2}; P_{i}, …, P_{i+k_{2}−1}), P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k}),

where

γ_{5} = n(k − k_{2})(2n − ∑_{j=i}^{i+k_{2}−1}(deg P_{j} + 1)) ≡ n(k − k_{2})(k_{2} + ∑_{j=i}^{i+k_{2}−1} deg P_{j}),
γ_{6} = (n + 1)(deg M_{1}(β_{2}; P_{i}, …, P_{i+k_{2}−1})

+ ∑_{j=1}^{i−1}(deg M_{1}(β_{2}; P_{i}, …, P_{i+k_{2}−1}) + ∑_{ℓ=1}^{j} deg P_{ℓ})

+ ∑_{j=i+k_{2}}^{k−1}(deg M_{1}(β_{2}; P_{i}, …, P_{i+k_{2}−1}) + ∑_{m=1}^{i−1} deg P_{m} + ∑_{ℓ=i+k_{2}}^{j} deg P_{ℓ}))

= (n + 1)((k − k_{2})(k_{2} + ∑_{m=i}^{i+k_{2}−1} deg P_{m}) + (k − i − k_{2})( ∑_{m=1}^{i−1} deg P_{m})

+ ∑_{j=1}^{i−1} ∑_{ℓ=1}^{j} deg P_{ℓ} + ∑_{j=i+k_{2}}^{k−1} ∑_{ℓ=i+k_{2}}^{j} deg P_{ℓ}).

Then an elementary calculation shows that γ_{1} + γ_{2} + γ_{3} + γ_{4} + γ_{5} + γ_{6} ≡ n + 1 (mod 2) .

226 FUKAYA, OH, OHTA, ONO

Hence we have found that M_{1}(β_{1}; M_{1}^{main}(β_{2}; P_{i}, …, P_{i+k_{2}−1}), P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k}) ⊂ (−1)^{n+1}∂M_{1}(β; P_{i}, …, P_{i+k_{2}−1}, P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k}).

Step 3. On the other hand, by using Lemma 25.3 again, we can see that M_{1}(β; P_{i}, …, P_{i+k_{2}−1}, P_{1}, …, P_{i−1}, P_{i+k_{2}}, …, P_{k})

= (−1)^{δ_{2}}M_{1}^{main}(β; P_{1}, …, P_{i−1}, P_{i}, …, P_{i+k_{2}−1}, P_{i+k_{2}}, …, P_{k}),

where

δ_{2} = ( ∑_{j=i}^{i+k_{2}−1}(deg P_{j} + 1))(∑_{j=1}^{i−1}(deg P_{j} + 1)).

Therefore we have

δ_{1} + n + 1 + δ_{2} ≡ n + 1 + ∑_{j=1}^{i−1}(deg P_{j} + 1) ( mod 2 ),

which proves Proposition 25.13 (1).

(2) We prove Proposition 25.13 (2). We recall that

M_{1}^{main}(β; P_{1}, …, P_{i}, …, P_{k}) = (−1)^{ϵ_{1}}M_{k+1}^{main}(β)_{(ev_{1},... ,ev_{k})} ×_{f_{1}×···×f_{k}} (∏_{j=1}^{k} P_{j})

with

ϵ_{1} = (n + 1) ∑_{j=1}^{k−1} ∑_{ℓ=1}^{j} deg P_{ℓ}.

By Lemma 22.3 (1), we find that

∂(M_{k+1}^{main}(β)_{(ev_{1},... ,ev_{k})} ×_{f_{1}×···×f_{k}} (∏_{j=1}^{k} P_{j}))

=∂M_{k+1}^{main}(β)_{(ev_{1},... ,ev_{k})} ×_{f_{1}×···×f_{k}} (∏_{j=1}^{k} P_{j})

⨆(−1)^{n+k+nk}M_{k+1}^{main}(β)_{(ev_{1},... ,ev_{k})} ×_{f_{1}×···×f_{k}} ∂(∏_{j=1}^{k} P_{j}),

because dim M_{k+1}^{main}(β) = n + μ(β) − 3 + k + 1 ≡ n + k. Moreover by using Lemma 22.3 again, it is easy to see that

∂(∏_{j=1}^{k} P_{j}) = ⨆_{i=1}^{k}(−1)^{∑_{j=1}^{i−1} dim P_{j}}P_{1} × ⋯ × ∂P_{i} × ⋯ × P_{k}.

On the other hand, Definition 25.1 yields that

M_{1}^{main}(β; P_{1}, …, ∂P_{i}, …, P_{k})
=(−1)^{ϵ_{2}}M_{k+1}^{main}(β)_{(ev_{1},... ,ev_{k})} ×_{f_{1}×···×f_{k}} (∏_{j=1}^{i−1} P_{j} × ∂P_{i} × ∏_{j=i+1}^{k} P_{j})

with

ϵ_{2} = (n + 1) ∑_{j=1}^{k−1} ∑_{ℓ=1}^{j} deg Pℓ^{′},

where

deg Pℓ^{′} = \{ \begin{matrix}deg P_{ℓ} & for ℓ ≠ i \\ deg P_{ℓ} + 1 & for ℓ = i. \end{matrix}

Therefore we can see that

ϵ_{1} + n + k + nk + ∑_{j=1}^{i−1}(n − deg P_{j}) + ϵ_{2} ≡ ∑_{j=1}^{i−1}(deg P_{j} + 1) + 1 (mod 2),

which proves Proposition 25.13 (2). □

Then Proposition 25.13 (1) and (2) implies (13.26.2) and (13.26.1) respectively. Therefore we complete proving that

d̂∘ d̂= 0 in (C(L; Λ_{0,nov}), 𝔪).

25.5. Orientation on the moduli space of constant maps.

Next, let us assume that P_{1} and P_{2} are oriented. Considering the moduli space of constant maps M_{3}^{main}(L, 0; P_{1}, P_{2}), we have M_{3}^{main}(L, 0; P_{1}, P_{2}) = P_{1} ∩ P_{2} as a set. We have to study the difference of orientations between them. To do this, we firstly confirm the convention on orientation on P_{1} ∩ P_{2}. Convention 25.14. We assume P_{1} and P_{2} are submanifolds in L and intersect transversally. We denote the dimensions of L, P_{1} and P_{2} by n, p_{1} and p_{2}. Suppose that L, P_{1} and P_{2} are oriented. Take x ∈ P_{1} ∩ P_{2}. The orientation on the normal bundle N_{P_{i}}L of P_{i} (i = 1, 2) in L is defined by

T_{x}L = N_{x,P_{i}}L × T_{x}P_{i}.

Since P_{1} and P_{2} intersect transversally, we may assume that N_{x,P_{1}}L ⊂ T_{x}P_{2} and N_{x,P_{2}} ⊂ T_{x}P_{1}. Then we define the orientation on P_{1} ∩ P_{2} by

T_{x}L = N_{x,P_{1}}L × N_{x,P_{2}}L × T_{x}(P_{1} ∩ P_{2}).

Remark 25.15. The convention defines “cohomology orientation” on P_{1} ∩ P_{2}. Namely we have P_{1} ∩ P_{2} = (−1)^{(n−p_{1})(n−p_{2})}P_{2} ∩ P_{1}.

Then we can show the following; Proposition 25.16. We can give an oriented Kuranishi structure on M_{3}^{main}(L; 0) so that we have the following oriented isomorphism

ev_{0} : M_{3}^{main}(L; 0)_{(ev_{1},ev_{2})} ×_{f_{1}×f_{2}} (P_{1} × P_{2}) → (−1)^{n(p_{1}+p_{2})+p_{1}(n−p_{2})}P_{1} ∩ P_{2}.

Note that since we have from Definition 25.1

M_{1}^{main}(L; 0; P_{1}, P_{2}) = (−1)^{(n+1) deg P_{1}}M_{3}^{main}(L; 0)_{(ev_{1},ev_{2})} ×_{f_{1}×f_{2}} (P_{1} × P_{2}),

we can immediately obtain the following consequence. Corollary 25.17. With the oriented Kuranishi structure on M_{3}^{main}(L; 0), we have

(M_{1}^{main}(L; 0; P_{1}, P_{2}), ev_{0}) = (−1)^{deg P_{1}(deg P_{2}+1)}P_{1} ∩ P_{2}.

Proof of Proposition 25.16: In order to show Proposition 25.16, we are going to explicitly describe some (weakly submersive) Kuranishi structures on M_{3}^{main}(L; 0), P_{1} × P_{2} and their fibre product. Firstly, we describe the Kuranishi structure on M_{3}^{main}(L; 0). We like to study orientations on the Kuranishi structure on M_{3}^{main}(L; 0). Since M_{3}^{main}(L; 0) consists of constant maps, the argument is a local problem. Here we shall consider the case L = ℝ^{n} and M = ℂ^{n} with the standard orientations. We put three marked points z_{0} = −1,_{1} = −\sqrt{−1},_{2} = 1 on the boundary of the unit disc D^{2}. The maps ev_{1} and ev_{2} on M_{3}^{main}(L; 0) are the evaluation maps at z_{1} and z_{2}.

Now we put

U := \{u_{a,b} : D^{2} → ℂ^{n} | u_{a,b}(z) = −\frac{1}{2}a\sqrt{−1}\overline{z} + b + \frac{1}{2}a\sqrt{−1}z, a, b ∈ ℝ^{n}\}
E := {(u_{a,b}, c) | z ∈ D^{2}, a, b, c ∈ ℝ^{n}}

and define a section s : U → E by s(u_{a,b}) = a. Then the moduli space M_{3}^{main}(L; 0) of constant maps to L is nothing but s^{−1}(0) ≃ ℝ_{b}^{n}.

Of course, U can be identified with ℝ_{a}^{n} ×ℝ_{b}^{n} which can be regarded as T_{∗}L×T_{∗}L, and E is a trivial vector bundle over U with fibre ℝ^{n} ≅ T_{∗}L. Applying the argument in 21.1 to M_{3}^{main}(L; 0), the orientation in this case is given by the orientation of L after identifying it with the target L via ev_{0}. We must provide the orientation on U and the fiberwise orientation on E respectively so that the orientation of M_{3}^{main}(L; 0) is consistent with the orientation of (s; E → U) in the sense of Kuranshi structure. From now on, we give U an orientation through identification with T_{∗}L × T_{∗}L and the fiber orientation of E through identification with ℝ^{n} ≅ T_{∗}L and (s; E → U) gives a Kuranishi structure on M_{3}^{main}(L; 0).

Note that the evaluation map (ev_{1}, ev_{2}) : U → L × L, which is given by

(ev_{1}, ev_{2})(u_{a,b}) = (a + b, b),

is an orientation preserving map. We have an orientation preserving identification

E_{∗} × T_{∗}M_{3}^{main}(L; 0) = T_{∗}U = T_{∗}L × T_{∗}L.

Secondly, we describe the Kuranishi structure on P_{1} × P_{2}. Let U_{i} ⊂ L be a tubular neighborhood of P_{i} in L (i = 1, 2). Let N_{P_{i}} be the normal bundle of P_{i} in L and s_{i} : U_{i} → N_{P_{i}} the canonical section. Then (s_{i}; N_{P_{i}} → U_{i}) gives a Kuranishi structure on P_{i}. Namely P_{i} = s_{i}^{−1}(0) and the orientation is given by

N_{P_{i}}_{x_{i}} × T_{x_{i}}s_{i}^{−1}(0) = T_{x_{i}}U_{i} ≅ T_{x_{i}}L

for x_{i} ∈ P_{i}. Thus P_{i} = s_{i}^{−1}(0) as an oriented space. The product of the Kuranishi structures (s_{1} ⊕ s_{2}; N_{P_{1}} ⊕ N_{P_{2}} → U_{1} × U_{2}) gives a Kuranishi structure on P_{1} × P_{2}. Then by Convention 22.1.(4), the orientation defined by the Kuranishi structure is just the same as the product orientation on P_{1} × P_{2} as following

P_{1} × P_{2} = (−1)^{p_{1}(n−p_{2})}(s_{1} ⊕ s_{2})^{−1}(0).

Because we have

N_{P_{1}}_{x_{1}} × N_{P_{2}}_{x_{2}} × T (s_{1} ⊕ s_{2})^{−1}(0)_{(x_{1},x_{2})} = T_{x_{1}}U_{1} × T_{x_{2}}U_{2}
= (N_{P_{1}}_{x_{1}} × T_{x_{1}}P_{1}) × (N_{P_{2}}_{x_{2}} × T_{x_{2}}P_{2})
= (−1)^{p_{1}(n−p_{2})}N_{P_{1}}_{x_{1}} × N_{P_{2}}_{x_{2}} × T_{x_{1}}P_{1} × T_{x_{2}}P_{2}

(See Convention 22.1 (4).)

Now, for two submersions;

(ev_{1}, ev_{2}) : U → U_{1} × U_{2}, identity : U_{1} × U_{2} → U_{1} × U_{2},

we can consider the fibre product of Kuranishi structures

(s ⊕ s_{1} ⊕ s_{2}; E ⊕ N_{P_{1}} ⊕ N_{P_{2}} → U_{(ev_{1},ev_{2})} ×_{identity} (U_{1} × U_{2})),

which gives a Kuranishi structure on M_{3}^{main}(L; 0)_{(ev_{1},ev_{2})} ×_{f_{1}×f_{2}} (P_{1} × P_{2}). As for the oreientation, we have from Convention 22.1 (4),

M_{3}^{main}(L; 0)_{(ev_{1},ev_{2})} ×_{f_{1}×f_{2}} (P_{1} × P_{2})

= (−1)^{rank(N_{P_{1}} ⊕N_{P_{2}} )(dim M_{3}^{main}(L;0)−dim U_{1}×U_{2})} × (−1)^{p_{1}(n−p_{2})}(s ⊕ s_{1} ⊕ s_{2})^{−1}(0)

= (−1)^{n(p_{1}+p_{2})+p_{1}(n−p_{2})}(s ⊕ s_{1} ⊕ s_{2})^{−1}(0). (25.18)

Now we are going to compare the orientation on (s ⊕ s_{1} ⊕ s_{2})^{−1}(0) with that on P_{1} ∩ P_{2}. The orientation on (s ⊕ s_{1} ⊕ s_{2})^{−1}(0) is defined by

E_{∗} × N_{P_{1}}_{∗} × N_{P_{2}}_{∗} × T_{∗}(s ⊕ s_{1} ⊕ s_{2})^{−1}(0) = T_{∗}(U_{(ev_{1},ev_{2})} ×_{identity} (U_{1} × U_{2})).

Then we will show Lemma 25.19. The isomorphism

ev_{0} : (s ⊕ s_{1} ⊕ s_{2})^{−1}(0) → P_{1} ∩ P_{2}

respects the orientation. Proof of Lemma 25.19: The orientation of P_{1} ∩ P_{2} and (s ⊕ s_{1} ⊕ s_{2})^{−1}(0) are given by the following exact sequences respectively:

0 → T_{∗}(P_{1} ∩ P_{2}) → T_{∗}L → N_{P_{1}} ⊕ N_{P_{2}} → 0

and

0 → T_{∗}(s ⊕ s_{1} ⊕ s_{2})^{−1}(0) → T_{∗}(U_{ev_{1},ev_{2}} ×_{id} (U_{1} × U_{2})) → E ⊕ π_{1}^{∗}N_{P_{1}} ⊕ π_{2}^{∗}N_{P_{2}} → 0.

We consider the following commutative diagram, which relates two exact sequences above.

E_{∗}  ⟶^{id}
⊕
0 ⟶ T_{∗}(P_{1} ∩ P_{2}) ⟶ T_{∗}L ⟶
I ↑  Φ ↑
0 ⟶ T_{∗}(s ⊕ s_{1} ⊕ s_{2})^{−1}(0) ⟶ T_{∗}(U_{ev_{1},ev_{2}} ×_{id} (U_{1} × U_{2})) ⟶

⟶^{id}  E_{∗}
⊕
⟶ N_{P_{1}} ⊕ N_{P_{2}} ⟶ 0
Ψ ↑
⟶ E ⊕ π_{1}^{∗}N_{P_{1}} ⊕ π_{2}^{∗}N_{P_{2}} ⟶ 0

where vertical isomorphisms are given as follows: I is nothing but the evaluation map

T_{∗}ev_{0} : T_{∗}(s ⊕ s_{1} ⊕ s_{2})^{−1}(0) → T_{∗}(P_{1} ∩ P_{2})

and can be explicitly written as

I((0, v), v, v) = v.

The maps Φ and Ψ are defined as

Φ((a, b), v_{1}, v_{2}) = (a, pr_{1}(v_{1}) + pr_{2}(v_{2}) + pr_{3}(b)) with v_{1} = a + b, v_{2} = b,

and

Ψ(u, v_{1}, v_{2}) = (u, v_{1}, v_{2}).

Here pr_{i} is the i-th factor projection of T_{∗}L = N_{P_{1}} ⊕ N_{P_{2}} ⊕ (P_{1} ∩ P_{2}). Because the isomorphisms Φ and Ψ preserve their orientations, it follows from the diagram chasing argument that the isomorphism I also preserves orientations. This finishes the proof. □

Therefore, combining (25.18) and Lemma 25.19, we complete the proof of Proposition 25.16.

25.6. The Bott-Morse case II.

We discuss the orientation problem for the space of connecting orbits in general Bott-Morse setting. First of all, we review some notations in Section 16. Let L_{0} and L_{1} be two Lagrangian submanifolds which intersect cleanly. We denote by R_{h} connected components of L_{0} ∩ L_{1}. Write V_{h} = {(T L_{0} + T L_{1})/(T L_{0} + T L_{1})^{⊥_{ω}}}|_{R_{h}}, a vector bundle on R_{h}. For each point p in R_{h}, we make the following construction. Denote by U and U′ the image of T_{p}L_{0} and T_{p}L_{1} in (V_{h})_{p}. We consider a path γ_{t} of Lagrangian subspaces in (V_{h})_{p} from U to U′ so that the path γ_{t} ⊕ T R_{h} is a path of oriented Lagrangian subspaces in T M. (Note that R_{h} itself may not be orientable.) We also choose a trivialization of this family over [0, 1] as in section 21.3.

In this section we consider a connecting orbit as a holomorphic map u : D^{2} → M such that u(∂_{0}D^{2}) ⊂ L_{0}, u(∂_{1}D^{2}) ⊂ L_{1}, u(−1) = q ∈ R_{h′} and u(1) = p ∈ P, where ∂_{0}D^{2} is the arc with positive imaginary part and ∂_{1}D^{2} is the arc with negative imaginary part. (This is opposite to the setting in section 25.2, because the role of marked points ±1 has been changed so that it becomes consistent with other conventions in A_{∞}-notation.) Then we consider the index of Dolbeault operator on a half disc with Lagrangian boundary condition as follows.

We write

Θ_{−} = (D^{2} ∩ {Rez ≤ 0})♯([0, ∞) × [0, 1])
Θ_{+} = ((−∞, 0] × [0, 1])♯(D^{2} ∩ {Rez ≥ 0}).

Take the Dolbeault operator

\overline{∂}_{(±,γ_{t}⊕T R_{h})} : W_{γ_{t}⊕T R_{h}}^{1,p}(Θ_{±}) → L^{p}(Θ_{±})

on the space

W_{γ_{t}⊕T R_{h}}^{1,p}(Θ_{±}) := \{ζ ∈ W^{1,p}(Θ_{±}, ℂ^{n}) | ζ(z) ∈ γ_{t} ⊕ T R_{h} for z = e^{∓π\sqrt{−1}(t+ \frac{1}{2} )},
ζ(τ, 1) ∈ T_{p_{j}}L_{1}, ζ(τ, 0) ∈ T_{p_{j}}L_{0}}.

Then we get a family of operators on the space of paths γ_{t}. Note that we consider the operators on Θ_{±}, which have cylindrical ends and the solutions asymptotic to the zero eigensection of J \frac{d}{dt} are not contained in W^{1,p}, L^{p}-space. Taking this into account, a similar argument as in Remark 21.12 (1) yields that a choice of orientation of \overline{∂}_{(+,γ_{t}⊕T R_{h})} and a choice of orientation of \overline{∂}_{(−,γ_{t}⊕T R_{h})} determines the orientation of T_{p}R_{h}. Glue the operators \overline{∂}_{(±,γ_{t}⊕T R_{h})} to get a Dolbeault operator on the disc D^{2}, the index of which is canonically oriented in Theorem 21.3. Then this orientation determines the orientation of index \overline{∂}_{(−,γ_{t}⊕T R_{h})} ⊕ T_{p}R_{h} ⊕ index \overline{∂}_{(+,γ_{t}⊕T R_{h})} by the index sum formula.

Moreover, for two paths γ_{t} and σ_{t} in (V_{h})_{p}, the orientation of \overline{∂}_{(+,γ_{t}⊕T R_{h})} determines the orientation of \overline{∂}_{(+,σ_{t}⊕T R_{h})} in the following way. Consider two Lagrangian boundary conditions given by γ_{t} and σ_{t} for T_{p}M-valued functions on Θ_{+}. Then there exists a complex vector bundle E of rank n on S^{2} such that after gluing this vector bundle at an interior point of Θ_{+}, namely changing the trivialization, the boundary condition γ_{t} ⊕ T R_{h} changes to the boundary condition σ_{t} ⊕ T R_{h}. By the index sum formula, the orientation of the index of \overline{∂}_{(+,γ_{t}⊕T R_{h})} and the orientation of the index of the Dolbeault operator twisted by E determines the orientation of \overline{∂}_{(+,σ_{t}⊕T R_{h})}. Note that the index of the Dolbeault operator twisted by E has a canonical orientation as a complex virtual vector bundle. Hence the orientation of \overline{∂}_{(+,γ_{t}⊕T R_{h})} determines the orientation of \overline{∂}_{(+,σ_{t}⊕T R_{h})}. In other words, Proposition 21.3 and tbe index sum formula imply that the orientation of index \overline{∂}_{(+,γ_{t}⊕T R_{h})} determines the orientation of the T_{p}R_{h} ⊕ index \overline{∂}_{(−,σ_{t}⊕T R_{h})}, which determines the orientation of index \overline{∂}_{(+,σ_{t}⊕T R_{h})} as we discussed before. Hence the orientation bundle of the family index of these operators descends to a local system on R_{h}, which we denote by Θ_{R_{h}}^{±}. In finite dimensional Bott-Morse situation, Θ^{+} and Θ^{−} correspond to the orientation bundles of the negative definite part (unstable direction) and the positive definite part (stable direction) respectively.

Note that R_{h} is not necessarily orientable and the space of connecting orbits may not be orientable. For instance, we deal with fiber product over a non-orientable space. In order to treat them, we introduce the notion of the orientation bundle of a space with Kuranishi structure and present its fundamental properties.

Definition 25.20. For a local chart (s; E → U) of a Kuranishi structure, we call detE ⊗ detT U the orientation bundle.

Remark 25.21. The orientation bundles of local charts are naturally glued to a line bundle over the space with Kuranishi structure. We call it the orientation bundle of the space with Kuranishi structure.

Next we give our convention on the identification of orientation bundles of fiber products.

Convention 25.22. Let O_{X} denote the orientation bundle of X. For f : X → B and g : Y → B, O_{X_{f} ×_{g}Y} is identified with pr_{X}^{∗}(0_{X} ⊗f ∗O_{B})⊗π^{∗}O_{B} ⊗pr_{Y}^{∗} (g^{∗}O_{B} ⊗ O_{Y}), which is naturally identified with pr_{X}^{∗}O_{X} ⊗ π^{∗}O_{B} ⊗ pr_{Y}^{∗} O_{Y}, where pr_{X} and pr_{Y} are projections to X and Y respectively and π is the projection to B.

Remark 25.23. Here the order of factors in tensor product is essential. For instance, the orientation bundle of a product of manifolds X and Y, the identification above corresponds to the isomorphism O_{X} ⊗ O_{Y} → O_{X×Y} , s ⊗ t ↦ s ∧ t.

We have similar properties as in Lemma 22.3.

Lemma 25.24. We have the following identifications.

(1) O_{∂(X×_{B}Y)}|_{∂X×_{B}Y} and 0_{∂X×_{B}Y}, O_{∂(X×_{B}Y)}|_{X×_{B}∂Y} and (−1)^{x+y}O_{X×_{B}∂Y}.

(2) O_{(X_{1}×_{B_{1}} X_{2})×_{B_{2}} X_{2}} and O_{X_{1}×_{B_{1}} (X_{2}×_{B_{2}} X_{3})}.

(3) O_{X_{1}×_{B_{1}×B_{2}} (X_{2}×X_{3})} and (−1)^{b_{2}(b_{1}+x_{2})}O_{(X_{1}×_{B_{1}} X_{2})×_{B_{2}} X_{3}}. More generally,

O_{X_{1}×_{B_{1}×···×B_{l}} (X_{2}×···X_{l+1})}

and

(−1)^{∑_{k=2}^{l} b_{k} ∑_{j=2}^{k}(b_{j−1}+x_{j})}O_{(···(X_{1}×_{B_{1}} X_{2})··· )×_{B_{l}} X_{l+1}}.

Recall that we are working with cohomology theory and an ordinary cochain is represented by a chain with an oriented normal bundle, in other words, a chain with coefficients in the orientation bundle when the space is a (not necessarily orientable) manifold.

Let P be a chain in R_{h} with coefficient in det T R_{h} ⊗ Θ_{R_{h}}^{+}, which is canonically isomorphic to Θ_{R_{h}}^{−}. Recall that a connecting orbit is a holomorphic map u : D^{2} → M such that u(∂_{0}D^{2}) ⊂ L_{0}, u(∂_{1}D^{2}) ⊂ L_{1}, u(−1) = q ∈ R_{h′} and u(1) = p ∈ P, where ∂_{0}D^{2} is the arc with positive imaginary part and ∂_{1}D^{2} is the arc with negative imaginary part.

At the point p, we take the operator \overline{∂}_{(+,γ_{t}⊕T R_{h})}. Because P is a chain with coefficient in det T R_{h} ⊗ Θ_{R_{h}}^{+}, we have the corresponding orientation to T_{p}R_{h} ⊕ index \overline{∂}_{(+,γ_{t}⊕T R_{h})}. For the gluing argument for orientations, one may need a choice of orientation of T_{p}R_{h}, which corresponds to the zero eigensection of J \frac{d}{dt}. However, according to our Convention 25.22, the orientation bundles of R_{h} and P appear. These two effects cancel each other and our orientation does not depend on the orientation bundle of R_{h}.

For the point q, we pick a local section of det T R_{h′} ⊗ Θ_{h′}^{+}, i.e., the orientation bundle of T_{q}R_{h′} ⊕ \overline{∂}_{(+,γ′_{t}⊕T R_{h′})}, so the corresponding orientation of \overline{∂}_{(−,γ′_{t}⊕T R_{h′})}. To get a local section of the orientation bundle, we need a local section of the orientation bundle of \overline{∂}_{(−,γ′_{t}⊕T R_{h′})} ⊕ T_{q}R_{h′}. However, according to Convention 25.22, the orientation bundle of R_{h′} appears once again. So it is only necessary to have a local section of the orientation bundle of \overline{∂}_{(−,γ′_{t}⊕T R_{h′})}. Then we consider the corresponding local section of the orientation bundle of the fiber product, i.e., a local orientation of the tangent space , at u, of the moduli space of connecting orbits and we locally get an orientation on the image of the evaluation map at −1. This is local, because we may not take a global section of Θ_{R_{h′}}. Combining the local orientation depending on the local section and the local section itself, we get a chain with coefficient in det T R_{h′} ⊗ Θ_{R_{h′}}^{+} .

For more general situation of “stable connecting orbits”, we must combine the argument here and the argument in section 25.2. Let M_{N+2}(λ) be the space of connecting orbits in homotopy class λ with N + 2 marked points. We have discussed the orientation bundle of it. For “stable connecting orbits”, we identify the orientation bundle of

M_{1}(λ − ∑_{k=1}^{ℓ} β_{i_{k}} − ∑_{k=1}^{m} β_{j_{k}}; B_{i_{1}}^{0}, …, B_{i_{ℓ}}^{0}, B_{j_{1}}^{1}, …, B_{j_{m}}^{1}; [P, f])

and the orientation bundle of

\begin{matrix}(−1)^{ϵ}(M_{ℓ+m+2}(λ − ∑_{k=1}^{ℓ} β_{i_{k}^{0}} − ∑_{k=1}^{m} β_{j_{k}^{1}}) \\ _{(ev_{1},ev_{2}... ,ev_{ℓ+1},ev_{ℓ+2}... ,ev_{ℓ+m+1})} ×_{f×...} (P × ∏_{k=1}^{ℓ} B(β_{i_{k}^{0}}) × ∏_{k=1}^{m} B(β_{j_{k}^{1}}))), \end{matrix}

where ϵ = (d_{h} +1)(deg P)(ℓ+m)+(n+1) ∑_{k=1}^{ℓ+m−1} ∑_{j=1}^{k} 1 and deg P = d_{h} −dim P. Then we define the coboundary operator as follows.

δ = (−1)^{μ(R_{h′} ,γ′_{t})}{(−1)^{d_{h′}} δ_{0} + ∑_{λ≠0} δ_{λ}},

where μ(R_{h′}, γ′_{t}) is the index of \overline{∂}_{(+,γ′_{t}⊕T R_{h′})}, δ_{0} is the classical coboundary operator, and δ_{λ} is the union of

ev_{0}(M_{1}(λ − ∑_{k=1}^{ℓ} β_{i_{k}} − ∑_{k=1}^{m} β_{j_{k}}; B_{i_{1}}^{0}, …, B_{i_{ℓ}}^{0}, B_{j_{1}}^{1}, …, B_{j_{m}}^{1}; [P, f])).

The reason for the sign (−1)^{μ(R_{h′} ,γ′_{t})} is the following. For the bubbling ends, our arguments before depends on Proposition 23.2. In this case, the dimension of M_{m_{1}+1} is n + m_{1} modulo 2 because of evenness of the Maslov index. However, in the case of connecting orbits, the dimension of M_{N+2}(λ) is μ(R_{h′}, γ′_{t}) − μ(R_{h}, γ_{t}) + d_{h′} + N + 1. We leave the readers to check the cancellation in δ ∘ δ, which is similar to the previous argument.

As for the degree of the Floer cochain complex in Bott-Morse setting, we can give the degree of a chain P in R_{h} assigned with a homotopy class of a Lagrangian path γ_{t} by index \overline{∂}_{(+,γ_{t}⊕T R_{h})} + deg P, where deg P is the codimension of P in R_{h}.

25.7. Orientation of M_{k+1}^{main}(L, j : β; para).

In section 15, we used the moduli space M_{k+1}^{main}(L, j : β; para) in the definition of the filtered A_{∞} homomorphism in Proposition 15.26. Here we give an orientation of M_{k+1}^{main}(L, j : β; para). We recall an element of M_{k+1}^{main}(L, j : β; para). Let ((Σ, z⃗), (u_{α}), (s_{α})) be a system satisfying the following properties

(15.22.1) u_{α} : (Σ_{α}, ∂Σ_{α}) → (M, L) is a J_{s_{α}} holomorphic map. (15.22.2) If α_{1} ≤ α_{2} then s_{α_{1}} ≤ s_{α_{2}} (15.22.3) ((Σ, z⃗), (u_{α})) is stable in the sense of Definition 3.8. (15.22.4) The homology class of u_{α} is β(α) and ∑_{α} β(α) = β. (15.22.5) If z ∈ Σ_{α} ∩ Σ_{α′} then u_{α}(z) = u_{α′}(z).

Denote by M_{1}^{main}(β_{1}; P_{1}, …, P_{k}; J_{s}) the moduli space of objects in (15.22) such that u is J_{s}-holomorphic on the component containing the 0-th marked point. We put the orientation on top dimensional strata by induction on the number of irreducible components of the domain of u. When the domain is irreducible, we follow the convention in Section 25.4, i.e.,

M_{1}^{main, irred}(β; P_{1}, …, P_{k}; para) = ⋃_{0≤s≤1}{s} × M_{1}^{main, irred}(β; P_{1}, …, P_{k}; J_{s}),

and

M_{1}^{main, irred}(β; P_{1}, …, P_{k}; J_{s}) = (−1)^{ϵ_{1}}M_{k+1}^{main}(L; β; J_{s})_{(ev_{1},... ,ev_{k})}×_{f_{1}×···×f_{k}} (∏_{i=1}^{k} P_{i})

where

ϵ_{1} = (n + 1) ∑_{j=1}^{k−1} ∑_{i=1}^{j} deg P_{i}.

236 FUKAYA, OH, OHTA, ONO

Note that P appearing in M_{1}^{main, irred}(β; …; para) are chains in L. Then Proposition 25.13 (1) implies that

(−1)^{ϵ_{2}+1} ⋃_{0≤s≤1}{s} × M_{1}^{main, irred}(β_{1}; P_{1}, …, P_{i−1}, M_{1}^{main, irred}(β_{2}; P_{i}, …, P_{i+l−1}; J_{s}),

P_{i+l}, …, P_{k}) ⊂ ∂M_{1}^{main, irred}(β; P_{1}, …, P_{k}; para) where

ϵ_{2} = n + 1 + ∑_{j=1}^{i−1}(deg P_{j} + 1).

On the other hand, Proposition 25.13 (2) implies that

⋃_{0≤s≤1}{s} × M_{1}^{main, irred}(β_{1}; P_{1}, …, P_{i−1}, M_{1}^{main, irred}(β_{2}; P_{i}, …, P_{i+l−1}; J_{t}), P_{i+l}, …, P_{k})

⊂(−1)^{ϵ_{3}+1}M_{1}^{main, irred}(β_{1}; P_{1}, …, P_{i−1}, ∂Q_{i,ℓ}(β_{2}), P_{i+l}, …, P_{k}; para),
⊂(−1)^{ϵ_{3}+1}∂M_{1}^{main, irred}(β_{1}; P_{1}, …, P_{i−1}, Q_{i,ℓ}(β_{2}), P_{i+l}, …, P_{k}; para),

where

ϵ_{3} = 1 + ∑_{j=1}^{i−1}(deg P_{j} + 1).

and

Q_{i,ℓ}(β) = ⋃_{0≤s≤1} Q_{i,ℓ}^{s}(β) ⊂ [0, 1] × L;
Q_{i,ℓ}^{s}(β) := ⋃_{0≤u≤s}{u} × M_{1}^{main, irred}(β; P_{i}, …, P_{i+l−1}; J_{u})

Hence the orientation of

⋃_{0≤s≤1}{s}×M_{1}^{main, irred}(β_{1}; P_{1}, …, P_{i−1}, M_{1}^{main, irred}(β_{2}; P_{i}, …, P_{i+l−1}; J_{s}), P_{i+l}, …, P_{k})

as the boundary of

M_{1}^{main, irred}(β; P_{1}, …, P_{k}; para)

and the orientation as the boundary of (−1)^{ϵ_{2}+ϵ_{3}+1}M_{1}^{main, irred}(β_{1}; P_{1}, …, P_{i−1}, Q_{i,ℓ}(β_{2}), …, P_{i+l−1}; J_{s}), P_{i+l}, …, P_{k}; para) are opposite. Thus we modify the orientation on

M_{1}^{main, irred}(β_{1}; P_{1}, …, P_{i−1}, Q_{i,ℓ}(β_{2}), P_{k+l}, …, P_{k}; para) by multiplying (−1)^{ϵ_{2}+ϵ_{3}+1} = (−1)^{n+1}. More generally, for the moduli space of objects in (15.22) with p + 1 components adjacent to the component containing the 0-th marked point, we put the orientation as follows:

(−1)^{d(n+1)} ⋃_{0≤s≤1}{s} × M_{1}^{main, irred}(β; Q_{1}, …, Q_{p+r}; J_{s}),

where d is the number of irreducible components of the domain of u in (15.22),

Q_{i_{j}} = ⋃_{0≤u≤s}{u} × M_{1}^{main, irred}(β_{j}; P_{1}^{(j)}, …, P_{q_{j}}^{(j)}; J_{u}) j = 1, …, p,

other Q_{i} are P_{i} directly attached to the holomorphic disc with the 0-th marked point, and P_{a}^{(j)} are similar objects to P′_{i} in the “lower stages”. These orientations match on codimension 1 strata and give a compatible orientation on the moduli space M_{1}^{main}(β; P_{1}, …, P_{k}; para).

Next we disuss the sign problem for the map φ̂ in Theorem 15.19 and Proposition 15.26. We put

𝔣_{k,β}(P_{1}, …, P_{k}) = (−1)^{n+1}(M_{1}^{main}(β; P_{1}, …, P_{k}; para), ev_{0}),

for (k, β) ≠ (1, β_{0}) and

𝔣_{1,β_{0}}(P) = P.

A part of the boundary of M_{1}^{main}(β; P_{1}, …, P_{k}; para) given by s = 1 corresponds to the sum of 𝔪_{k′,β_{1}}^{(2)} ∘ φ̂_{β_{2}}(P_{1}, …, P_{k}) with β = β_{1} + β_{2}, β_{1} ≠ β_{0} and k′ ≤ k. There are other boundaries occuring when s = 0 and

P′_{i′} = M_{1}^{main, irred}(β_{2}; P_{i}, …, P_{i+l−1}; para)

for some i′, i.e., the {u = 0} part of the boundary of

(−1)^{d(n+1)}M_{1}^{main, irred}(β_{1}; P′_{1}, …, P′_{i′−1}, Q_{i,ℓ}(β_{2}), P′_{i′+1}, …, P′_{p+r}; para),

where β_{1} ≠ β_{0}. Since we have

∑_{j=1}^{i′−1}(deg P′_{j} + 1) = ∑_{j=1}^{i−1}(deg P_{j} + 1),

this boundary corresponds to the sum of

(−1)^{ϵ_{3}}𝔣_{k−l+1,β_{1}}(P_{1}, …, P_{i−1}, 𝔪_{l,β_{2}}^{(0)}(P_{i}, …, P_{i+l−1}), P_{i+l}, …, P_{k})

with β = β_{1} + β_{2}. On the other hand, the same thing, i.e., the boundary of M_{1}^{main}(β; P_{1}, …, P_{k}; para), corresponds to (−1)^{n}(−1)^{n+1}𝔪_{1,β_{0}}^{(2)} ∘ 𝔣_{k,β}(P_{1}, …, P_{k}).

Finally we consider the boundary of M_{1}^{main, irred}(β; P; para). The boundary {s = 0, 1} corresponds to 𝔪_{1,β} ∘ 𝔣_{1,β_{0}}(P) − 𝔣_{1,β_{0}} ∘ 𝔪_{1,β}(P). On the other hand, the same thing contributes to (−1)^{n}(−1)^{n+1}𝔪_{1,β_{0}} ∘ 𝔣_{1,β}(P).

Combining these discussions, we have derived

d̂^{(2)} ∘ φ̂ − φ̂ ∘ d̂^{(0)} = 0.

Chapter 7. Lagrangian Surgery.

§26. Construction of examples by Lagrangian surgery.

In this chapter, we study the relation between Lagrangian surgery and pseudoholomorphic discs. Combining this with the results from [Fu7], we obtain various examples of the construction we made in the previous chapters.

We first briefly review the Lagrangian surgery. See [Po1] for more details. Let L_{1} and L_{2} be Lagrangian submanifolds in M. We assume for simplicity that they intersect each other at p ∈ L_{1} ∩ L_{2} transversely. We can find a neighborhood U of p and a symplectic diffeomorphism I : U → V ⊂ ℂ^{n} such that I(p) = 0, I(L_{1} ∩ U) = ℝ^{n} ∩ V, I(L_{2} ∩ U) = \sqrt{−1}ℝ^{n} ∩ V.

Let ϵ be a small real number. We choose a smooth real valued function f_{ϵ} on ℝ^{n}\{0} such that

f_{ϵ}(x) = ϵ log |x|

and denote by H_{ϵ} ⊂ ℂ^{n} the graph of df_{ϵ}(x), which is a Lagrangian submanifold in T^{∗}(ℝ^{n}\\{0\}) ≅ ℂ^{n}\\sqrt{−1}ℝ^{n} and asymptotic to \sqrt{−1}ℝ^{n} as |x| → 0.

Let τ : ℂ^{n} → ℂ^{n} be the reflection along the diagonal

Δ = \{(x_{1} + \sqrt{−1}y_{1}, ⋯, x_{n} + \sqrt{−1}y_{n})|x_{i} = y_{i}\},

i.e., be the map

(x_{1} + \sqrt{−1}y_{1}, ⋯, x_{n} + \sqrt{−1}y_{n}) ↦ (y_{1} + \sqrt{−1}x_{1}, ⋯, y_{n} + \sqrt{−1}x_{n}).

Next we consider the function

f̃_{ϵ}(x) = \{ \begin{matrix}ϵ log |x| & if |x| < |ϵ| \\ 0 & if |x| > 2|ϵ| \end{matrix}

and its graph Graph df̃_{ϵ}. We define a Lagrangian submanifold H′_{ϵ} such that the following holds:

(26.1.1) τ(H′_{ϵ}) = H′_{ϵ}. (26.1.2)

\{(x_{1} + \sqrt{−1}y_{1}, ⋯, x_{n} + \sqrt{−1}y_{n}) | ∀i x_{i} ≥ y_{i}\} ∩ H′_{ϵ}
= \{(x_{1} + \sqrt{−1}y_{1}, ⋯, x_{n} + \sqrt{−1}y_{n})|∀i x_{i} ≥ y_{i}\} ∩ Graph df̃_{ϵ}.

Then, by construction, H′_{ϵ} is invariant under τ and H′_{ϵ} = ℝ^{n} ∪ \sqrt{−1}ℝ^{n} outside a small neighborhood of 0. Therefore we can construct a Lagrangian submanifold L_{ϵ} ⊂ M such that

L_{ϵ} − U = L_{1} ∪ L_{2} − U, I(L_{ϵ} ∩ U) = H′_{ϵ} ∩ V.

We call L_{ϵ} the Lagrangian submanifold obtained from L_{1} and L_{2} by Lagrangian surgery. We remark that there are two different choices, ϵ > 0 and ϵ < 0 up to the Lagrangian isotopies (but not by Hamiltonian isotopies). In fact if n > 2, L_{ϵ} are not isotopic to L_{−ϵ}. On the other hand, using the fact that the local models are Hamiltonian isotopic, one can easily verify that L_{ϵ_{1}} is Lagrangian isotopic to L_{ϵ_{2}} if ϵ_{1}/ϵ_{2} > 0. However they are not Hamiltonian isotopic to one another in general.

Let L_{0} be another Lagrangian submanifold and x ∈ L_{1} ∩ L_{0}, y ∈ L_{2} ∩ L_{0}. We assume that L_{0} intersect both L_{1} and L_{2} transversely and that L_{0} ∩ L_{1} ∩ L_{2} is empty. If ϵ is small enough, then we may assume x, y ∈ L_{ϵ}.

We are going to study the moduli space M(L_{0}, L_{ϵ}; x, y) by comparing it with the moduli space of holomorphic triangles, which we define below.

We fix three points u_{1}, u_{2}, u_{3} on ∂D^{2} in the counter-clockwise order and define

M(L_{0}, L_{1}, L_{2}; x, y, p)

= \{φ : D^{2} → M | \begin{matrix}φ is pseudo-holomorphic, \\ φ(u_{1}) = x, φ(u_{2}) = p, φ(u_{3}) = y, \\ φ(\overline{u_{1}u_{2}}) ⊂ L_{0}, φ(\overline{u_{2}u_{3}}) ⊂ L_{1}, φ(\overline{u_{3}u_{1}}) ⊂ L_{2}. \end{matrix}\}

Here \overline{u_{i}u_{j}} denotes the part of S^{1} = ∂D^{2} between u_{i} and u_{j}.

Gromov’s compactness theorem implies, roughly speaking, that if φ_{ϵ} ∈ M(L_{0}, L_{ϵ}; x, y) is a family of elements, it will have a subsequence converging to an element of M(L_{0}, L_{1}, L_{2}; x, y, p), as ϵ → 0. To make this statement more precise is the purpose of §§27, 29. In this section, we will just state the result.

Take φ ∈ M(L_{0}, L_{1}, L_{2}; x, y, p). We assume that the virtual dimension of the moduli space M(L_{0}, L_{1}, L_{2}; x, y, p) is 0 in a neighborhood of φ and that φ is transversal. (Although we can remove those assumptions using the by now standard technique, we discuss only this simplest case here to avoid the complicated notations that would occur.) Here is the main result which will be proved in §§27, 29.

Theorem 26.2. Let ϵ be sufficiently small and consider the set of all elements of M(L_{0}, L_{ϵ}; x, y) sufficiently close to φ. Then M(L_{0}, L_{ϵ}; x, y) is described as follows:

(1) If ϵ > 0 then it consists of one point which is transversal.

(2) If ϵ < 0 then it is diffeomorphic to S^{n−2} all of whose elements are transversal.

Remark 26.3. Theorem 26.2 (1) was mentioned as a conjecture in the preprint version of [Fu7].

Postponing the proof of Theorem 26.2, we now apply Theorem 26.2 to give several illustrations of the construction in the earlier chapters. We also need the result from [Fu7] on the number of holomorphic polygons in the flat symplectic tori. We remark that in [Fu7] the first named author took an axiomatic way in place of the way using a rigorous construction of the moduli space of holomorphic discs. Namely in [Fu7] he gave the list of the properties that the number of holomorphic discs are supposed to satisfy, and carried out some calculation of the numbers based on the axioms. In fact, those properties that the first named author assumed in [Fu7] are now proved in this paper. (Strictly speaking, we did not include the proof for the case where there are more than 2 Lagrangian submanifolds. However it is a minor modification of the argument presented in this paper and so will be omitted.) Hence, the counting of holomorphic discs in [Fu7] holds true for the numbers that we are using in this paper. We will discuss the 4 and 6 dimensional flat tori separately.

26.1. The case of 4 dimensional flat tori.

We represent the flat 4-torus as ℂ^{2}/(ℤ[\sqrt{−1}])^{2} and let z^{i} = x^{i} + \sqrt{−1}y^{i}, i = 1, 2 be its coordinate. Let A = (a_{ij}) be a symmetric 2 × 2 matrix and put

ω_{A} = ∑ a_{ij}dx^{i} ∧ dy^{j}.

ω_{A} is nondegenerate and so become a symplectic structure on ℂ^{2}/(ℤ[\sqrt{−1}])^{2}, if A is invertible. The standard complex structure of ℂ^{2}/(ℤ[\sqrt{−1}])^{2} is tamed by ω_{A} if A is positive definite. Hereafter we assume that A is positive definite. We put three Lagrangian submanifolds

\begin{aligned}L_{0} & = \{(z_{1}, z_{2}) | y_{1} = y_{2} = 0\}, \\ L_{1} & = \{(z_{1}, z_{2}) | x_{1} = x_{2} = 0\}, \\ L_{2} & = \{(z_{1}, z_{2}) | x_{1} = y_{1}, x_{2} = y_{2}\}. \end{aligned}

These are Lagrangian sub-tori of T^{4}. Let v = (v_{1}, v_{2}) ∈ T^{2} = ℝ^{2}/ℤ^{2} and put

L_{1}(v) = {(z_{1}, z_{2}) | x_{1} = v_{1}, x_{2} = v_{2}}.

Then we have

\begin{aligned}L_{1}(v) ∩ L_{2} & = \{(v_{1}(1 + \sqrt{−1}), v_{2}(1 + \sqrt{−1}))\}, \\ L_{2} ∩ L_{0} & = \{(0, 0)\}, \\ L_{0} ∩ L_{1}(v) & = \{(v_{1}, v_{2})\}. \end{aligned}

We denote p(v) = (v_{1}(1 + \sqrt{−1}), v_{2}(1 + \sqrt{−1})), x = (0, 0), y(v) = (v_{1}, v_{2}).

We now perform Lagrangian surgery of L_{1}(v) and L_{2} and obtain L_{ϵ} = L_{ϵ}(v). We can prove that the virtual dimension of every component of M(L_{0}, L_{ϵ}; x, y(v)) is zero, for both ϵ > 0 and ϵ < 0. Also we can check that the virtual dimension of components of M(L_{0}, L_{ϵ}; y(v), x) are negative. Moreover the Maslov index of each elements of π_{2}(T^{4}, L_{i}) is zero. Therefore (using the fact n = 2) the virtual dimension of the moduli space of the holomorphic discs bounding L_{i} is 0+2−3 < 0. It follows that all the obstruction classes vanish automatically. Hence the Floer homology HF (L_{0}, L_{ϵ}(v)) is well defined. Furthermore the only nontrivial matrix element of the boundary operator is ⟨δx, y(v)⟩, which we now compute below.

We first consider the set of maps φ : D^{2} → T^{4} such that :

(26.4)  \begin{matrix}φ(u_{1}) = x, φ(u_{2}) = y(v), φ(u_{3}) = p(v), \\ φ(\overline{u_{1}u_{2}}) ⊂ L_{0}, φ(\overline{u_{2}u_{3}}) ⊂ L_{1}(v), φ(\overline{u_{3}u_{1}}) ⊂ L_{2}. \end{matrix}

Lemma 26.5. The homotopy class of the maps φ : D^{2} → T^{4} satisfying (26.4) corresponds one to one to ℤ^{2}.

Proof. Let ṽ = (ṽ_{1}, ṽ_{2}) ∈ ℝ^{2} be a lift of v. We put

\begin{aligned}L̃_{0} & = \{(z_{1}, z_{2}) ∈ ℂ^{2} | y_{1} = y_{2} = 0\}, \\ L̃_{1}(ṽ) & = \{(z_{1}, z_{2}) ∈ ℂ^{2} | x_{1} = ṽ_{1}, x_{2} = ṽ_{2}\}, \\ L̃_{2} & = \{(z_{1}, z_{2}) ∈ ℂ^{2} | x_{1} = y_{1}, x_{2} = y_{2}\}. \end{aligned}

Their pairwise intersections are denoted by x̃, p̃(ṽ), ỹ(ṽ) respectively. We consider maps φ̃ : D^{2} → ℂ^{2} such that

(26.6)  \begin{matrix}φ̃(u_{1}) = x̃, φ̃(u_{2}) = ỹ(ṽ), φ̃(u_{3}) = p̃(ṽ), \\ φ̃(\overline{u_{1}u_{2}}) ⊂ L̃_{0}, φ̃(\overline{u_{2}u_{3}}) ⊂ L̃_{1}(ṽ), φ(\overline{u_{3}u_{1}}) ⊂ L̃_{2}. \end{matrix}

For each map φ : D^{2} → T^{4} satisfying (26.4), there exists a unique ṽ and a unique lift φ̃ of φ satisfying (26.6). Also the maps satisfying (26.6) are homotopic to one another. Therefore the set of homotopy classes of φ corresponds one to one to that of the lifts of v. Hence follows the lemma. □

Now we quote the following result that was proved in [Fu6,7], based on the result of [FOh], which we will use in the later discussion.

Lemma 26.7. The order counted with sign of the moduli space of M(L̃_{0}, L̃_{1}(ṽ), L̃_{2}; x̃, p̃(ṽ), ỹ(ṽ)) is 1.

We next remark that the symplectic volume ∫ φ̃^{∗}ω_{A} is \frac{1}{2}ṽ^{t}Aṽ. Thus combining this with Theorem 26.2 and Lemma 26.7, we have proved the following.

Theorem 26.8. Let ϵ > 0 be sufficiently small and π : ℂ^{2} → T^{4} be the projection. Then we have :

(26.9)  ⟨δx, y(v)⟩ = ∑_{ṽ:π(ṽ)=v} T^{\frac{1}{2} ṽ^{t}Aṽ}.

In particular, the Floer homology HF (L_{0}, L_{ϵ}(v)) vanishes with the universal Novikov ring as its coefficients.

The last statement follows because the right hand side of (26.9) is invertible in the universal Novikov ring.

Remark 26.10. We may put T = e^{−1} to (26.9) and also consider the “convergent power series version” of the Floer homology. (26.9) then will become a theta function ϑ_{A}(v). This relation of the theta function to Floer homology is discovered by M. Kontsevich in [K3] in the case of elliptic curve. It is further studied by Polishchuk-Zaslow [PZ], and was partially generalized by the first named author [Fu6,7] to the higher dimension.

Remark 26.11. Remark 26.10 implies that the “convergent power series version of Floer homology” is nonzero if and only if ϑ_{A}(v) = 0. Note that it is not known in general whether the power series expression of ⟨δx, y⟩ converges or not after we put T = e^{−1}. In our case of flat Lagrangian tori L_{0},L_{ϵ}(v), the convergence follows from Theorem 26.8. However some of the properties of Floer homology, especially its invariance under Hamiltonian isotopies, do not hold for the convergent power series version of Floer homology. In fact, in our example, one can find a Hamiltonian diffeomorphism ϕ_{v} : T^{4} → T^{4} for each v, such that ϕ_{v}(L_{ϵ}(v)) ∩ L_{0} = ∅, provided ϵ > 0. Namely HF (ϕ_{v}(L_{ϵ}(v)), L_{0}) = 0 but HF (L_{ϵ}(v), L_{0}) ≠ 0 in case ϑ_{A}(v) = 0 in the convergent version. (The case A = I is illustrated below.)

Figure 26.1

To obtain something invariant under Hamiltonian perturbations out of the convergent power series version of Floer homology, we need to include the 1-loop effect. Namely we need to include the order of zero and/or infinity of the Floer homology analog of Reidemeister torsion (or more precisely the invariant that Hutchings constructed in [Hu], [HL]).

Next we consider the case when ϵ < 0. We did not check the orientation of the elements in Theorem 26.2. So we work over ℤ_{2} coefficient. (By the argument of Chapter 4, we can work over ℤ_{2} in case the dimension is 2, which implies that all Lagrangian submanifolds are semi-positive.) Then Theorem 26.2 implies the following.

Theorem 26.12. Let ϵ < 0 be sufficiently small. Then we have :

⟨δx, y(v)⟩ ≡ 0 mod 2.

In particular, we have

HF (L_{ϵ}(v), L_{0}) ≃ Λ_{ℤ_{2}} ⊕ Λ_{ℤ_{2}}

Using the invariance of Floer homology under the Hamiltonian deformation, we immediately obtain

Corollary 26.13. If ϵ < 0, then ϕ(L_{ϵ}(v)) ∩ L_{0} ≠ ∅ for any Hamiltonian diffeomorphism ϕ.

We remark that there exists a Hamiltonian diffeomorphism ϕ such that ϕ(L_{−ϵ}(v))∩ L_{0} = ∅, if −ϵ > 0. Therefore it follows from Corollary 26.13 that L_{ϵ}(v) and L_{−ϵ}(v) are not Hamiltonian isotopic. It is an interesting problem to check whether they are Lagrangian isotopic.

26.2. The case of 6 dimensional flat tori.

We now consider 6 dimensional torus T^{6} = ℂ^{3}/(ℤ[\sqrt{−1}])^{3}. We take positive definite 3×3 matrix A = (a_{ij}) and define ω_{A} = ∑ a_{ij}dx^{i} ∧dy^{j}. We define L_{0}, L_{1}(v), L_{2} in the same way, as before. Namely

\begin{aligned}L_{0} & = \{(z_{1}, z_{2}, z_{3}) ⊂ ℂ^{3} | y_{1} = y_{2} = y_{3} = 0\}, \\ L_{1}(v) & = \{(z_{1}, z_{2}, z_{3}) ⊂ ℂ^{3} | x_{1} = v_{1}, x_{2} = v_{2}, x_{3} = v_{3}\}, \\ L_{2} & = \{(z_{1}, z_{2}, z_{3}) ⊂ ℂ^{3} | x_{1} = y_{1}, x_{2} = y_{2}, x_{3} = y_{3}\}. \end{aligned}

We perform Lagrangian surgery at all of x, p(v), y(v), the pairwise intersection points of the three. We then obtain a Lagrangian submanifold which we denote by L_{ϵ_{1},ϵ_{2},ϵ_{3}}. Here ϵ_{1}, ϵ_{2}, ϵ_{3} are the parameters entering in the Lagrangian surgery at these three points respectively.

In a way similar to the proof of Lemma 26.5, we find that the set of homotopy classes of maps from D^{2} → T^{3} satisfying (26.4) corresponds one to one to ℤ^{3}. We can also identify it with the set π^{−1}(v) = {ṽ ∈ ℝ^{3}|π(ṽ) = v}, where π : ℝ^{3} → ℝ^{3}/ℤ^{3} is the projection. It also follows that there exists a unique holomorphic triangle in each homotopy class. We write this unique holomorphic triangle by φṽ : D^{2} → T^{6} for each ṽ ∈ π^{−1}(v). Using these observations and Theorem 26.2, we will study the obstruction classes for the well-definedness of Floer homology of L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v). We denote by

(26.14)  Cṽ ∈ H_{2}(T^{6}, L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v); ℤ)

the relative homology class represented by a map φ′ ; ṽ : (D^{2}, ∂D^{2}) → (T^{6}, L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v)) close to the map φṽ.

We first consider the case ϵ_{1}, ϵ_{2}, ϵ_{3} > 0. Let ṽ_{0} be the element of π^{−1}(v) for

which ∫ φ∗ ; ṽω = \frac{1}{2}ṽ^{t}Aṽ is the smallest. We put ℓ_{0} = φ′_{ṽ_{0}∗}(∂D^{2}) ∈ H_{1}(L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v)).

Using the assumption ϵ_{1}, ϵ_{2}, ϵ_{3} > 0, we find that the Maslov index of ℓ_{0} is 0. It also follows that L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) is oriented. Since all oriented 3-manifolds are spin, we can use the rational coefficient. In fact, we can even use ℤ-coefficients noting that all 3 dimensional Lagrangian submanifolds are semi-positive.

It is proved in [Fu6,7] that M(L̃_{0}, L̃_{1}(ṽ), L̃_{2}; x̃, ỹ(ṽ), p̃(ṽ_{0})) consists of a single point. Therefore the moduli space M(T^{6}, L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v); [Cṽ_{0}]) also consists of a single point by Theorem 26.2. From this, we have

Theorem 26.15. If ϵ_{1}, ϵ_{2}, ϵ_{3} > 0, then the first obstruction o_{1} to the well definedness of the Floer homology of L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) is [ℓ_{0}] ∈ H_{1}(L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v); ℤ).

In the same way, we prove that, if ϵ_{1}, ϵ_{2}, ϵ_{3} are sufficiently small positive numbers, then M(T^{6}, L_{ϵ_{1},ϵ_{2},ϵ_{3}}; [Cṽ]) consists of a single point for any v ∈ ℝ^{3}.

Unfortunately we do not know how to count the order of M(T^{6}, L_{ϵ_{1},ϵ_{2},ϵ_{3}}; [kCṽ]). (Namely what the contribution of each multiple covered disc is.) In fact this number is not well-defined. Namely it depends on how we deform the cup product 𝔪_{2,0} as an A_{∞} algebra. Let us recall how we perturbed the cup product into an A_{∞} algebra in Theorem 20.6: We choose diffeomorphisms φ_{0}, φ_{1} : L → L and modify x ∩ y to φ_{0}(x) ∩ φ_{1}(y). Let us consider a simple pseudo-holomorphic disc u : (D^{2}, ∂D^{2}) → (M, L), with [u] = β. To consider the contribution of double cover of u, we need to take φ_{0}, φ_{1} so that φ_{0}(u(∂D^{2})) is transverse to φ_{1}(u(∂D^{2})). Such choice is not unique up to homotopy. If we take φ′_{0} and connect it to φ_{0} by an isotopy φ_{0}^{t} such that φ_{0}^{0} = φ_{0}^{t}, φ^{′0} = φ_{0}^{1}, we remark that φ_{0}^{t}(u(∂D^{2})) ∩ φ_{1}(u(∂D^{2})) = ∅ for generic t but that there is a codimension one set of t ∈ [0, 1] such that φ_{0}^{t}(u(∂D^{2})) ∪ φ_{1}(u(∂D^{2})) is a one point. Thus if we make two different choices of φ_{0} the number of the pseudo-holomorphic discs of homology class 2β could be different.

Of course, the A_{∞} algebra (C(L), 𝔪) as a whole is independent of such choices up to homotopy equivalence.

We now consider the case where one of the ϵ_{i}, say ϵ_{1} is negative. In this case the Maslov index of ℓ is 1. In particular L is not oriented. So we use ℤ_{2} coefficient. Theorem 26.2 in this case implies that the moduli space of holomorphic discs (D^{2}, ∂D^{2}) → (T^{3}, L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v)) of homology class Cṽ is diffeomorphic to S^{1}. The fundamental cycle of the evaluation map : S^{1} × S^{1} → L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) is equal to the homology class of 2 sphere S_{1}^{2} which is supported in a neighborhood of L_{1}(v) ∩ L_{2}. (Recall that ϵ_{1} < 0 is the parameter corresponding to the surgery at L_{1}(v) ∩ L_{2}.) Furthermore one can check that this homology class is independent of the choice of ṽ with π(ṽ) = v. By a dimension counting, except the class [2Cṽ], we can prove that all other obstruction classes are trivial. For the class [2Cṽ], it defines a top dimensional class, which, however, cannot be surjective from the above description of the fundamental cycle and hence again gives null contribution. Hence we have :

Theorem 26.17. If ϵ_{2}, ϵ_{3} > 0 > ϵ_{1}, then

𝔪_{0}(1) = ∑_{ṽ:π(ṽ)=v}[S_{1}^{2}] · (T^{\frac{1}{2} ṽ^{t}Aṽ}e^{\frac{1}{2}} ) ∈ H_{2}(L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v); Λ_{nov,ℤ_{2}}).

In particular, the Lagrangian submanifold L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) is obstructed in the sense of Chapter 5.

Next we consider the case when ϵ_{3} > 0 > ϵ_{1}, ϵ_{2}. Then the Maslov index of [ℓ] is 2. Hence L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) is oriented. The moduli space of holomorphic discs (D^{2}, ∂D^{2}) → (T^{3}, L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v)) of the homology class Cṽ is diffeomorphic to S^{1} ×S^{1}. The image of the evaluation map S^{1} × S^{1} × S^{1} → L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) is homotopic to a subset of ℓ ∪ S_{1}^{2} ∪ S_{2}^{2}. Hence it is homologous to zero. We can also show that the other obstruction classes also vanish in this case.

Theorem 26.18. If ϵ_{3} > 0 > ϵ_{1}, ϵ_{2} or 0 > ϵ_{1}, ϵ_{2}, ϵ_{3}, then all the obstructions to the well-definedness of the Floer homology of L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) vanish.

Sketch of the proof. We already discussed the case ϵ_{3} > 0 > ϵ_{1}, ϵ_{2}. In the case 0 > ϵ_{1}, ϵ_{2}, ϵ_{3}, the Maslov index of ℓ is 3. Hence L_{ϵ_{1},ϵ_{2},ϵ_{3}}(v) is unoriented. So we use ℤ_{2} coefficient. We then find that obstruction is in H_{4}(L_{ϵ_{1},ϵ_{2},ϵ_{3}}) and hence vanish automatically. See §27 for the detail. □

Theorem 26.18 gives an example where the obstruction vanishes while the moduli space of pseudo-holomorphic discs is nonempty.

It seems possible to study Lagrangian submanifolds obtained by Lagrangian surgery from the flat Lagrangian tori with more complicated configuration in a similar way. (We refer to the last section of the preprint version of [Fu7] for some related study.) We leave it to the future research.

§27. Local model of holomorphic discs in ℂ^{n}.

The proof of Theorem 26.2 uses a gluing argument as many other results in the symplectic geometry or in the gauge theory do. We first need to find a local model solution of the Cauchy-Riemann equation in ℂ^{n}. Then we glue this with the given pseudo-holomorphic triangle near the vertex of the triangle and obtain pseudo-holomorphic strips or discs bounding the surgered Lagrangian submanifold L_{ϵ}. As in the case of [FOh], solutions of the Cauchy-Riemann equation in ℂ^{n} can be constructed explicitly but the construction is more non-trivial than the one in [FOh]. In this section, we will give a construction of such local solutions.

Let H_{ϵ} be the Lagrangian submanifold in ℂ^{2}

H_{ϵ} = Graph df_{ϵ}, f_{ϵ}(x) := ϵ log |x|

as in §26. The intersection H_{ϵ} ∩ (ℂ × {0}) divides ℂ × {0} into three components. We denote by E the component containing the part Im z, Re z large when ϵ > 0, and the intersection of the component containing the part with Im z, Re z large and {(z, 0)| Re z, Im z > 0} when ϵ < 0.

Figure 27.1

We can easily find a biholomorphic map φ_{0} : Int D^{2} → E such that

(27.1.1) φ_{0} extends continuously to D^{2}\\{1\} → \overline{E}, here \overline{E} is a closure of E in ℂ^{2}. (27.1.2) φ_{0}(∂D^{2}) ⊂ H_{ϵ}. (27.1.3) As u → 1, φ_{0}(u) is asymptotic to the plane ℂ × {0} ⊂ ℂ^{2} with both Im φ_{0}(u) → +∞ and Re φ_{0}(u) → +∞.

We will consider the set of holomorphic maps φ : D^{2}\{1} → ℂ^{2} that satisfy (27.1.2) and (27.1.3) and the following

(27.2) For each ϵ, there exists a neighborhood U of 1 in D^{2} such that φ(U\{1}) lies in the ϵ neighborhood of E.

Proposition 27.3. The set of holomorphic maps φ : D^{2}\{1} → ℂ^{2} that satisfy (27.1.2), (27.1.3) and (27.2) is described by the following:

(27.4.1) If ϵ > 0, then φ = φ_{0} ∘ h for some h ∈ P SL(2; ℝ) ≃ Aut(D^{2}) with h(1) = 1. (We denote by Aut(D^{2}, 1) the group of all such h hereafter.) (27.4.2) If ϵ < 0, there exists exactly two such φ modulo the action of Aut(D^{2}, 1). They switch under the biholomorphic map (z_{1}, z_{2}) ↦ (z_{1}, −z_{2}).

Before we give the proof of Proposition 27.3, we need to describe the way how we parametrize the solutions φ. We choose the functions f̃: ℂ^{2} → ℂP^{1}, g̃ : ℂ^{2}\{0} → ℂP^{1} defined by

\begin{aligned}f̃(z_{1}, z_{2}) & = z_{1}^{2} + z_{2}^{2}, \\ g̃(z_{1}, z_{2}) & = z_{2}/z_{1}. \end{aligned}

We then put f = f̃∘ φ, g = g̃ ∘ φ. The boundary condition (27.1.2) implies

f(∂D^{2}) ⊂ {z| Im z = 2ϵ}, (27.5.1)

g(∂D^{2}) ⊂ ℝ. (27.5.2)

The boundary condition (27.1.3) and (27.2) imply that f, g can be extended to a continuous map on D^{2} and satisfy .

(27.6.1)  f(1) = ∞, g(1) = 0.

Since φ is a continuous map to ℂ^{2} defined on D^{2}\{1} it follows that

(27.6.2)  f^{−1}(∞) = {1}.

We use the reflexion principle to define the doubles of f (strictly speaking, f − 2ϵ\sqrt{−1}), g i.e., real maps from ℂP^{1} → ℂP^{1}. f maps 1 to ∞. We again use the boundary condition (27.2) to obtain

(27.6.3)  f has a simple pole at 1.

In fact, if the order of infinity at 1 of f would be larger than 1, then f(U) would contain a neighborhood of ∞ in ℂP^{1}. (Here U is as in (27.2).) This would contradict (27.2).

Conversely, let f, g be holomorphic maps D^{2} → ℂP^{1} satisfying (27.5), (27.6). Then we have :

Lemma 27.7. Suppose that f has only simple zeros in D^{2}. Then, there exists φ satisfying the assumption of Proposition 27.3 if and only if the following holds: Let u ∈ D^{2}. Then f(u) = 0 if and only if 1 + g(u)^{2} = 0 with u being a simple zero of 1 + g^{2}.

If g is not identically zero, each pair (f, g) corresponds to two such φ’s.

Proof. When we have φ = (φ_{1}, φ_{2}) as in Proposition 27.3, we define f and g by

(27.8.1)  g(u) = φ_{2}(u)/φ_{1}(u),
(27.8.2)  f(u) = (1 + g(u)^{2})φ_{1}(u)^{2}

Then the above discussion proves all the required properties of f and g.

Now we prove the “if” direction. When (f, g) are given, we put φ = (φ_{1}, φ_{2}) where φ_{1}, φ_{2} will be given by solving (27.8.1) and (27.8.2). By the assumptions, we can solve (27.8.2) to obtain φ_{1} which is single valued and holomorphic. Then we obtain φ_{2} from (27.8.1). Consideration of (f, −g) gives rise to another solution (φ_{1}, −φ_{2}) if g ≢ 0. □

We remark that Lemma 27.7 implies that 0 ∈ Image φ. Hence f and g are well-defined everywhere.

We are now in the position to finish the proof of Proposition 27.3.

Proof of Proposition 27.3. Let φ : D^{2}\{1} → ℂ^{2} be a map that satisfy (27.1.2), (27.1.3) and (27.2) and define f and g as above. It is easy to see that holomorphic maps f : D^{2} → ℂP^{1} satisfying (27.5.1), (27.6.2) and (27.6.3) are necessarily biholomorphic and hence is unique up to the action of Aut(D^{2}, 1). Based on Lemma 27.7, it would be enough to explicitly describe all f and g that satisfy the conditions imposed in the lemma. We will start with the description of g.

We first consider the case when ϵ > 0. In this case f^{−1}(0) is empty. Therefore, by Lemma 27.7, 1 + g^{2} will never become 0 on D^{2}. It follows from (27.5.2) that g must be constant. Hence follows g ≡ 0 from (27.6.1). We have thus proved that f, g are unique up to Aut(D^{2}, 1) action. Since g is identically zero, we have only one φ corresponding to the pair (f, g), that is nothing but φ_{0}. This finishes the proof for the case ϵ > 0.

We next consider the case when ϵ < 0. In this case f^{−1}(0) = {u_{0}} is a single point. We have exactly two choices g_{1},g_{2} of g that satisfy (27.5.2), (27.6.1) and

(27.9)  g_{1}(u_{0}) = \sqrt{−1}, g_{2}(u_{0}) = −\sqrt{−1}.

We have two solutions of (27.8) for each choice of g. (Hence we have 4 different choices of φ’s altogether.) Let us write them by φ^{i}, i = 1, 2, 3, 4. We may order them so that φ^{i+2} = −φ^{i}. Then only one of φ^{1}, φ^{3}, say φ^{1}, will satisfy the boundary condition (27.1.2). Similarly we may assume φ^{2} but not φ^{4} satisfies the boundary condition (27.1.2). Thus we have exactly two solutions modulo the action of Aut(D^{2}, 1). The proof of Proposition 27.3 is now complete. □

We next consider the case when dimension is higher than 2. The argument is similar. We take a domain E in ℂ × {(0, ⋯, 0)} ⊂ ℂ^{n} similar to the case of n = 2. Let us consider the holomorphic map φ = (φ_{1}, ⋯, φ_{n}) : D^{2} → ℂ^{n} satisfying (27.1.2), (27.1.3) and (27.3). Then we have Proposition 27.10. (27.10.1) If ϵ > 0, then φ is equal to φ_{0} ∘ h, where h ∈ P SL(2; ℝ) ≃ Aut(D^{2}), with h(1) = 1. (We denote by Aut(D^{2}, 1) the group of all such h hereafter.) (27.10.2) If ϵ < 0, the set of such φ, modulo the action of Aut(D^{2}, 1), is parametrized by S^{n−2}.

Proof. We put φ = (φ_{1}, ⋯, φ_{n}) and

\begin{aligned}f(u) & = ∑_{k=1}^{n} φ_{k}^{2}(u), \\ g(u) & = (g_{2}(u), ⋯, g_{n}(u)) = (\frac{φ_{2}(u)}{φ_{1}(u)}, ⋯, \frac{φ_{n}(u)}{φ_{1}(u)}). \end{aligned}

Here f : D^{2} → ℂP^{1}, g : D^{2} → ℂP^{n−1} are holomorphic maps. The boundary conditions for f and g imply

(f − 2ϵ\sqrt{−1})(∂D^{2}) ⊂ ℝP^{1}, g(∂D^{2}) ⊂ ℝP^{n−1}.

We also have (27.6.2), (27.6.3) and

(27.6.1′)  f(1) = ∞, g(1) = (0, ⋯, 0).

It follows that

(27.11)  (1 + g(u) · g(u))φ_{1}(u)^{2} = f(u).

Here we denote

z⃗ · z⃗ := z_{2}^{2} + ⋯ + z_{n}^{2}

for z⃗ = (z_{2}, ⋯, z_{n}) ∈ ℂ^{n−1} and extend the map z⃗ ↦ z⃗ · z⃗ to ℂP^{n−1} by putting it to be infinity outside ℂ^{n−1} ⊂ ℂP^{n−1}.

We first consider the case when ϵ > 0. Then f will not become 0. Then using (27.11) in a way similar to the case of n = 2, we find that there is no u such that 1 + g(u) · g(u) = 0. It easily follows from this that g must be constant. Hence g ≡ (0, ⋯, 0) by (27.6.1′). (27.10.1) follows from this.

Next we consider the case ϵ < 0. In this case, there is a unique u_{0} ∈ D^{2} such that f(u_{0}) = 0. Also u_{0} is a simple zero of f. It follows from (27.11) that u_{0} is a unique zero of 1 + g(u) · g(u) on D^{2} which is a simple zero. If we consider the double ĝ : ℂP^{1} → ℂP^{n−1} of g, ĝ is defined over ℝ and of degree one. The latter fact comes from g(∂D^{2}) ⊂ ℝP^{n−1} and the simpleness of the zero u_{0} of 1 + g(u) · g(u). It follows that the degree of ĝ is 1. Recall that the set of degree one rational curve on ℂP^{n−1} that is defined over ℝ and containing (0, ⋯, 0) is parametrized by ℝP^{n−2}. Therefore the set of φ is parametrized by S^{n−2}. The proof of Proposition 27.10 is now complete. □

§28. Wall crossing and monodromy.

When n = 2, ℂ^{2} has a hyper-Kähler rotation: for each θ ∈ (−π, π], we define a diffeomorphism r_{θ} : ℂ^{2} → ℂ^{2} by

r_{θ}(x_{1} + \sqrt{−1}y_{1}, x_{2} + \sqrt{−1}y_{2})
= (x_{1} + \sqrt{−1}(y_{1} cos θ + y_{2} sin θ), x_{2} + \sqrt{−1}(−y_{1} sin θ + y_{2} cos θ)).

Although it is not a symplectic diffeomorphism (with respect to standard sympletic structure), a straightforward calculation shows that this rotation preserves the Lagrangian property of the particular Lagrangian submanifold H_{ϵ}, i.e, all of

H_{ϵe^{\sqrt{−1}θ}} = r_{θ}(H_{ϵ})

stay Lagrangian and so we have S^{1}-family of Lagrangian submanifold each of which is asymptotic to ℝ^{2} ∪ \sqrt{−1}ℝ^{2}. We would like to use this S^{1}-family to define a S^{1}-family of Lagrangian surgery at each double point of an immersed Lagrangian surface. However this rotation does not preserve the Liouville class. (See below.) In particular, H_{ϵ} is not exact unless ϵ is real. This makes it impossible to interpolate it, by a Lagrangian surface, to ℝ^{2} ∪\sqrt{−1}ℝ^{2} which is exact. So unlike the case where ϵ is real and so H_{ϵ} is exact, we cannot localize the other surgery for θ ≠ 0, π, but need to use a more global construction for them.

We first analyze how the Liouville class of H_{ϵ} changes over ϵ = re^{\sqrt{−1}θ} with θ ∈ (−π, π]. When ϵ is positive real (i.e., when θ = 0), we consider the circle ν_{0} : S^{1} → ℂ^{2} ≅ T^{∗}ℝ^{2} defined by

ν_{0}(ϕ) = df_{ϵ}(\sqrt{ϵ}(cos ϕ, sin ϕ))

For ϵ = re^{\sqrt{−1}θ}, θ ∈ (−π, π], we put

ν_{ϵ} = ν_{θ} := r_{θ} ∘ ν_{0}

where r_{θ} : ℂ^{2} → ℂ^{2} is the diffeomorphism defined in §26. We denote by [ν_{ϵ}] the homotopy classs H_{2}(ℂ^{2}, H_{ϵ}) through the natural isomorphism H_{2}(ℂ^{2}, H_{ϵ}) ≅ H_{1}(H_{ϵ}) ≅ ℤ. This generates H_{2}(ℂ^{2}, H_{ϵ}). Then a simple computation shows that

ν_{θ}^{∗}α = −r sin θdϕ

where α is the Liouvile form α = −(y_{1}dx_{1} + y_{2}dx_{2}). It follows from this that

ω[ν_{ϵ}] = 0 for θ = 0, π
< 0 for θ ∈ (0, π)
> 0 for θ ∈ (−π, 0).

This in particular shows that H_{ϵ} is not exact, unless ϵ is real, i.e, θ = 0, π.

Before proceeding further, we study the moduli space M(ℂ^{2}, H_{ϵ} : [ν_{ϵ}]) for each ϵ ∈ (−π, π].

Proposition 28.1. M(ℂ^{2}, H_{ϵ}; [ν_{ϵ}]) consists of one point if θ = −π/2 and is empty otherwise. Similarly M(ℂ^{2}, H_{ϵ}; [−ν_{ϵ}]) consists of one point if θ = π/2 and is empty otherwise.

Proof. We define Π : ℂ^{2} → ℝ^{2} by

Π(x_{1} + y_{1}\sqrt{−1}, x_{2} + y_{2}\sqrt{−1}) = (x_{1} − y_{1}, x_{2} − y_{2}).

Π defines a Lagrangian fibration which we can use to identify ℂ^{2} ≃ T^{∗}ℝ^{2}. We use (q_{1}, q_{2}, p_{1}, p_{2}) for the canonical coordinate under the identification ℂ^{2} ≃ T^{∗}ℝ^{2}. We denote ϵ = re^{iθ}. In this coordinates, a simple calculation shows that the image Π(H_{ϵ}) lies outside the ball that is centered at 0 and which has the maximum radius \sqrt{2r} when θ = ±π/2, and the minimum radius zero when θ = 0, π. We denote the radius of the ball by \sqrt{2r}c_{θ}, 0 ≤ c_{θ} ≤ 1 for θ ∈ (−π, π] in general. (Thus the boundary of the ball B(\sqrt{2r}c_{θ}) is the “caustics” of our Lagrangian submanifold.) Therefore there exists a linear map φ_{0} : D^{2} → ℂ^{2} such that φ_{0}(∂D^{2}) ⊂ H_{ϵ} and that the composition Π ∘ φ : D^{2} → ℝ^{2} is given by the formula Π(φ_{0}(u)) = \sqrt{2r}c_{θ}u. It is not difficult to check that φ is a minimal disc (in the Riemannian geometric sense) and in fact area minimizing among the maps representing the homotopy class ν_{ϵ} ∈ π_{2}(ℂ^{2}, H_{ϵ}). Therefore if there exists a holomorphic map representing the class ν_{ϵ}, the map φ_{0} must be holomorphic. However it is easy to see that φ_{0} can be holomorphic only if θ = −π/2 and is anti-holomorphic only if θ = π/2. The proof of Proposition 28.1 follows. □

Now we want to implant the S^{1}-family of local models H_{ϵ} into a given compact sympletic manifold at each double point of a Lagrangian immersion.

Let Ψ : Σ → M be a Lagrangian immersion into a symplectic 4-manifold. We assume that Ψ is an embedding on Σ\{p, q} and has an ordinary double point at x = Ψ(p) = Ψ(q). As in §26, we choose a small neighborhood U of x and take a symplectic diffeomorphism I : U → D^{4} ⊂ ℂ^{2} where D^{4} is the unit ball in ℂ^{2}. I maps the two branches of the immersion to (ℝ^{2} ∪ \sqrt{−1}ℝ^{2}) ∩ D^{4}. We put A = D^{4}\\frac{1}{2}D^{4}. Here \frac{1}{2}D^{4} is the four ball of radius \frac{1}{2} centered at the origin. Let A_{r} = A ∩ ℝ^{2}, A_{i} = A ∩ \sqrt{−1}ℝ^{2}. We may identify the neighborhoods U_{r}, U_{i} of A_{r}, A_{i} in ℂ^{2} with neighborhoods of the zero section of the cotangent bundle of A_{r}, A_{i} respectively.

We consider the Lagrangian submanifolds H_{ϵ} in ℂ^{2} for ϵ = |ϵ|e^{\sqrt{−1}}θ defined in §26 for θ ∈ (−π, π]. Since H_{ϵ}∩A converge to (ℝ^{2}∪\sqrt{−1}ℝ^{2})∩A as |ϵ| → 0 and so are contained as a Lagrangian graph in the above Darboux neighborhood of A_{r} ∪ A_{i}, there exist closed one forms u_{r}, u_{i} on A_{r}, A_{i} such that H_{ϵ} ∩ A can be identified with the graphs of u_{r}, u_{i} respectively on the cotangent bundle. In fact, with respect to the polar coordinates of A_{r} ≅ [\frac{1}{2}, 1] × S^{1}, we have

(28.2.1)  u_{r} = −r sin θdϕ.

Similarly due to the symmetry of H_{ϵ} along the diagonal Δ, we also have

(28.2.2)  u_{i} = −r sin θdϕ

on A_{i} ≅ [\frac{1}{2}, 1] × S^{1}.

We denote

Σ_{out} = Σ\Ψ^{−1}(\frac{1}{2}D^{4}) ≅ Ψ(Σ)\\frac{1}{2}D^{4}

It is easy to see that we can implant the local model H_{ϵ} ∩ D^{4} at the double point to produce the required Lagrangian surgery L_{ϵ}, if we can find a closed one form u on Σ_{out} such that the restrictions of u to A_{r}, A_{i} are u_{r}, u_{i} respectively. By considering the exact sequence

⟶ H^{1}(Σ_{out}; ℝ) ⟶ H^{1}(∂Σ_{out}; ℝ) ⟶^{δ} H^{2}(Σ_{out}, ∂Σ_{out}; ℝ) →,

we can find such a closed one form if δ([u_{r}] ⊕ [u_{i}]) = 0.

Recall that

H^{2}(Σ_{out}, ∂Σ_{out}; ℝ) ≅ (H^{0}(Σ_{out}; ℝ))^{∗}

by the Lefschetz duality. Therefore if we assume in addtion that Σ is connected, we have H^{2}(Σ_{out}, ∂Σ_{out}; ℝ) ≅ ℝ and so in this case, δ([u_{r}] ⊕ [u_{i}]) is characterized by the real number

δ([u_{r}] ⊕ [u_{i}])[Σ_{out}] = [u_{r}] ⊕ [u_{i}](∂Σ_{out}),

which vanishes by (28.2). (Note that to make the parametrization of ∂Σ_{out} consistent with the induced boundary orientation of ∂Σ_{out}, we should parametrize the two components ∂Σ_{out} ∩ A_{r} and ∂Σ_{out} ∩ A_{i} in the opposite directions.) Therefore [u_{r}] ⊕ [u_{i}] ∈ H^{1}(A_{r} ∪ A_{i}; ℝ) is always in the image of H^{1}(Σ; ℝ). This proves the following proposition.

Proposition 28.3. Suppose that Σ is connected. Then for each small ϵ = |ϵ|e^{\sqrt{−1}θ} there exists a Lagrangian submanifold L_{ϵ} ⊆ M such that :

(28.3.1) L_{ϵ} ∩ (M\U) will converge to Ψ(Σ) ∩ (M\U) in C^{∞} topology as ϵ goes to zero. (28.3.2) I(L_{ϵ} ∩ V) = (\frac{1}{2}D^{4} ∩ H_{ϵ}) where x ∈ V ⊂ \overline{V} ⊂ U.

Remark 28.4. (1) In case p belongs to a component different from that of q in Σ, the above argument shows that [u_{r}] ⊕ [u_{i}] ∉ H^{1}(A_{r} ∪ A_{i}; ℝ), unless ϵ is real. So in this case, we cannot find L_{ϵ} satisfying (28.3) in Lemma 28.3. This is the case, for example, when M = T^{4}, Ψ(Σ) = T^{2} ∪ T^{2} where two tori intersects transversely at one point.

On the other hand, we have the following example where the assumption of Lemma 28.3 is satisfied. Let M be an elliptic K3 surface and π : M → N be the projection. We take a symplectic structure so that the fibers of π are (special) Lagrangian submanifold. (Note that K3 surface is hyper-Kähler. Hence we can find such a symplectic structure among the S^{2} parametrized family of symplectic structures.) Suppose M has a type-one singular fiber F = π^{−1}(p_{0}) in the sense of Kodaira classification. Then F is the image of a Lagrangian immersion from S^{2}.

In this case, the Lagrangian submanifold L_{ϵ} in Lemma 28.3 can be taken as π^{−1}(p) where p is in a small circle around the point p_{0}.

(2) By generalizing the above argument to a configuration of several Lagrangian submanifolds, one can derive a sufficient condition for a simultaneous surgery at several intersections points in terms of the data at the intersection points and of the topology of the configuration. Since we do not need this in this paper, we do not discuss this point further here.

Now we continue our discussion from §26 and consider the space M(L_{0}, L_{ϵ}; x, y) for the Lagrangian submanifolds L_{ϵ} constructed in Lemma 28.3 for all ϵ = |ϵ|e^{\sqrt{−1}θ} not necessarily real, provided such a family indeed exists as in the case of local models in ℂ^{2} we considered in §26 . We assume that the almost complex structure J is integrable near the intersections and that there exists a pseudo-holomorphic triangle φ ∈ M(L_{0}, L_{1}, L_{2}; x, p, y). We first remark that the surgered Lagrangian submanifold L_{ϵ} has π_{2}(M, L_{ϵ}) = ℤ. Therefore the homotopy class of the maps φ′ that satisfy the same boundary condition as the elements of M(L_{0}, L_{ϵ}; x, y) and that is close to φ, is parametrized by ℤ.

Let M(L_{0}, L_{ϵ}; x, y; ξ) be the set of elements of M(L_{0}, L_{ϵ}; x, y) belonging to this homotopy class ξ. When θ = 0, M(L_{0}, L_{ϵ}; x, y; ξ) is nonempty only for one homotopy class, say ξ_{0}. Furthermore M(L_{0}, L_{ϵ}; x, y; ξ_{0}) consists of a single point. We study how the family M(L_{0}, L_{ϵ}; x, y; ξ) changes as θ varies. The family will define a chain and its bifurcation occurs when M(ℂ^{2}, H_{ϵ}; [ν_{ϵ}]) becomes nonempty. We also know from Theorem 26.2 that when θ = π, there exist exactly two homotopy class ξ_{0}, ξ_{0} + 1 such that M(L_{0}, L_{ϵ}; x, y; ξ) is nonempty. Both M(L_{0}, L_{ϵ}; x, y; ξ_{0}), M(L_{0}, L_{ϵ}; x, y; ξ_{0} + 1) consist of a single point respectively.

If we trace the homotopy class ξ_{θ} continuously as θ varies from −π to π, the class ξ_{2π} will become ξ_{0} + 1 because of the presence of monodromy. Note that ϵ = re^{iθ} = re^{i(θ+2π)}. From these observations, we find that M(L_{0}, L_{ϵ}; x, y) pictorially looks like Figure 28.1 below.

Figure 28.1

So we have a wall crossing of the Floer homology at θ = ±π/2.

However the line where this wall crossing occurs is not well-defined in that it depends on the choice of almost complex structure J. To get something invariant under the perturbations of J and etc., we need to consider the moduli space of Lagrangian submanifolds. Then “the homology class” of the line where wall crossing occurs will have some invariant meaning. (This phenomenon was used in [Fu7] for some calculation.)

§29. Gluing.

We now use the result of §27 to prove Theorem 26.2. We will prove only (1) in detail. The proof of (2) is similar and so will be omitted. We will follow the same notations as in §26 and assume, for simplicity, that our almost complex structure is integrable in a neighborhood p and I is biholomorphic. Let φ ∈ M(L_{0}, L_{1}, L_{2}; x, p, y) be a pseudo-holomorphic triangle.

We regard it as a map ℍ → M where ℍ is the upper half plane. We will construct an element of M(L_{0}, L_{ϵ}; x, y) for small ϵ. Note φ(u_{2}) = p. In a neighborhood of u_{2}, we write

(29.1)  I(φ(u + u_{2})) = u^{1/2+2k}F (u^{2}).

where k ≥ 0 is an integer, and F is holomorphic in a neighborhood of 0 with F (0) ≠ 0. (It is easy to see that there exist unique such k and F.) By composing a complex linear isomorphism with I if necessary, we may assume that F (0) = (1, 0).

We first consider the case k = 0. In this case, we can glue φ with the solution of (27.4.1) or (27.9.1) by partitions of unity, and obtain an approximate solution of the Cauchy-Riemann equation. Since we assume that the given pseudo-holomorphic triangle is transversal and we can easily check that the local model constructed in §27 is transversal, the standard argument (which has been used in many literature like [FOh], for example) proves that the linearization (of the Cauchy-Riemann equation) at the glued approximate solution is surjective. Hence using the method similar to that of [FOh], we prove that there exists a unique solution nearby to the approximate one.

We now turn to the case k > 1. As a first step we will interpolate φ with φ′ which is defined as follows: We choose a sufficiently small δ > 0 and put

I(φ′(u + u_{2})) = u^{1/2}(u^{2k} + δ)F (u^{2}).

Note that φ′ behaves like φ with given k when |u| is large and like the case k = 0 above when |u| is small. We glue φ′ and φ around u + u_{2} with |u| ∼ μ to obtain the first approximation φ′′. Here we take ϵ, δ and μ so that

0 < ϵ ≪ δ^{\frac{1}{2k}} ≪ μ ≪ 1.

Then at u + u_{2} with |u| ∼ \frac{ϵ}{10}, our map φ′′ is close to

u + u_{2} ↦ I^{−1}(δu^{1/2}F (u^{2})).

Then we can apply the same argument to glue φ′′ with φ_{0} as in the case k = 1 and obtain the final approximate solution. The rest of the proof is similar and is omitted.

To complete the proof of Theorem 26.2, we also need to check that there is no other solution near the constructed ones. The proof of this fact is similar to that of [FOh] and is omitted. The proof of Theorem 26.2 is now complete.

Appendices.

§A1 Smooth correspondence.

In this book, we used the smooth correspondence several times to define various operations. For our purpose, it was important to carry out this in the chain level rather than in (co)homology. In this appendix, we explain how to perform this constructions precisely. We start with a definition. Let M_{1}, M_{2} be smooth manifolds and N be a manifold with corners. Let ϕ : N → M_{1} × M_{2} be a smooth proper map. We call 𝔑 = (N, ϕ) a smooth correspondence between M_{1} and M_{2}. Unless otherwise stated, we assume that all manifolds are oriented. Let ϕ_{i} : N → M_{i} be the composition of ϕ and the projection.

Let us first suppose that M_{i} are compact and N has no boundary. Then we obtain a map

𝔑_{∗} : H_{∗}(M_{1}; ℤ) → H_{∗+k}(M_{2}; ℤ)

by

(A1.1)  𝔑_{∗}(u) = ϕ_{2∗} ∘ P D. ∘ ϕ_{1}^{∗} ∘ P D.(u).

Here P D. denotes the Poincaré duality and k = dim N − dim M_{1}.

The purpose of this appendix is to generalize this map to the case where N may have boundaries and/or corners. In that case, (A1.1) does not make sense and we have to work in the chain level.

Let us first explain two ideas to define an analogy of (A1.1) in the chain level. If u is represented by a singular chain ∑ c_{i}σ_{i} where σ_{i} : Δ^{m} → M_{1} is smooth

then 𝔑_{∗}(u) is realized by

(A1.2)  ∑ c_{i} Δ^{m}_{σ_{i}} ×_{ϕ_{1}} N.

More precisely, we take a triangulation of the fiber product Δ^{m}_{σ_{i}} ×_{ϕ_{1}} N and put

(A1.3)  Δ^{m}_{σ_{i}} ×_{ϕ_{1}} N = ∑_{j}(Δ^{m+k}, σ_{i,j}).

Then

(A1.4)  𝔑_{∗}(u) = [∑_{i,j}(σ_{i,j}, ϕ_{2} ∘ π_{2})] .

Here π_{2} : Δ^{m}_{σ_{i}} ×_{ϕ_{1}} N → N is the second projection.

There are two troubles to justify (A1.4). One is that the fiber product Δ^{m}_{σ_{i}}×_{ϕ_{1}}N has a good triangulation and has the fundamental chain only under some transversality condition.

The other is that the triangulation (A1.3) is not unique even if it is well-defined. When we work in the homology level, this second point does not cause any trouble, since the homology class is independent of the triangulation. However for the purpose of this book, we need to define various operations in the chain level.

Another possible approach to realize (A1.1) in the chain level is to use De-Rham cohomology. Namely for a closed form ω realizing the Poincaré dual to u, we put

𝔑^{∗}(ω) = ϕ_{2!}ϕ_{1}^{∗}(ω).

Here ϕ_{1}^{∗} is the pull back of the differential form and ϕ_{2!} is the integration along the fiber.

In this approach, ϕ_{1}^{∗} does not cause any trouble but ϕ_{2!} is problematic. In fact, integration along the fiber of a differential form defines a current which is not necessarily smooth in general. This point is especially serious in our situation where N has non-empty boundary. In fact in our situation, ϕ_{2!} rarely sends a smooth form to a smooth form.

One might try to use currents (more precisely distribution valued forms) instead of (smooth) differential forms. Then ϕ_{2!}, the push-forward of the current, is always well-defined. But the pull back ϕ_{1}^{∗} of the current is not necessary well-defined.

The route we are going to take is some kind of mixture of the two approaches outlined above. Namely we take an appropriate countably generated set of currents realized by geometric chains.

Let S_{n−k}(M; R) be the set of all codimension k smooth singular chains with R as a coefficient. We first consider the case R ⊂ ℝ. Let D^{′k}(M; ℝ) be the set of all distribution valued k forms on M. We remark that an element T ∈ D^{′k}(M; ℝ) given by

T (u) = ∫_{M} T ∧ u,

defines a linear map C^{∞}(M, Λ^{n−k}(M)) → ℝ which is continuous with respect of the compact C^{∞} topology.) For a given smooth singular chain σ : Δ^{n−k} → M and smooth n − k form ω, we put

T (σ)(ω) = ∫_{Δ^{k}} σ^{∗}ω.

It is obvious that T (σ) ∈ D^{′k}(M; ℝ). σ ↦ T (σ) is extended to a homomorphism T : S_{n−k}(M; R) → D^{′k}(M; ℝ) in an obvious way.

Definition A1.5. \overline{S}^{k}(M; R) is the image of the homomorphism T : S_{n−k}(M; R) → D^{′k}(M; ℝ).

We remark that T is not injective. Hence \overline{S}^{∗}(M; R) will be isomorphic to an appropriate quotient complex of S_{∗}(M; R).

Let N_{i} be manifolds with corners and ϕ_{i} : N_{i} → M be smooth maps. We say that ϕ_{1} is transverse to ϕ_{2} if the restriction of ϕ_{1} to codimension a corner of N_{1} is transverse to the restriction of ϕ_{2} to codimension b corner of N_{2}, for every a and b. Since a simplex is a manifold with corner, we can define transversality between a smooth singular simplex and a map from a manifold with corners as above.

Definition A1.6. Let 𝔑 = (N, ϕ) be a smooth correspondence from M_{1} to M_{2}. Let c ∈ \overline{S}^{k}(M; R). We say c is transverse to 𝔑 if there exists an element ∑ c_{i}σ_{i} ∈ S_{n−k}(M; R) such that c = T (∑ c_{i}σ_{i}) and that each σ_{i} is transverse to ϕ_{1}.

Now suppose that c = T (∑ c_{i}σ_{i}) is transverse to 𝔑. We take a simplicial decomposition of the fiber product Δ^{m}_{σ_{i}} ×_{ϕ_{1}} N as in (A1.3) and put

(A1.7)  𝔑_{∗}(c) = ∑_{i,j} c_{i} · T (σ_{i,j}, ϕ_{2} ∘ π_{2})

It is easy to see that the right hand side of (A1.7) is independent of the simplicial decomposition (A1.3). However to show that 𝔑_{∗} is well-defined we need to check the following :

Lemma A1.8. Let T (∑ c′_{i}σ′_{i}) = T (∑ c_{i}σ_{i}) ∈ D^{′k}(M; ℝ). We assume that σ_{i}, σ′_{i} are both transverse to ϕ_{1} : N → M_{1}. We choose simplicial decompositions of Δ^{m}_{σ_{i}} ×_{ϕ_{1}} N and Δ^{m}_{σ′_{i}} ×_{ϕ_{1}} N respectively as in (A1.3). Then we have

∑_{i,j} c_{i} · T (σ_{i,j}, ϕ_{2} ∘ π_{2}) = ∑_{i,j} c′_{i} · T (σ′_{i,j}, ϕ_{2} ∘ π_{2}).

We prove Lemma A1.8 at the end of this section. By Lemma A1.8, (A1.7) defines a homomorphism 𝔑_{∗} from a sub-complex of \overline{S}_{∗}(M_{1}; R) to \overline{S}_{∗}(M_{2}; R). Now the proof of the following lemma is a standard application of transversality theorem and is omitted.

Lemma A1.9. There is a countably generated sub-complex C of \overline{S}_{∗}(M_{1}; R) such that each element of C is transverse to 𝔑 and that the cohomology of C is isomorphic to the cohomology H^{∗}(M; R).

Actually we used a statement more complicated than Lemma A1.9. Namely we are working in the situation where there are countably many correspondences, and have to construct a countably generated sub-complex which is invariant by the correspondences and whose homology is isomorphic to the singular homology. But the argument we need for this purpose was already discussed in §19.

Let 𝔑 = (N, ϕ) be a smooth correspondence. Let ∂N be the disjoint union of the codimension one strata of N. (∂N, ϕ) is again a smooth correspondence which we denote by ∂𝔑.

The following lemma (which we have used frequently in this book) is now easy to show.

Lemma A1.10. ∂ ∘ 𝔑_{∗} − 𝔑_{∗} ∘ ∂ = (∂𝔑)_{∗}.

We next consider the case when the coefficient ring is ℤ_{2}. (This case was used in Chapter 3 in an essential way.) Let π : S_{∗}(M; ℤ) → S_{∗}(M; ℤ_{2}) be the quotient map.

Definition A1.11. Let u, v ∈ S_{∗}(M; ℤ_{2}). We say u ∼ v if there exists ũ, ṽ ∈ S_{∗}(M; ℤ) such that π(ũ) = u, π(ṽ) = v and T (ũ) = T (ṽ).

It is easy to see that ∼ is an equivalence relation. Let \overline{S}^{n−∗}(M; ℤ_{2}) be the quotient space and T : S_{∗}(M; ℤ_{2}) → \overline{S}^{n−∗}(M; ℤ_{2}) be the obvious quotient map. It is then easy to see that Lemma A1.8 in the case R = ℤ implies the case R = ℤ_{2}. Hence we can define a smooth correspondence with ℤ_{2} coefficients. Note that we do not need orientation of M_{i} and N since orientation is used only to define orientation of the singular chain in (A1.3).

Before proving Lemma A1.8, we give some more remarks. We have used the chain complex \overline{S}^{∗}(M; R) rather than the singular chain complex S_{∗}(M; R). There are two reasons for this. One is that the simplicial decomposition (A1.3) is not unique so the operation 𝔑_{∗} is not-well defined in the chain level. This was previously pointed out by the first named author. In [Fu3], he used the notion of geometric chains due to Gromov [Gr1] for this purpose, where one can use the fiber product itself without taking its simplicial decomposition.

However there is another trouble in the situation where the correspondences are defined by moduli spaces and evaluation maps, which forces us to take the quotient \overline{S}^{∗}(M; R) rather than S_{∗}(M; R) or geometric chains. In this situations, we consider N = M where M is an appropriate moduli space of pseudo-holomorphic maps. We have studied its boundary ∂M and used Lemma A1.10 to deduce various identities among the operators induced by the correspondence. In the various cases we studied in this book, some components of ∂M are of codimension one (in M) but did not appear in the formula, since the image of those components by the evaluation map (which is ϕ_{2} in our case) is of smaller dimension than its virtual dimension. Thus, in this case, those component gives 0 as a current but not as a singular or geometric chain. Therefore those terms can be omitted only after taking quotient which leads us to work with \overline{S}^{∗}(M; R).

We finally prove Lemma A1.8. We first reduce to the case where N is a subset of M_{1}.

We embed N to a manifold without boundary N_{+} (but maybe noncompact). Let M ⊂ Int N_{+−} ⊂ N_{+−} ⊂ N_{+}, such that N_{+−} is compact with smooth boundary. Then we consider N ⊂ N_{+} × M_{1}. Here the inclusion is by x ↦ (x, ϕ_{1}(x)). We take a simplicial decomposition of N_{+−} × Δ^{k}. This defines a pull backs of both ∑ c′_{i}σ′_{i} and ∑ c_{i}σ_{i} to N_{+} × M_{1}. Thus we may replace M_{1} with N_{+} × M_{1} and assume N ⊂ M_{1}.

To prove Lemma A1.8, we use a result by Hörmander [Hö] on the well-definedness of the product of distribution. He used the notion of wave front set of the distribution (due to M. Sato). For a current T on M, its wave front set W F (T) is defined and is an ℝ_{+} invariant subset of T^{∗}M − {0}. In the case where T is represented by a submanifold M′, its wave front set is nothing but the conormal bundle T_{M′}^{∗}M. For a given smooth map ϕ : K → M from a closed manifold K, we have a current T_{K} and its wave front set W F (T_{K}) is

T_{K}^{∗}M = {(v, ϕ(p)) | v ∈ T_{ϕ(p)}^{∗}M, ∀V ∈ T_{p}K, v(ϕ_{∗}(V)) = 0}.

If K is a manifold with corners, then the wave front set W F (T_{K}) of T_{K} is contained in the union of T_{K_{i}}^{∗}M where K_{i} is the codimension i corner of K. We consider the following condition.

(A1.12)  If V_{i} ∈ W F (T_{i}), V_{i} ∈ T_{p}M, then V_{1} + V_{2} ≠ 0.

We recall the following result by Hörmander [Hö].

Lemma A1.13. If T_{i} satisfies condition (A1.12), then there exists a current T_{1} ∧ T_{2} with the following properties. If ω_{i,j} is a sequence of smooth forms such that lim_{j→∞} ω_{i,j} = T_{i} as current, then the sequence ω_{1,j} ∧ ω_{2,j} converges to T_{1} ∧ T_{2} as current.

Now we take T_{1} = T_{N}, T_{2} = T (∑ c_{i}σ_{i}). Condition (A1.12) is satisfied by the assumption that σ_{i} is transversal to N. Then T_{N} ∧ T (∑ c_{i}σ_{i}) is well defined in the sense above. It is easy to see that this current is equal to ∑ c_{i}T (N ∩ σ_{i}). (Note that each N ∩ σ_{i} is a manifold with corner and hence defines a current.) Thus by T (∑ c_{i}σ_{i}) = T (∑ c′_{i}σ′_{i}) we have

(A1.14) ∑ c_{i}T (N ∩ σ_{i}) = T_{N} ∧ T (∑ c_{i}σ_{i}) = T_{N} ∧ T (∑ c′_{i}σ′_{i}) = ∑ c′_{i}T (N ∩ σ′_{i}).

Since the push-forward of currents is always well-defined, (A1.14) implies Lemma A1.8. □

§A2. Review of Kuranishi structure.

Our moduli spaces are not necessarily transversal and we use the machinery developed in [FOn2]. (See [LiT], [Ru], [Sie] for related works.) We introduced the notion of Kuranishi structure in [FOn2]. Since the moduli spaces we study in the previous chapters are obtained as various fiber products, we need to define the fiber product of spaces with Kuranishi structure.

First of all, we review the definition of Kuranishi structure with corners from [FOn2]. (Hereafter, for simplicity, we will just call the Kuranishi structure for the Kuranishi structure with corners.) Let X be a compact metrizable space. A Kuranishi structure assigns (V_{p}, E_{p}, Γ_{p}, ψ_{p}, s_{p}) to each p ∈ X and (V_{pq}, ϕ̂_{pq},ϕ_{pq},h_{pq}) to each given pair of points p, q ∈ X. They are required to satisfy the following properties:

(A2.1.1) V_{p} is a smooth manifold with corners, and E_{p} is a smooth vector bundle on it. (A2.1.2) Γ_{p} is a finite group acting smoothly on E_{p} → V_{p}. (A2.1.3) s_{p} is a Γ_{p} equivariant smooth section of E_{p}. (A2.1.4) ψ_{p} is a homeomorphism from s_{p}^{−1}(0)/Γ_{p} to a neighborhood of p in X. (A2.1.5) V_{pq}, ϕ̂_{pq}, ϕ_{pq} h_{pq} are defined if q ∈ ψ_{p}(s_{p}^{−1}(0)). (A2.1.6) V_{pq} is an open subset of V_{q} containing ψ_{q}^{−1}(q). (A2.1.7) h_{pq} is a homomorphism Γ_{q} → Γ_{p}. (A2.1.8) (ϕ̂_{pq}, ϕ_{pq}) is an h_{pq} equivariant map of vector bundles E_{q}|_{V_{pq}} → E_{p}. (A2.1.9) ϕ̂_{pq}s_{q} = s_{p}ϕ_{pq}. (A2.1.10) ψ_{q} = ψ_{p}ϕ_{pq}. (A2.1.11) If r ∈ ψ_{q}(s_{q}^{−1}(0) ∩ V_{pq}), then

ϕ̂_{pq} ∘ ϕ̂_{qr} = ϕ̂_{pr}

in a neighborhood of ψ_{r}^{−1}(r). (A2.1.12) dim V_{p} − rank E_{p} is independent of p and is called the virtual dimension of the Kuranishi structure.

V_{p} or U_{p} = V_{p}/Γ_{p} is said to be a Kuranishi neighborhood of p. (More precisely (V_{p}, E_{p}, Γ_{p}, ψ_{p}, s_{p}) is a Kuranishi neighborhood.) E_{p} is called the obstruction bundle and s_{p} is called the Kuranishi map.

The definition given above is actually the Kuranishi structure together with good coordinate system in the sense defined in [FOn2]. It is proved in [FOn2] that every space with Kuranishi structure has a good coordinate system.

Definition A2.2. Let Y be a topological space. A family {f_{p}} of continuous maps f_{p} : V_{p} → Y is said to be a strongly continuous map if

f_{p} ∘ ϕ_{pq} = f_{q},

on V_{pq}. A strongly continuous map induces a continuous map f : X → Y. We will ambiguously denote f = {f_{p}} when the meaning is clear.

When Y is a smooth manifold, a strongly continuous map f : X → Y is defined to be smooth if all f_{p} : V_{p} → Y are smooth. We say that it is weakly submersive if each of f_{p} is a submersion.

We now assume that Y is a smooth manifold. Let X^{i} be the spaces with Kuranishi structures (V_{p}^{i}, E_{p}^{i}, Γ_{p}^{i}, ψ_{p}^{i}, s_{p}^{i}; V_{pq}^{i}, ϕ̂_{pq}^{i}, ϕ_{pq}^{i}, h_{pq}^{i}). Let {f_{p}^{i}} : X^{i} → Y be smooth strongly continuous maps. Let P be a manifold with corner and f : P → Y be a smooth map. (P, f) defines a current on Y^{n}. Let f^{i} : X^{i} → Y be the continuous map induced from {f_{p}^{i}}. We put

(A2.3) ∏_{i} X^{i}_{(f^{i})} ×_{f} P := {((p^{i}), p) ∈ ∏_{i} X^{i} × P | (f^{1}(p^{1}), ⋯, f^{n}(p^{n})) = f(p)}

and will define a natural Kuranishi structure on it.

We first consider the case when {f_{p}^{i}} are weakly submersive. Let ((p^{i}), p) ∈ ∏_{i} X^{i}_{(f^{i})} ×_{f} P. Let V_{p_{i}}^{i} be Kuranishi neighborhoods of p^{i}. We put

(A2.4) V_{((p^{i}),p)}^{S} := {((x^{i}), x) ∈ ∏_{i} V_{p_{i}}^{i} × P | (f_{p^{1}}^{1}(x^{1}), ⋯, f_{p^{n}}^{n}(x^{n})) = f(x)} .

Since f_{p^{i}}^{i} are submersions, it follows that V_{((p^{i}),p)}^{S} is a smooth manifold for each ((p_{i}), p). We pull back the exterior product ∏ E_{p^{i}}^{i} → ∏ V_{p^{i}}^{i} to V_{((p^{i}),p)}^{S} to obtain E_{((p^{i}),p)}^{S}. We next put Γ_{((p^{i}),p)} = ∏ Γ_{p^{i}}^{i}. It is easy to see that Γ_{((p^{i}),p)} acts on E_{((p^{i}),p)}^{S} → V_{((p^{i}),p)}^{S}. We can define a section s_{((p^{i}),p)}^{S} to E_{((p^{i}),p)}^{S} → V_{((p^{i}),p)}^{S} from s_{p^{i}}^{i} in an obvious way. A map ψ_{((p^{i}),p)}^{S} : (s_{((p^{i}),p)}^{S})^{−1}(0) → ∏_{i} X^{i}_{f^{i}} ×_{f} P is induced from ψ_{p^{i}}^{i}.

It is straightforward to construct V_{((p^{i}),p)((q^{i}),q)}^{S} etc. and a Kuranishi structure on ∏_{i} X^{i}_{(f^{i})} ×_{f} P. We call this Kuranishi structure S-Kuranishi structure on the fiber product (A2.2).

The proof of the following lemma is straightforward and omitted. (See [FOn2] §5 for the notations used in its statement.)

Lemma A2.5. If the Kuranishi structures on X^{i} have tangent bundles, so does the S-Kuranishi structure on Π_{i}X^{i}_{(f^{i})} ×_{f} P. If the Kuranishi structures on X^{i}, Y and P are oriented, so is the S-Kuranishi structure on Π_{i}X^{i}_{(f^{i})} ×_{f} P.

We next consider the case when f^{i} are not necessarily weakly submersive. In this case, we assume that P is a submanifold and f is an embedding. Let π : N_{P} Y^{n} → P be the normal bundle of P in Y^{n}. We identify the total space N_{P} Y^{n} with a tubular neighborhood of P in Y^{n}. We put

(A2.6)  \begin{aligned}V_{((p^{i}),p)}^{L} = \{((x^{i}), x) ∈ Π_{i}V_{p_{i}}^{i} × P | & (f_{p^{1}}^{1}(x^{1}), ⋯, f_{p^{n}}^{n}(x^{n})) ∈ N_{P} Y^{n}, \\ & π(f_{p^{1}}^{1}(x^{1}), ⋯, f_{p^{n}}^{n}(x^{n})) = f(x)\}. \end{aligned}

It is easy to see that V_{((p^{i}),p)}^{L} is a smooth manifold. We define a vector bundle E_{((p^{i}),p)}^{L} on V_{((p^{i}),p)}^{L} by

(A2.7)  (E_{((p^{i}),p)}^{L})_{((x^{i}),x)} = ⨁ E_{x^{i}}^{i} ⊕ N_{f(x),P} Y^{n}.

Γ_{((p^{i}),p)} acts on this bundle. We define a section s_{((p^{i}),p)}^{L} to E_{((p^{i}),p)}^{L} → V_{((p^{i}),p)}^{L} by

(A2.8)  s_{((p^{i}),p)}^{L}((x^{i}), x) = ((s^{i}(x^{i})), (f_{p^{1}}^{1}(x^{1}), ⋯, f_{p^{n}}^{n}(x^{n})))

Here we remark that (f_{p^{1}}^{1}(x^{1}), ⋯, f_{p^{n}}^{n}(x^{n}))) is an element of the fiber N_{P,x}Y^{n} by (A2.6). We define

(A2.9)  ψ_{((p^{i}),p)}^{L}((x^{i}), x) = ((ψ(x^{i})), f(x)).

It is straightforward to construct V_{((p^{i}),p)((q^{i}),q)}^{L} etc. and a Kuranishi structure on Π_{i}X^{i}_{(f^{i})} ×_{f} P. We call this Kuranishi structure L-Kuranishi structure on the fiber product (A2.3). We can prove the same statement as Lemma A2.5 for the L-Kuranishi structure.

It seems possible to generalize the construction to the case when Y is not necessary a manifold but has a Kuranishi structure. We would then have a category whose objects are the spaces with Kuranishi structure and whose morphisms are a kind of strongly continuous smooth maps. Such a category would be a smooth analogue of the category of schemes or stacks. (We remark that the fiber product of schemes is always well-defined and plays an important role in the theory of schemes.) We are not trying to rigorously define it in this paper since we do not need it.

§A3. Normally polynomial perturbations.

In this section, we complete the proof of Theorem 9.5. We are going to construct a single valued perturbation 𝔰_{ϵ} so that the moduli space M_{1}^{𝔰}(β : P_{1}, ⋯, P_{k}) has a triangulation. We use the notations as in §9.

We recall that we are in the following situation : We constructed a section 𝔰_{ϵ} on a neighborhood of M_{1}(β : P_{1}, ⋯, P_{k})_{sing} so that the compatibility condition (9.10) is satisfied and must construct its extension so that transversality is satisfied for the elements without non-trivial automorphisms.

We need to remark that the section 𝔰_{ϵ} constructed on M_{1}(β : P_{1}, ⋯, P_{k})_{sing} may not be transversal. Although the section \overline{𝔰}_{ϵ} defined in a neighborhood of reduced model is transversal. The later and Condition (9.10) however do not imply that 𝔰_{ϵ} is transversal on M_{1}(J, β : P_{1}, ⋯, P_{k})_{sing}.

Let us explain how the transversality fails. We use the same notation as in (9.9). We put Γ = Aut(Σ, φ). Γ acts on V and the image I_{V̄ ,V} (V̄) is contained in the

fixed point set of Γ. The group Γ acts also on the bundle (pull back) E → V. Hence it acts on I_{V̄ ,V}^{∗} E. The subbundle Ē of I_{V̄ ,V}^{∗} E is also contained in the invariant part of the action.

The transversality of 𝔰_{ϵ} on (\overline{U}, \overline{E}, \overline{s}, \overline{ψ}) only implies that

d\overline{𝔰}_{ϵ} : T I_{V̄ ,V} (V̄) → Ē

is surjective. One may use it to show that the Γ invariant part of

d𝔰_{ϵ} : T V → E

is identified with d\overline{𝔰}_{ϵ} and is surjective. However it does not imply the surjectivity of the other part.

We can make 𝔰_{ϵ} to be transverse to zero on the part where Γ acts freely, just by applying standard transversality theorem. However, then, for the perturbed section 𝔰_{ϵ}, the behavior of 𝔰_{ϵ}^{−1}(0) near the singular locus is hard to control. Hence we need to do something more to find a triangulation asserted in Theorem 9.5.

This is rather an essential point of equivariant transversality, which requires a careful treatment. We borrow an idea from [FOn3] for this purpose.

We first recall some notations and lemmas from [FOn3]. We repeat the proof of the facts we need for reader’s convenience.

Let M be a manifold and Γ be a finite group. Let E_{1}, E_{2} be real vector bundles on which Γ acts. We assume that the action of Γ on E_{1} is effective. We pull back E_{2} to the total space of E_{1} by the projection π : E_{1} → M and denote it by Ẽ_{2}. Let D be a sufficiently large integer.

Definition A3.1. A section s of Ẽ_{2} is said to be a fiberwise polynomial section of degree ≤ D, if its restriction to each fiber π^{−1}(x) ⊂ E_{1} is a polynomial of degree ≤ D.

We put I_{v} = {γ ∈ Γ | γv = v}. For each subgroup Γ′ ⊆ Γ we put

E_{1}(Γ′) = {x ∈ E_{1} | I_{x} = Γ′}.

Let V_{1} = E_{1x}, V_{2} = E_{2x} be the fibers. Let V_{i}^{Γ′} be the Γ′ invariant part of V_{i}. We define

(A3.2)  i(Γ′) = dim V_{1}^{Γ′} − dim V_{2}^{Γ′}.

The main technical result we use in this section is the following :

Proposition A3.3. Let D be sufficiently large. For any generic normally polynomial section s of degree ≤ D, we have

(A3.4)  dim(s^{−1}(0) ∩ E_{1}(Γ′)) = dim M + i(Γ′),

and that s^{−1}(0) ∩ E_{1}(Γ′) is a smooth submanifold of E_{1}(Γ′).

Moreover s^{−1}(0)/Γ has a triangulation compatible with smooth structures on s^{−1}(0) ∩ E_{1}(Γ′).

Proof. Let Poly_{Γ}^{D}(V_{1}, V_{2}) be the set of all Γ-equivariant polynomial maps P : V_{1} → V_{2} of degree ≤ D. There is an evaluation map

ev : Poly_{Γ}^{D}(V_{1}, V_{2}) × V_{1} → V_{2}.

We first consider the case Γ′ = {1}.

Lemma A3.5. If the action of Γ on V_{1} is effective, then, for sufficiently large D, the space s^{−1}(0) ∩ V_{1}(1) is a smooth manifold of dimension

dim(s^{−1}(0) ∩ V_{1}(1)) = dim V_{1} + dim Poly_{Γ}^{D}(V_{1}, V_{2}) − dim V_{2}.

In other words, ev is a submersion on Poly_{Γ}^{D}(V_{1}, V_{2}) × V_{1}(1).

Lemma A3.5 follows easily from the following Sublemma A3.6.

Sublemma A3.6. Let p ∈ V_{1} and w ∈ V_{2}. We assume I_{p} = {1}. Then there exists a Γ-equivariant polynomial map P : V_{1} → V_{2} such that P (p) = w.

Proof. We may assume that V_{2} is an irreducible (over ℝ) Γ module. We put

W = ⨁_{γ∈Γ} ℝ[γ].

and define a Γ action on it by

g(∑ c_{γ}[γ]) = ∑ c_{γ}[γg^{−1}].

Since W is a regular representation of Γ, there exists a surjective Γ linear map Ψ : W → V_{2}. We choose w_{γ} ∈ ℝ such that :

Ψ (∑ w_{γ}[γ]) = w.

Since I_{p} = {1}, there exists a (ℝ valued) polynomial f on V_{1} such

f(γp) = w_{γ}

for each γ ∈ Γ. We put

P (x) = Ψ (∑_{γ} f(γx)[γ]) .

It is straightforward to see that P has the required property. □

Two bundles E_{1}, E_{2} → M induce a bundle Poly_{Γ}^{D}(E_{1}, E_{2}) → M whose fiber is Poly_{Γ}^{D}(V_{1}, V_{2}). We may identify fiberwise polynomial section of degree ≤ D with a section of the bundle Poly_{Γ}^{D}(E_{1}, E_{2}). We consider the evaluation maps

C^{∞}(M; Poly_{Γ}^{D}(E_{1}, E_{2})) ×_{M} E_{1}(1) →^{ev_{1}} Poly_{Γ}^{D}(E_{1}, E_{2}) ×_{M} E_{1}(1) →^{ev_{2}} E_{2}.

By Lemma A3.4 ev_{2}^{−1}(0) is a smooth submanifold of Poly_{Γ}^{D}(E_{1}, E_{2}) ×_{M} E_{1}(1) of codimension dim V_{2}. Since ev_{1} is a submersion we find that ev_{1}^{−1}(ev_{2}^{−1}(0)) is a smooth submanifold of codimension dim V_{2}.

We next consider the map

π : ev_{1}^{−1}(ev_{2}^{−1}(0)) → C^{∞}(M; Poly_{Γ}^{D}(E_{1}, E_{2})),

induced from the projection. Then this map is a Fredholm map of index dim E_{1} − dim V_{2}. Let us take a generic section of C^{∞}(M; Poly_{Γ}^{D}(E_{1}, E_{2})). Then it is a regular value s of π by Sard-Smale theorem. It corresponds to a normally polynomial section s of Ẽ_{2}. We can identify

s^{−1}(0) ∩ E_{1}(1) = ev_{1}^{−1}(ev_{2}^{−1}(0)) ∩ π^{−1}(0).

Hence s^{−1}(0) ∩ E_{1}(1) is a smooth manifold and (A3.4) holds for Γ′ = {1}.

To prove (A3.4) for general Γ′ we need a similar but a bit more complicated argument. We define

E_{2}(Γ′) = {v ∈ E_{2}|I_{v} ⊇ Γ′}.

We pull it back to E_{1}(Γ′) and denote it by Ẽ_{2}(Γ′). We put

V_{2}^{Γ′} = {v ∈ V_{2} | I_{v} ⊇ Γ′}.

We obtain an evaluation map

(A3.7)  ev : Poly_{Γ}^{D}(V_{1}, V_{2}) × V_{1}(Γ′) → V_{2}^{Γ′},

by restriction. Recall that

V_{1}(Γ′) = {v ∈ V_{1}|I_{v} = Γ′} (≠ V_{1}^{Γ′}).

Lemma A3.8. The map (A3.7) is a submersion.

Proof. Let x ∈ V_{1}(Γ′), it suffices to show that the linear map : Poly_{Γ}^{D}(V_{1}, V_{2}) → V_{2}^{Γ′}, P ↦ P (x) is surjective. Let N(Γ′) be the normalizer of Γ′ in Γ. We put H(Γ′) = N(Γ′)/Γ′. H(Γ′) acts freely on V_{1}(Γ′). Also H(Γ′) acts on V_{2}^{Γ′}. Let w ∈ V_{2}^{Γ′}. By Sublemma A3.6, we find an H(Γ′) invariant polynomial map P_{1} : V_{1}(Γ′) → V_{2}^{Γ′} such that P_{1}(x) = w. P_{1} is N(Γ′)-invariant too.

Since I_{x} = Γ′, it follows that γ(x) ∈ V_{1}(Γ′) if and only if γ ∈ N(Γ′). Therefore there exists a polynomial map P_{2} : V_{1} → V_{2}^{Γ′} such that

(A3.9)  P_{2}(γx) = \{ \begin{matrix}P_{1}(γx) = γP_{1}(x) & if γ ∈ N(Γ′) \\ 0 & otherwise. \end{matrix}

(P_{2} is not Γ invariant.) We define P ∈ Poly_{Γ}^{D}(V_{1}, V_{2}) by

P (v) = \frac{1}{♯N(Γ′)} ∑_{γ∈Γ} γ^{−1}P_{2}(γv)

N(Γ′) invariance of P_{1} and (A3.9) implies that P (x) = w. The proof of Lemma A3.7 is complete. □

Using (A3.9) we find that s^{−1}(0) ∩ E_{1}(Γ′) is a smooth submanifold of E_{1}(Γ′) and it satisfies (A3.4) in a way similar to the case Γ′ = {1} we already discussed.

We finally prove that s^{−1}(0) has a triangulation compatible to the smooth structures on each stratum. We again consider the evaluation map ev : Poly_{Γ}^{D}(V_{1}, V_{2}) × V_{1} → V_{2}. This map is real analytic. Hence Z = ev^{−1}(0) has a Whitney stratification. We have a fiber bundle over M whose fiber is Z. We denote it Z → M. The Whitney stratification of Z induces a Whitney stratification of Z. Therefore Z has a simplicial decomposition such that each stratum is a sub-complex. (See [GM].) Then the evaluation map

C^{∞}(M; Poly_{Γ}^{D}(E_{1}, E_{2})) ×_{M} E_{1} → Poly_{Γ}^{D}(E_{1}, E_{2}) ×_{M} E_{1}

is a submersion. Hence for generic s ∈ C^{∞}(M; Poly_{Γ}^{D}(E_{1}, E_{2})) the map : E(1) = M×_{M} E(1) → Poly_{Γ}^{D}(E_{1}, E_{2})×_{M} E_{1}, (x, v) ↦ (s(x), v) is transverse to each simplex of Z ⊂ Poly_{Γ}^{D}(E_{1}, E_{2})×_{M} E_{1}. For such s, the set s^{−1}(0) has a compatible triangulation. The proof of Proposition A3.3 is now completed. □

We next “semi-globalize” Proposition A3.3 to Proposition A3.11 below. Let X be an orbifold and E be an orbi-bundle on it. We put

X(Γ) = {x ∈ X | I_{x} ≅ Γ}.

Here I_{x} is defined as follows: We write a neighborhood of x locally as V_{x}/Γ_{x}: Let x̄ be the element of V_{x} corresponding to x: Then I_{x} is the group of elements of Γ_{x} which fix x̄.

X(Γ) is a smooth manifold. Let N_{X(Γ)}X be the normal orbi-bundle of X(Γ) in X.

Definition A3.10. A (single valued) section s of E is said to be normally polynomial if the following holds. Let Γ be a group. We identify N_{X(Γ)}X/Γ to a tubular neighborhood of X(Γ). We pull back s to N_{X(Γ)}X. Then, in a neighborhood of zero section of N_{X(Γ)}X, s coincides with a fiberwise polynomial section. (Here we put E_{1} = N_{X(Γ)}X, E_{2} = E|_{X(Γ)}.)

For each connected component X(Γ)_{k} of X(Γ), we define i(Γ; k) as in Formula (A3.2) on X(Γ)_{k}.

Proposition A3.11. Let s be a generic normally polynomial section of E. Then X(Γ; k) ∩ s^{−1}(0) is a smooth manifold of dimension dim X(Γ)_{k} + i(Γ; k). Moreover s^{−1}(0) has a triangulation compatible with the smooth structures on X(Γ; k)∩s^{−1}(0).

Proof. We construct such s on the tubular neighborhoods of X(Γ) by downward induction on the order of Γ.

In case the order of Γ is maximal then X(Γ) is a closed submanifold of X. Hence we can apply Proposition A3.3 to obtain a desired s.

Suppose we have defined s on the union of tubular neighborhoods of X(Γ) over Γ with ♯Γ > ♯Γ_{0}.

Note that if s is a fiberwise polynomial section on N(Γ_{0}) and if Γ ⊂ Γ_{0} then s induces a fiberwise polynomial section on the intersection N(Γ) ∩ N(Γ_{0}).

Hence we already have s on N(Γ_{0}) minus compact sets. We can then extend it by a relative version of Proposition A3.3, which can be proved in the same way. The proof of Proposition A3.11 is complete. □

Now we consider the case of Kuranishi structure. Let (V_{p}, E_{p}, Γ_{p}, ψ_{p}, s_{p}), (V_{pq}, ϕ̂_{pq}, ϕ_{pq} h_{pq}) be as in §A2. (In other words they are good coordinate system , in the sense of [FOn2] Definition 6.1, for a space with Kuranishi structure X.) We assume that this Kuranishi structure has a tangent bundle in the sense of [FO2] Definition 5.10.

Definition A3.12. Let s_{p}^{ϵ} be the system of single valued sections of E_{p}. We say that it is a normally polynomial section if ϕ̂_{pq}s_{q}^{ϵ} = s_{p}^{ϵ}ϕ_{pq} and if it is a normally polynomial section for each p. For each Γ and a connected component V_{p}(Γ)_{k} of V_{p}(Γ), we define i(Γ; p; k) as in Formula (A3.2).

Proposition A3.13. There exists a family of normally polynomial sections 𝔰^{ϵ} = (s_{p}^{ϵ}) such that X^{𝔰^{ϵ}} = ∪_{p}(s_{p}^{ϵ})^{−1}(0) has the following properties.

(A3.14.1) X^{𝔰^{ϵ}}(Γ) = ∪_{p}(s_{p}^{ϵ})^{−1}(0) ∩ V_{p}(Γ) is a smooth manifold. (A3.14.2) The dimension of X^{𝔰^{ϵ}}(Γ) is dim V_{p}(Γ)_{k}+i(Γ; p; k), which depends only on the connected component of X^{𝔰^{ϵ}}(Γ). (A3.14.3) ∪_{p}(s_{p}^{ϵ})^{−1}(0) has a triangulation compatible with smooth structures of X^{𝔰^{ϵ}}(Γ). (A3.14.4) lim_{ϵ→0} 𝔰^{ϵ} = s, where s is a Kuranishi map.

Proof. We order p so that q < p implies that q ∈ ψ_{p}(s_{p}^{−1}(0)) (and hence (V_{pq}, ϕ̂_{pq}, ϕ_{pq} h_{pq}) exists). We construct the desired s_{p}^{ϵ} by induction on p according to this order <.

If p is minimal we apply Proposition A3.11 to obtain s_{p}^{ϵ}. Let us assume that we have s_{q}^{ϵ} for every q < p.

We consider s_{q}^{ϵ} and the image ϕ_{pq}(V_{pq}). We restrict s_{q}^{ϵ} on the image ϕ_{pq}(V_{pq}) and use the embedding ϕ̂_{pq} to obtain a section s_{pq,1} of E_{q}|_{ϕ_{pq}(V_{pq})} → V_{pq}. We next consider the normal bundle N_{ϕ_{pq}(V_{pq})} of ϕ_{pq}(V_{pq}) in V_{p}. By the definition of existence of tangent bundle we have a canonical isomorphism

(A3.15)  N_{ϕ_{pq}(V_{pq})} ⊕ ϕ̂_{pq}(E_{q}|_{V_{pq}}) = E_{p}|_{N_{ϕpq (Vpq)}}.

Here, in the right hand side, we identify N_{ϕ_{pq}(V_{pq})} with the tubular neighborhood. Now we use the projection N_{ϕ_{pq}(V_{pq})} → ϕ_{pq}(V_{pq}) and compose it with s_{pq,1}. We then obtain a section of ϕ̂_{pq}(E_{q}|_{V_{pq}}) on N_{ϕ_{pq}(V_{pq})}. We denote it by s_{pq,1} also. On the other hand, there is an obvious section of the pull back of N_{ϕ_{pq}(V_{pq})} to the total space of N_{ϕ_{pq}(V_{pq})} itself. We denote it by s_{pq,2}. Then by (A3.15), we obtain a section (s_{pq,2}, s_{pq,1}) of E_{p}|_{N_{ϕpq (Vpq)}}. (In fact this is the way how we constructed multi section in [FOn2]. Here we use it to construct a single valued section.) We remark that this section (s_{pq,2}, s_{pq,1}) is normally polynomial since the part we added is linear. Moreover the required properties (A3.14.1), (A3.14.2) are satisfied on N_{ϕ_{pq}(V_{pq})} if it is satisfied by s_{q}^{ϵ}. Hence we can use a relative version of Proposition A3.3 to obtain the section s_{p}^{ϵ}. The proof of Proposition A3.13 is complete. □

Remark A3.16. In the proof of Proposition A3.11 we use the downward induction on the order of Γ. In the proof of Proposition A3.13 the induction is of opposite direction. Namely if q < p then the order of Γ_{q} is not greater than the order of Γ_{p}.

Now we go back to the proof of Theorem 9.5. We consider the semi-positive Lagrangian submanifold L of M. We consider the moduli space X = M_{1}(β : P_{1}, ⋯, P_{k}) as in §9. We constructed a Kuranishi structure on X in Chapter 6. We apply Proposition A3.13 to obtain 𝔰^{ϵ}. Then the perturbed moduli space M_{1}^{𝔰^{ϵ}}(β : P_{1}, ⋯, P_{k}) has a triangulation. We put

M_{1}^{𝔰^{ϵ}}(β : P_{1}, ⋯, P_{k})_{smooth} = X^{𝔰^{ϵ}}(1).

Then this space is smooth with correct dimension. Let Γ ≠ {1}. To complete the proof of Theorem 9.5, it suffices to show that

(A3.17)  dim X^{𝔰^{ϵ}}(Γ) ≤ dim X^{𝔰^{ϵ}}(1) − 2.

Let (Σ, φ) ∈ X^{𝔰^{ϵ}}(Γ). Let (Σ̄, φ̄) be its reduced model. Let (Σ̄, φ̄) ∈ M_{1}^{𝔰^{ϵ}}(J, β̄ : P_{1}, ⋯, P_{k}). (More precisely (Σ̄, φ̄) is contained in a Kuranishi neighborhood of elements of M_{1}(β̄ : P_{1}, ⋯, P_{k}).) Then, it follows from the definition of i(Γ; p; k) that the virtual dimension of M_{1}(β̄ : P_{1}, ⋯, P_{k}) is dim V_{p}(Γ)_{k} + i(Γ; p; k).

Therefore (9.8) and (A3.14.2) imply (A3.17). The proof of Theorem 9.5 is now complete. □

Remark A3.18. Remark that the semi-positivity is essentially used in the proof above. Namely we use it to show (9.8). The idea in [FOn3] is different from that of this paper. There we use (almost) complex structure (of the moduli space), instead of dimension counting (9.8) to construct a fundamental chain over ℤ of the moduli space of pseudo-holomorphic curves in general situation. We will pursue this line of ideas and prove the general Arnold-Givental’s conjecture elsewhere.

§A4 Construction of the spectral sequence from a continuous family of filtrations

The purpose of this section is to construct a spectral sequence from a filtered differential Λ_{0,nov} module and to apply the spectral sequence to prove Theorem F.

We first start from the following situation. Let V = (Λ_{0,nov}^{(0)})^{⊕I} be a free Λ_{0,nov}^{(0)} module, which is not necessarily finitely generated. (Here we denote by Λ_{0,nov}^{(0)} the degree 0 part of Λ_{0,nov}.) We define a filtration on V by F^{λ}V = (F^{λ}Λ_{0,nov}^{(0)})^{⊕I} which will induce a topology on V. Let V̂ be the completion of V. We call such V̂ a completed free filtered Λ_{0,nov}^{(0)} module generated by energy zero elements, or in short c.f.z. If V is finitely generated in addition, we say that it is a finite c.f.z.

We define a function, which we call the energy function,

E : V̂ → ℝ_{≥0}

such that

𝔳 ∈ F^{E(𝔳)}V, 𝔳 ∉ F^{λ}V if λ > E(𝔳).

Let \overline{V} = V/Λ_{0,nov}^{+,(0)}V ≅ ℚ^{I}. We fix an embedding \overline{V} ⊂ V as the energy 0 part of V so that its composition with the projection V → \overline{V} is the identity map.

Let 𝔳 ∈ V. We put

𝔳 = ∑ T^{λ_{i}}v_{i},

Here v_{i} ∈ \overline{V}, λ_{i} < λ_{i+1}, lim_{i→∞} λ_{i} = ∞, and v_{i} ≠ 0. We call v_{i} the components of 𝔳, T^{λ_{1}}v_{1} the leading component and v_{1} the leading coefficient of 𝔳. We denote the leading coefficient v_{1} of 𝔳 by σ(𝔳). We also define the leading term and the leading coefficient of an element of Λ_{0,nov}^{(0)} in the same way.

Definition A4.1. A (finite) set {𝔳_{i}}_{1≤i≤m} of elements 𝔳_{i} ∈ V̂, i = 1, ⋯, m which generates the subspace W ⊂ V̂ is called a standard basis of W if the following holds: When we put E(𝔳_{i}) = λ_{i} and let σ(𝔳_{i}) be the leading coefficient of 𝔳_{i}, we have

(A4.1.1) λ_{i} ≤ λ_{i+1}. (A4.1.2) σ(𝔳_{1}), ⋯, σ(𝔳_{m}) are linearly independent over ℚ.

It is easy to see that W is a free Λ_{0,nov}^{(0)} module if it has a standard basis.

Lemma A4.2. Any finitely generated submodule W of V̂ has a standard basis.

Proof. Assume that W is generated by the set of m elements 𝔳_{i}, i = 1, ⋯, m. We put E(𝔳_{i}) = λ_{i}. We may assume

λ_{i} ≤ λ_{i+1}

by reordering them, if necessary.

We will modify the set so that it becomes a standard basis.

Let us first take k such that

(A4.3)  λ_{1} = ⋯ = λ_{k} < λ_{k+1}.

If we consider the leading coefficients σ(𝔳_{j}) of 𝔳_{j} for j = 1, ⋯, k, we can find an invertible matrix (a_{i,i′}) with rational entries such that the linear combinations

∑_{i′=1}^{k} a_{i,i′}σ(𝔳_{i′}),

are linearly independent for i = 1, ⋯, ℓ ≤ k and become zero for i > ℓ. By changing a part of the generators, 𝔳_{1}, ⋯, 𝔳_{k} by the matrix (a_{i,i′}), we find a generating set {𝔳_{i}} of W that satisfies (A4.4.1), where the condition (A4.4.n) is defined as follows:

(A4.4.n) When we define an integer k_{e}’s by

λ_{1} = ⋯ = λ_{k_{1}} < λ_{k_{1}+1} = ⋯ = λ_{k_{2}} < λ_{k_{2}+1} = ⋯,

σ(𝔳_{j})’s for j = 1, ⋯, k_{n} are linearly independent.

Now we will modify the given elements 𝔳_{i}’s inductively over n so that (A4.4.n) is satisfied. Let us assume (A4.4.n). By applying a filtered automorphism of V, we may assume that

𝔳_{j} = T^{λ_{j}}σ(𝔳_{j}),

for j = 1, ⋯, λ_{k_{n}}.

Let X ⊂ \overline{V} is the ℚ linear subspace generated by σ(𝔳_{j}), j = 1, ⋯, λ_{k_{n}}. We consider the following two cases separately:

(A4.5.1) There exists i ∈ {k_{n} + 1, ⋯, k_{n+1}} such that σ(𝔳_{i}) ∉ X. (A4.5.2) σ(𝔳_{i}) ∈ X for all i ∈ {k_{n} + 1, ⋯, k_{n+1}}.

We first consider the case where (A4.5.1) is satisfied. Then we have an invertible rational matrix (a_{i,i′}) (i, i′ ∈ {k_{n} + 1, ⋯, k_{n+1}}) and b_{i,j} ∈ ℚ, (i ∈ {k_{n} + 1, ⋯, k_{n+1}}, j ∈ {1, ⋯, k_{n}}) satisfying the condition (A4.6) below, if we put

v′_{i,1} = ∑_{j=1}^{k_{n}} b_{i,j}σ(𝔳_{j}) + ∑_{i′=k_{n}+1}^{k_{n+1}} a_{i,i′}σ(𝔳_{i′}) :

(A4.6.1) There exists k_{n} < ℓ ≤ k_{n+1} such that the union {σ(𝔳_{j})}_{1≤j≤k_{n}} ∪ {v′_{i,1}}_{k_{n}+1≤i≤ℓ} is linearly independent. (A4.6.2) v′_{i,1} are zero for i > ℓ.

Now for i = k_{n} + 1, ⋯, k_{n+1}, we replace 𝔳_{i} by

𝔳′_{i} = ∑_{j} b_{i,j}T^{λ_{kn+1}−λ_{j}}𝔳_{j} + ∑_{i′} a_{i,i′}𝔳_{i′}.

Note that the leading coefficient of 𝔳′_{i} is equal to v′_{i,1} for k_{n} + 1 ≤ i ≤ ℓ. Hence, (A4.6.2) implies E(𝔳′_{i}) > λ_{k_{n}+1} for ℓ < i ≤ k_{n+1}. Now (A4.4.n + 1) follows from (A4.6.1). Thus the induction works for the first case.

In case (A4.5.2) is satisfied, for all k_{n} + 1 ≤ i ≤ k_{n+1} we have

v′_{i,1} = ∑_{j=1}^{k_{n}} b_{i,j}σ(𝔳_{j}).

for some b_{i,j}. We replace 𝔳_{i} for i = k_{n} + 1, ⋯, k_{n+1} by

𝔳′_{i} = 𝔳_{i} − ∑_{j=1}^{k_{n}} b_{i,j}T^{λ_{kn+1}−λ_{j}}𝔳_{j}.

It then follows from the above that the coefficient of 𝔳′_{i} for the energy level λ_{k_{n}+1} becomes

σ(𝔳_{i}) − ∑_{j=1}^{k_{n}} b_{i,j}σ(𝔳_{j}) = σ(𝔳_{i}) − σ(𝔳_{i}) = 0

for all k_{n} + 1 ≤ i ≤ k_{n+1}. Therefore we have

E(𝔳′_{i}) > λ_{k_{n}+1} = λ_{k_{n+1}}

for all k_{n} + 1 ≤ i ≤ k_{n+1}. We now re-number 𝔳_{i}’s for i = k_{n} + 1, ⋯ according to their energy.

We again divide the cases into (A4.5.1) and (A4.5.2) and repeat the same process. If we end up with the (A4.5.1), we go to the next step of the induction. Otherwise we continue. This process will continue infinitely many times only if all the components of 𝔳_{i}, i = k_{n} + 1, ⋯ are contained in X. In this case it is easy to see that we can eliminate all 𝔳_{i}, i = k_{n} + 1, ⋯ from the given generating set without changing the module generated by them. The proof of Lemma A4.2 is now complete. □

An immediate corollary of Lemma A4.2 and the remark right before it is the following

Corollary A4.7. Any finitely generated submodule W of V̂ is free and closed.

Now we consider the case of graded Λ_{0,nov}^{(0)} modules. Let Ĉ be a graded Λ_{0,nov} module. We assume Ĉ^{k} is c.f.z. for each k.

Definition A4.8. A differential graded c.f.z. (abbreviated as d.g.c.f.z) is a pair (Ĉ, δ) with δ : V̂ → V̂ a degree 1 operator such that

δ ∘ δ = 0, δ(F^{λ}Ĉ) ⊆ F^{λ}Ĉ.

We call the pair a finite d.g.c.f.z. if each Ĉ^{k} is finitely generated c.f.z.

We now prove the following proposition which will be essential to prove some convergence properties of the spectral sequence which we will construct in this section.

Proposition A4.9. Let W be a finitely generated Λ_{0,nov}^{(0)} submodule of Ĉ^{k}. Then there exists a constant c independent of λ such that

δ(W) ∩ F^{λ}Ĉ^{k+1} ⊂ δ(W ∩ F^{λ−c}Ĉ^{k}).

Proof. Obviously δ(W) is a finitely generated submodule of V̂ and so has a standard basis by Lemma A4.2. Let {𝔳_{i}} be a standard basis of δ(W) and let 𝔴_{i} ∈ W satisfy δ𝔴_{i} = 𝔳_{i}. Suppose 𝔲 ∈ δ(W) with E(𝔲) = λ and

𝔲 = ∑ a_{i}𝔳_{i}.

We put σ(a_{i}) ∈ ℚ to be the leading coefficient of a_{i} ∈ Λ_{0,nov}^{(0)} and let E(𝔳_{i}) = λ_{i} and E(a_{i}) = μ_{i}.

Lemma A4.10. If λ > λ_{i} for all i, then we have λ_{i} + μ_{i} ≥ λ.

Proof. Suppose the contrary that there exists some i with λ_{i} + μ_{i} < λ. Let

λ − a = inf_{i}{λ_{i} + μ_{i}}

with a > 0. We will then have

∑_{i:λ−a=λ_{i}+μ_{i}} σ(a_{i})σ(𝔳_{i}) = 0

because E(u) = λ > λ − a. This contradicts to the fact that the leading coefficients of 𝔳_{i}’s are linearly independent. This finishes the proof. □

It follows from Lemma A4.10 that

E(a_{i}) = μ_{i} ≥ λ − λ_{i}.

We put c = max{λ_{i}}. Then there exists b_{i} ∈ Λ_{0,nov}^{(0)} such that T^{λ−c}b_{i} = a_{i}. Hence

𝔲 = δ(∑ T^{λ−c}b_{i}𝔴_{i}) ∈ δ(W ∩ F^{λ−c}Ĉ^{k}).

Proposition A4.9 is proved. □

Lemma A4.11. Let V be a finite c.f.z. and W be its submodule such that there exists 𝔴_{0} ∈ W satisfying T^{λ}𝔴_{0} = 𝔴 for any 𝔴 ∈ W with E(𝔴) = λ. Then W is finitely generated. Moreover it has a standard basis all of whose elements have energy 0.

Proof. We emphasize that Λ_{0,nov}^{(0)} is not Noetherian and so the finitely-generatedness of W does not trivially follow from a standard result on the ring theory.

Let W_{00} be the set of elements of \overline{V} which are leading coefficients of some elements of W. W_{00} ∪ \{0\} is obviously a finite dimensional subspace of \overline{V}, which we write as W_{0}. Let {w_{i}} be a basis of W_{0}. For each w_{i}, we choose an element 𝔴_{i} ∈ W whose leading coefficient is w_{i}. By the hypothesis of the lemma, we may assume E(𝔴_{i}) = 0. We will prove that {𝔴_{i}} generates W, which will in turn imply the lemma.

By applying a filtered automorphism on V, we may assume that

(A4.12)  𝔴_{i} = w_{i} ∈ \overline{V} .

Now let 𝔳 ∈ W and put

𝔳 = ∑ T^{λ_{i}}v_{i}.

We have v_{1} ∈ W_{0} by definition. (A4.12) implies v_{1} ∈ W and so ∑_{i=2} T^{λ_{i}}v_{i} is contained in W. Hence we have v_{2} ∈ W_{0}. Repeating the same argument, we prove that v_{i} ∈ W_{0} for all i. It is then easy to see that 𝔳 is contained in the Λ_{0,nov}^{(0)} submodule generated by 𝔴_{i}’s, since each of v_{i} is a linear combination of w_{i}’s. □

Corollary 4.13. Let (C, δ) be a finite d.g.c.f.z. Then Ker δ_{k} = Ker δ ∩ C^{k} is finitely generated for all k. Furthermore we can take their standard bases consisting of elements of energy 0.

Proof. It is easy to see that Ker δ satisfies the assumption of Lemma 4.11. □

Corollary A4.13 and Lemma A4.2 immediately imply the following.

Proposition A4.14. Let (C, δ) be a finite d.g.c.f.z. Then for each integer p, there exist finitely many constants μ_{i,p} > 0, i = 1, ⋯, k_{p} with ℓ ≤ rank C^{p} − k_{p} and ℓ_{p} such that

(A4.15)  H^{p}(C, δ) ≅ ( ⨁_{i=1}^{k_{p}} (Λ_{0,nov}^{(0)}/F^{μ_{i}}Λ_{0,nov}^{(0)})) ⨁(Λ_{0,nov}^{(0)})^{⊕ℓ_{p}}.

Proof. First note that since C^{p} is finite, Im δ_{p} is finitely generated, and so has a standard basis, say, \{T^{μ_{1}}u_{1}, T^{μ_{2}}u_{2}, ⋯, T^{μ_{kp}} u_{k_{p}}\} for u_{i} ∈ \overline{C}. Since we have Im δ_{p−1} ⊂ Ker δ_{p} and Ker δ_{p} has a basis in \overline{C} consisting of energy zero, u_{i} ∈ Ker δ_{p}. Therefore we may extend {u_{1}, ⋯, u_{k_{p}}} to a (standard) basis {u_{1}, u_{2}, ⋯, u_{k_{p}}, u_{k_{p+1}}, ⋯, u_{k_{p}+ℓ_{p}}}. Now (A4.15) immediately follows from this. □

Now let (Ĉ, δ) be a d.g.c.f.z. and Ĉ^{k} be a completion of C^{k}, which is free over Λ_{0,nov}^{(0)}. We put

\overline{C} = C/Λ_{0,nov}^{+,(0)}C ≅ Ĉ/Λ_{0,nov}^{+,(0)}Ĉ

and let \overline{δ} be the induced derivation on \overline{C}. We again embed \overline{C} ⊆ C ⊆ Ĉ as the energy 0 part. In general \overline{C} is not a differential graded subalgebra of Ĉ. Let \{𝔟_{i}\} be a basis of C (over Λ_{0,nov}^{(0)}) and \overline{𝔟}_{i} be the corresponding basis of \overline{C} over ℚ = Λ_{0,nov}^{(0)}/Λ_{0,nov}^{+,(0)}. We put

\overline{δ}(\overline{𝔟}_{i}) = ∑ δ_{0,ij}\overline{𝔟}_{j},

and define δ_{0} : Ĉ → Ĉ by

δ_{0}𝔟_{i} = ∑ δ_{0,ij}𝔟_{j}

Condition A4.16. We say that (Ĉ, δ) satisfies the gap condition if there exists λ′′ > 0 such that for any λ we have

δ𝔳 − δ_{0}𝔳 ∈ F^{λ+λ′′} Ĉ

for all 𝔳 ∈ F^{λ}Ĉ.

Assuming Condition A4.16, we take a constant λ_{0} with 0 < λ_{0} < λ′′ and define a filtration by

(A4.17)  𝔉^{n}Ĉ = F^{nλ_{0}} Ĉ.

We use this filtration to define our spectral sequence. Since construction of the spectral sequence from a filtered module over a filtered ring is not so standard, we present its construction here for reader’s convenience.

We put

(A4.18.1)  Z_{r}^{p,q}(Ĉ) = {x ∈ 𝔉^{q}Ĉ^{p}|δx ∈ 𝔉^{q+r−1}Ĉ^{p+1}} + 𝔉^{q+1}Ĉ^{p},

(A4.18.2)  B_{r}^{p,q}(Ĉ) = (δ(𝔉^{q−r+2}Ĉ^{p−1}) ∩ 𝔉^{q}Ĉ^{p}) + 𝔉^{q+1}Ĉ^{p},

(A4.18.3)  E_{r}^{p,q}(Ĉ) = \frac{Z_{r}^{p,q}(Ĉ)}{B_{r}^{p,q}(Ĉ)}.

We denote

Λ^{(0)}(λ) = Λ_{0,nov}^{(0)}/F^{λ}Λ_{0,nov}^{(0)}.

We define a filtration of Λ_{0,nov}^{(0)} by 𝔉^{n}Λ_{0,nov}^{(0)} = F^{nλ_{0}}Λ_{0,nov}^{(0)}. Then its associated graded module is given by

gr_{∗}(𝔉Λ_{0,nov}^{(0)}) = ⨁_{n∈ℤ_{≥0}} gr_{n}(𝔉Λ_{0,nov}^{(0)})

where each gr_{n}(𝔉Λ_{0,nov}^{(0)}) is naturally isomorphic to Λ^{(0)}(λ_{0}). We also have

gr_{∗}(𝔉Λ_{0,nov}) = gr_{∗}(𝔉Λ_{0,nov}^{(0)})[e, e^{−1}] = ⨁_{n∈ℤ_{≥0}} gr_{n}(𝔉Λ_{0,nov}^{(0)})[e, e^{−1}]

Recall that E_{r}^{p,q} has a natural structure of Λ^{(0)}(λ_{0}) module. Multiplication by e^{±1} ∈ Λ_{0,nov} defines a map

e^{±1} : E_{r}^{p,q}(Ĉ) → E_{r}^{p±2,q}(Ĉ)

which turns E_{r}(Ĉ) := ⨁_{p,q} E_{r}^{p,q}(Ĉ) into a gr_{∗}(𝔉Λ_{0,nov}) module and ⨁_{p,q} δ_{r}^{p,q} : ⨁_{p,q} E_{r}^{p,q}(Ĉ) → ⨁_{p,q} E_{r}^{p+1,q+r−1}(Ĉ) is a gr(𝔉^{∗}Λ_{0,nov}) module homomorphism.

Lemma A4.19. There exists a Λ^{(0)}(λ_{0}) homomorphism

δ_{r}^{p,q} : E_{r}^{p,q}(Ĉ) → E_{r}^{p+1,q+r−1}(Ĉ)

such that

(A4.20.1) δ_{r}^{p+1,q+r} ∘ δ_{r}^{p,q} = 0. (A4.20.2) Ker(δ_{r}^{p,q})/ Im(δ_{r}^{p−1,q−r}) ≅ E_{r+1}^{p,q}(Ĉ). (A4.20.3) e^{±1} ∘ δ_{r}^{p,q} = δ_{r}^{p±1,q} ∘ e^{±1}.

Proof. We define

δ_{r}^{p,q}[x] = [δx] ∈ E_{r}^{p+1,q+r−1}(Ĉ).

The proof of (A4.20) is standard. □

Of course the construction of E_{r}^{p,q}(Ĉ) is quite standard. What is different from the standard case is that our filtration is not bounded. Namely we do not have 𝔉^{n}Ĉ = 0 for large n. Hence the convergence property of our spectral sequence is far from being trivial in general. However it is stable from below in that 𝔉^{0}Ĉ = Ĉ. As a consequence we have :

Lemma A4.21. There exists an injection

E_{r+1}^{p,q}(Ĉ) → E_{r}^{p,q}(Ĉ)

if q − r + 2 ≤ 0.

Proof. The denominator δ(𝔉^{q−r+2}Ĉ^{p−1})∩𝔉^{q}Ĉ^{p}+𝔉^{q+1}Ĉ^{p} in the definition of E_{r}^{p,q}(Ĉ) is independent of r if q − r + 2 ≤ 0. □

Definition A4.22 We define

E_{∞}^{p,q}(Ĉ) = lim_{⟵} E_{r}^{p,q}(Ĉ)

as the projective limit which exists by Lemma A4.21.

OBSTRUCTION TO FLOER HOMOLOGY 275

Lemma A4.23. We have

E_{2}^{∗,∗}(Ĉ) ≅ H(\overline{C}; \overline{δ}) ⊗_{ℚ} gr_{∗}(𝔉Λ_{0,nov}).

as gr_{∗}(𝔉Λ_{0,nov}) module. Proof. By definition we have

E_{1}^{∗,∗}(Ĉ) ≅ \overline{C} ⊗_{ℚ} gr_{∗}(𝔉Λ_{0,nov}).

It follows from Condition A4.16 that δ_{1} = \overline{δ}. Hence finishes the proof. □ Definition A4.24. We define 𝔉^{q}H(V, δ) to be the image of H(𝔉^{q}V, δ) in H(V, δ).

To relate the limit E_{∞}^{p,q} of the spectral sequence and 𝔉^{q}H(V, δ), we need some finiteness assumption which we now describe. Let V and V′ be d.g.c.f.z. satisfying the gap condition. Let φ : V → V′ be a map such that φδ = δ′φ and let \overline{φ} : \overline{V} → \overline{V}′ be the map induced on \overline{V} = V/Λ_{0,nov}^{+}V and \overline{V}′ = V′/Λ_{0,nov}^{+}V′ respectively. \overline{φ} lifts to φ_{0} : V → V′ using the basis on V and V′. We say that φ satisfies a gap condition, if there exists λ′′ such that

(φ − φ_{0})(F^{λ}V̂) ⊂ F^{λ+λ′′} V̂.

Definition A4.25. (V̂, δ) is said to be weakly finite if there exists a finite d.g.c.f.z. (W, δ′), and a gapped filtered chain map φ : (V̂, δ) → (W, δ′) which induces isomorphisms on cohomology

\begin{aligned}φ_{∗} : H(V̂, δ) & ≅ H(W, δ′), \\ \overline{φ}_{∗} : H(\overline{V̂}, \overline{δ}) & ≅ H(\overline{Ŵ}, \overline{δ}′). \end{aligned}

Using these definitions, we prove Theorem A4.26. If C is weakly finite, then there exists r_{0} such that :

E_{r_{0}}^{p,q}(Ĉ) ≅ E_{r_{0}+1}^{p,q}(Ĉ) ≅ ⋯ ≅ E_{∞}^{p,q}(Ĉ) ≅ 𝔉^{q}H^{p}(Ĉ, δ)/𝔉^{q+1}H^{p}(Ĉ, δ)

as Λ^{0}(λ_{0}) = Λ_{0,nov}^{0}/F^{λ_{0}}Λ_{0,nov}^{0} modules. Proof. We first consider the case when C^{k} is finitely generated. (In this case we have C^{k} = Ĉ^{k}.) By Proposition A4.9 there exists c such that

δ(C^{k}) ∩ F^{λ}C^{k+1} ⊂ δ(F^{λ−c}C^{k}),

for any λ. We choose r_{0} so that

(r_{0} − 1)λ_{0} − c > λ_{0}.

Now let r ≥ r_{0} and

𝔵 ∈ Z_{r}^{p,q}(C).

We have 𝔵 = 𝔵_{0} + 𝔵_{1} such that

δ𝔵_{0} ∈ 𝔉^{r+q−1}C = F^{(r+q−1)λ_{0}}C, 𝔵_{1} ∈ 𝔉^{q+1}C.

Therefore, by Proposition A4.9, we have 𝔶 such that

δ𝔵_{0} = δ𝔶, 𝔶 ∈ F^{(r+q−1)λ_{0}−c}C ⊂ F^{(q+1)λ_{0}}C = 𝔉^{q+1}C.

In follows that

[𝔵] = [𝔵 − 𝔶] = [𝔵_{0} − 𝔶] ∈ E_{r}^{p,q}(C),

and

δ_{r}[𝔵] = δ_{r}[𝔵_{0} − 𝔶] = 0.

Thus we have

E_{r_{0}}^{p,q} ≅ ⋯ E_{r_{0}+1}^{p,q} ≅ ⋯ ≅ E_{∞}^{p,q}.

To prove E_{∞}^{p,q} ≅ 𝔉^{q}H^{p}(C, δ)/𝔉^{q+1}H^{p}(C, δ), we first construct a map

π_{p,q} : 𝔉^{q}H^{p}(C, δ) → E_{∞}^{p,q}.

Let 𝔵 ∈ 𝔉^{q}C^{p} with δ𝔵 = 0 and [𝔵] ∈ E_{r}^{p,q}(C) for any r ≥ r_{0}. Such [𝔵] defines an element of E_{∞}^{p,q}(C). Suppose that 𝔵 − 𝔵′ = δ𝔶 ∈ 𝔉^{q−r}V^{p−1}. If r > q, we have 𝔶 ∈ 𝔉^{q−r}V^{p−1} = V^{p−1}. Hence [𝔵] = [𝔵′] in E_{r}^{p,q}(C). Therefore this assignment defines a well-defined map which we denote by π_{p,q}.

We next prove that π_{p,q} is surjective. Let

[𝔵] ∈ E_{r_{0}}^{p,q}(C) ≅ E_{∞}^{p,q}(C).

Then, as we proved before, there exists 𝔶 ∈ 𝔉^{q+1}C such that

[𝔵 − 𝔶] ≡ [𝔵] ∈ E_{r}^{p,q}(C), δ(𝔵 − 𝔶) = 0.

Therefore 𝔵−𝔶 defines a cycle in 𝔉^{q}C^{p} and so [𝔵−𝔶] ∈ 𝔉^{q}H^{p}(C, δ) with π_{p,q}[𝔵−𝔶] = [𝔵] and hence π_{p,q} is surjective.

We next prove that

Ker π_{p,q} = 𝔉^{q+1}H^{p}(C, δ).

Let [x] ∈ 𝔉^{q}H^{p}(C, δ) and π_{p,q}([x]) = 0. Then for r > q + 2 we have

x ∈ δ(𝔉^{q−r+2}C^{p−1}) + 𝔉^{q+1}C^{p} ⊂ δ(C^{p−1}) + 𝔉^{q+1}C^{p}.

Hence [x] ∈ 𝔉^{q+1}H^{p}(C, δ) as required. This proves Ker π_{p,q} ⊂ 𝔉^{q+1}H^{p}(C, δ). On the other hand, 𝔉^{q+1}H^{p}(C, δ) ⊂ Ker π_{p,q} is obvious.

Finally we relax the condition that Ĉ is finitely generated. Let (C′, δ′) be a finite d.g.c.f.z. and φ : Ĉ → C′ be a gapped chain map which induces isomorphisms on homologies as in Definition A4.25. Then φ induces a morphism of spectral sequence. It is an isomorphism in the E_{2} term and hence an isomorphism at E_{∞} term. On the other hand, φ is assumed to induce an isomorphism on the cohomology H(Ĉ, δ) → H(C′, δ′). Therefore π_{p,q} is an isomorphism for C because it is so for C′ □

We also remark that the proof above also gives the following which we will use later.

Addendum A4.26 bis. Even if we do not assume that C is weakly finite, we still have a well-defined map

π_{p,q} : 𝔉^{q}H^{p}(Ĉ, δ)/𝔉^{q+1}H^{p}(Ĉ, δ) → E_{∞}^{p,q}(Ĉ)

which is injective.

Proof. Let 𝔵 ∈ 𝔉^{q}Ĉ^{p} with δ𝔵 = 0. By definition of Z_{r}^{p,q}(Ĉ), 𝔵 ∈ Z_{r}^{p,q}(Ĉ) for any r ≥ 1 and so defines an element [𝔵]_{r} ∈ E_{r}^{p,q}(Ĉ). Furthermore [𝔵]_{r+1} is mapped to [𝔵]_{r} by the injection E_{r+1}^{p,q}(Ĉ) → E_{r}^{p,q}(Ĉ) and hence the sequence [𝔵]_{r} defines an element E_{∞}^{p,q}(Ĉ). It can be easily seen that this defines a well-defined map which we denote by

π̃_{p,q} : 𝔉^{q}H^{p} → E_{∞}^{p,q}.

To prove the injectivity of this map, it suffices to show

Ker π̃_{p,q} = 𝔉^{q+1}H^{p}(C̃, δ)

which can be proven by the same argument as in the above proof of Theorem A4.26. □

We use Theorem A4.26 to prove the following proposition that we used in Chapter 4.

Proposition A4.27. Let C,C′ be two d.g.c.f.z. satisfying the gap condition and φ : C → C′ be a gapped chain map between them. Suppose that \overline{φ} induces an isomorphism : H(\overline{C}, \overline{δ}) → H(\overline{C}′, \overline{δ}′). If we assume in Addison that both C and C′ are weakly finite, then φ induces an isomorphism : H(Ĉ, δ) → H(Ĉ′, δ′).

Proof. The gap condition implies that φ induces a morphism of the spectral sequence. Then

E_{r}^{p,q}(C) → E_{r}^{p,q}(C′)

is an isomorphism because the hypothesis on \overline{φ} implies that so is for E_{2}-term. Hence so is for

E_{∞}^{p,q}(C) → E_{∞}^{p,q}(C′).

Proposition A4.27 then follows from Theorem A4.26. □

The following theorem enables us to apply Theorem A4.26 to the situation we are mainly interested in.

Let L be a relatively spin Lagrangian submanifold of M and (C(L, Λ_{0,nov}), 𝔪) be the associated A_{∞} algebra. We assume that L is unobstructed i.e., the moduli M̂(C(L, Λ_{0,nov}), 𝔪) of A_{∞} deformations is nonempty and let b be its element.

Theorem A4.28. For any b ∈ M̂(C(L, Λ_{0,nov}), 𝔪)), (C(L, Λ_{0,nov}), 𝔪_{1}^{b}) is weakly finite.

The proof will be postponed until later in the end of this section.

Now assuming this theorem, we use our spectral sequence to prove Theorem F in the general case. (Recall that we proved the theorem in §8 under the additional assumption that the image of the map π_{2}(N, L) → ℝ; h ↦ ∫_{D^{2}} h^{∗}ω is discrete in ℝ.)

First we state a relation between Floer homology over Λ_{nov} coefficient and over Λ_{0,nov} coefficient. We remark that Λ_{nov} = Λ_{0,nov}[T^{−1}] i.e., Λ_{nov} is a localization of Λ_{0,nov}. In particular Λ_{nov} is flat over Λ_{0,nov}. Therefore we have

(A4.29)  HF ((L, b), (L, b); Λ_{nov}) ≅ HF ((L, b), (L, b); Λ_{0,nov}) ⊗_{Λ_{0,nov}} Λ_{nov}.

Recall that the conclusion of A4.14 holds in our case because our C = C(L; Λ_{0,nov}) is weakly finite by Theorem A4.28. Hence we have

(A4.30) H^{p}(C(L; Λ_{0,nov}), 𝔪_{1}^{b}) ≅ ( ⨁_{i=1}^{m_{p}} (Λ_{0,nov}^{(0)}/F^{λ_{i,p}}Λ_{0,nov}^{(0)})) ⨁(Λ_{0,nov}^{(0)})^{⊕ℓ_{p}}

as a Λ_{0,nov}^{(0)} module. From this, it follows that

(A4.31)  HF^{p}((L, b), (L, b); Λ_{nov}) ≅ (Λ_{nov}^{(0)})^{⊕ℓ_{p}}

as a Λ_{nov}^{(0)} module, which is ℤ-graded. We recall Λ_{0,nov} ≅ Λ_{0,nov}^{(0)}[e, e^{−1}] and e has degree two. Therefore H^{p}(C(L; Λ_{0,nov}, 𝔪_{1}^{b}) is periodic with period 2 as a Λ_{0,nov}^{(0)} module and so is HF^{k}((L, b), (L, b); Λ_{nov}) as a Λ_{nov}^{(0)} module.

For the proof of Theorem F, we study the spectral sequence we have constructed in this section. Again we would like to emphasize that it is not obvious that E_{r}^{p,q}(C) is finitely generated, because our coefficient ring Λ_{0,nov}^{(0)} is not Noetherian. To prove the finite generatedness of E_{r}^{p,q}(C) is one of the main parts of our proof.

For this purpose we need to further study modules over Λ_{0,nov}^{(0)} in general.

Definition A4.32. A Λ_{0,nov}^{(0)} submodule W of V is called pure if it is generated by monomials T^{μ_{1}}a_{1}, ⋯, T^{μ_{n}}a_{n} for a_{i} ∈ \overline{V} and for μ_{i} ∈ ℝ_{≥0}.

Roughly speaking the pure submodule is a notion analogous to the graded module. The difference is that here our “grading” is provided by the real number rather than by the integer.

Lemma A4.33. Let V be a finite c.f.z. Then W is pure if and only if there exists a family of linear subspaces 𝔈_{λ}(W) ⊆ \overline{V}, λ ∈ ℝ_{≥0} such that

(A4.34.1) If λ ≤ λ′ then 𝔈_{λ}(W) ⊆ 𝔈_{λ′}(W). (A4.34.2) For each c ∈ ℝ, the set {λ| dim 𝔈_{λ}(W) ≥ c} is closed. (A4.34.3) ∑_{i=1}^{∞} T^{λ_{i}}v_{i}, (v_{i} ∈ \overline{V}, λ_{i} < λ_{i+1}, λ_{i} → ∞) is an element of W if and only if

v_{i} ∈ 𝔈_{λ_{i}}(W).

Proof. Suppose that W is pure, and let a_{i} and μ_{i} be as in Definition A4.32. We choose 𝔈_{λ}(W) to be the subspace spanned by {a_{i}|μ_{i} ≤ λ}. Condition (A4.34) is easy to see for this choice.

Conversely suppose that for any λ ≥ 0 there exist 𝔈_{λ}(W)’s that satisfy (A4.34). We remark that the ℚ vector space \overline{V} is finite dimensional since V is finitely generated. Therefore the nondecreasing function λ ↦ dim 𝔈_{λ}(W) is discontinuous only at finitely many points λ_{i}. We choose dim 𝔈_{λ_{i}}(W)−dim 𝔈_{λ_{i}−ϵ}(W) = k_{i} and vectors a_{i,1} ⋯ a_{i,k_{i}} ∈ 𝔈_{λ_{i}}(W) that induce a basis of 𝔈_{λ_{i}}(W)/𝔈_{λ_{i}−ϵ}(W). It is easy to see that the finitely many elements T^{λ_{i}}a_{i,j} for j = 1, ⋯, i_{k} and i = 1, ⋯ generate W. This proves the pureness of W. □

We remark that existence of 𝔈_{λ}(W) satisfying (A4.34) implies that W is finitely generated.

Lemma A4.35. If W_{1}, W_{2} ⊂ V is pure then W_{1} ∩ W_{2}, W_{1} + W_{2} are pure.

Proof. It is easy to see that the choice

\begin{aligned}𝔈_{λ}(W_{1} ∩ W_{2}) & = 𝔈_{λ}(W_{1}) ∩ 𝔈_{λ}(W_{2}) \\ 𝔈_{λ}(W_{1} + W_{2}) & = 𝔈_{λ}(W_{1}) + 𝔈_{λ}(W_{2}) \end{aligned}

will satisfy all the properties for W_{1} ∩ W_{2} and W_{1} + W_{2} respectively. □

Let \overline{A} : \overline{V} → \overline{V}′ be a ℚ linear map and \overline{A} ⊗ 1 : V = \overline{V} ⊗_{ℚ} Λ_{0,nov}^{(0)} → V′ = \overline{V}′ ⊗_{ℚ} Λ_{0,nov}^{(0)} be its obvious extension. We denote this extension by A.

Definition A4.36. We call a Λ_{0,nov}^{(0)} module homomorphism φ : V → V′ pure if there exists \overline{A} : \overline{V} → \overline{V}′ and a constant λ ≥ 0 such that φ = T^{λ}A.

Lemma A4.37. If φ : V → V′ is pure and if W ⊆ V′ is pure, then φ^{−1}W is pure.

Proof. Let φ = T^{λ}A and 𝔈_{λ}(W) be the subspace of \overline{V} satisfying (A4.34). Then the choice

𝔈_{μ}(φ^{−1}W) = \overline{A}^{−1}(𝔈_{μ+λ}(W))

will satisfy (A4.34) for φ^{−1}W and hence the proof. □

Now we go back to our situation. Let (C, δ) be a finite d.g.c.f.z. We take a constant c such that

(A4.38)  δ(C) ∩ F^{λ}Ĉ^{k+1} ⊂ δ(F^{λ−c}Ĉ^{k}).

(Proposition 4.9.) Since C is finitely generated, we can find \overline{A}_{i} : \overline{C} → \overline{C} such that

(A4.39)  δ = ∑ T^{μ_{i}}A_{i},  0 = μ_{1} < μ_{2} < ⋯ → ∞.

We choose k ∈ ℤ_{+} with

(A4.40)  μ_{k} > c + 1.

and take λ_{0} such that

(A4.41)  0 < λ_{0} < \frac{1}{2} min\{1, inf\{μ_{i+1} − μ_{i}|i = 1, ⋯, k\}\}.

We construct our spectral sequence using this λ_{0} and the filtration

𝔉^{n}C = F^{nλ_{0}}C.

By the choice, μ_{2} > λ_{0} and so Lemma A4.23 holds. We prove

Proposition A4.42. Let Z_{r}^{p,q}(C) and B_{r}^{p,q}(C) be the spaces defined by (A4.18). Then both Z_{r}^{p,q}(C) and B_{r}^{p,q}(C) are pure.

Proof. The proof is by induction on r. The case r = 1 is trivial. Suppose the proposition for 1, ⋯, r − 1. We will now prove the proposition for r.

We consider i, if any, such that

(A4.43)  μ_{i} ∈ [(r − 2)λ_{0}, (r − 1)λ_{0}).

We consider three cases separately.

(A4.44.1) (r − 2)λ_{0} > c + 1. (A4.44.2) (r − 2)λ_{0} ≤ c + 1, and there exists i satisfying (A4.43). (A4.44.3) (r − 2)λ_{0} ≤ c + 1, and there does not exist i satisfying (A4.43).

We first consider the case (A4.44.1). Let 𝔵 ∈ Z_{r−1}^{p,q}(C). Then it follows from the definition of Z_{r−1}^{p,q}(C) that we can write as 𝔵 = 𝔵_{1} + 𝔵_{2} with

δ(𝔵) ∈ F^{(q+r−2)λ_{0}}C, 𝔵_{2} ∈ 𝔉^{q+1}C.

By (A4.38), there exists 𝔶 ∈ F^{(q+r−2)λ_{0}−c}C such that

δ(𝔵_{1}) = δ(𝔶).

Since

(q + r − 2)λ_{0} − c > (q + 1)λ_{0}

it follows that 𝔶 ∈ 𝔉^{q+1}C. Hence 𝔵 ∈ Z_{r}^{p,q}(C), which proves Z_{r}^{p,q}(C) = Z_{r−1}^{p,q}(C). By the induction hypothesis, Z_{r}^{p,q}(C) is pure.

Next we consider the case (A4.44.2).

Lemma A4.45. Let i be as in (A4.43). Then

Z_{r}^{p,q}(C) = (T^{μ_{i}}A_{i})^{−1}B_{r−1}^{p+1,q+r−2} ∩ Z_{r−1}^{p,q}(C).

Proof. Let 𝔳 = T^{λ}v be a generator of Z_{r−1}^{p,q}(C). Here v ∈ \overline{C}. We may assume

λ ∈ [qλ_{0}, (q + 1)λ_{0}].

For j < i, we have

μ_{j} < μ_{i} − 2λ_{0} < (r − 3)λ_{0}.

Therefore

(A4.46)  T^{μ_{j}}A_{j}(𝔳) = T^{λ+μ_{j}} \overline{A}_{j}v,  λ + μ_{j} < (q + r − 2)λ_{0}.

Since

δ𝔳 ∈ 𝔉^{q+r−2}C,

(A4.46) implies

(A4.47)  A_{j}v = 0,  for j < i.

On the other hand, we can easily find

(T^{μ_{i+1}}A_{i+1} + T^{μ_{i+2}}A_{i+2} + ⋯)(𝔳) ∈ 𝔉^{q+r−1}C.

Thus we find, for 𝔳 ∈ Z_{r−1}^{p,q}(C),

δ(𝔳) ∈ 𝔉^{q+r−1}C ⟺ T^{μ_{i}}A_{i}(𝔳) ∈ 𝔉^{q+r−1}C.

The lemma follows. □

Lemma A4.45, our induction hypothesis and Lemma A4.35 imply that Z_{r}^{p,q}(C) is pure for the case (A4.44.2).

Finally for the case (A4.44.3), we can prove that Z_{r}^{p,q}(C) = Z_{r−1}^{p,q}(C) in the same way as above. This finishes our induction and proves that Z_{r}^{p,q}(C) is pure.

We next show that B_{r}^{p,q}(C) is pure. In case (A4.44.1) holds, we can prove

δZ_{r−2}^{p,q}(C) ⊆ δZ_{r−1}^{p,q}(C)

by the same argument as before. Hence B_{r}^{p,q}(C) = B_{r−1}^{p,q}(C) and hence B_{r}^{p,q}(C) is pure by the induction hypothesis.

Next we assume (A4.44.2). Then by the same argument as in the proof of Lemma A4.45, we can show

(A4.48) \begin{aligned}B_{r}^{p,q}(C) & = δ(Z_{r−1}^{p,q−r+2}(C)) + 𝔉^{q+1}C^{p} = T^{μ_{i}}A_{i}(Z_{r−1}^{p,q−r+2}(C)) + 𝔉^{q+1}C^{p} \\ & = (T^{μ_{i}}A_{i})(Z_{r−1}^{p,q−r+2}(C)) ∩ Z_{r}^{p,q}(C), \end{aligned}

where i is as in (A4.43). Lemma A4.35, (A4.48) and the induction hypothesis imply that B_{r}^{p,q}(C) is pure.

In the case when (A4.44.3) holds, we can again prove that B_{r}^{p,q}(C) = B_{r−1}^{p,q}(C). The proof of Proposition A4.42 is now complete. □

Definition A4.49. Let W ⊆ C be pure and 𝔈_{μ}(W) be as in Lemma A4.33. We denote

dim_{μ} W = dim_{ℚ} 𝔈_{μ}(W).

Then we define

dim_{μ} E_{r}^{p,q}(C) = dim_{μ} Z_{r}^{p,q}(C) − dim_{μ} B_{r}^{p,q}(C),
Avdim E_{r}^{p,q}(C) = \frac{1}{λ_{0}} ∫_{qλ_{0}}^{(q+1)λ_{0}} dim_{μ} E_{r}^{p,q}(C)dμ.

and call Avdim the average dimension. It is a nonnegative real number and may not be an integer.

We note that

F^{qλ_{0}}C ⊇ Z_{r}^{p,q}(C) ⊇ B_{r}^{p,q}(C) ⊇ F^{(q+1)λ_{0}}C.

Therefore we have

dim_{μ} Z_{r}^{p,q}(C) = dim_{μ} B_{r}^{p,q}(C) = rank C

for μ ≥ (q + 1)λ_{0} and

dim_{μ} Z_{r}^{p,q}(C) = dim_{μ} B_{r}^{p,q}(C) = 0,

for μ ≤ qλ_{0}.

Now let (C, δ) be the finite d.g.c.f.z. that is weakly equivalent to (C(L; Λ_{0,nov}), 𝔪_{1}^{b}), which is provided by Theorem A4.28.

Lemma A4.50. Let (C, δ) be as above. Then we have

Avdim E_{2}^{0,q}(C) = ∑_{p} rank H^{2p}(L; ℚ),
Avdim E_{2}^{1,q}(C) = ∑_{p} rank H^{2p+1}(L; ℚ).

This lemma is an immediate consequence of Lemma A4.23.

The following lemma also immediately follows from Lemma A4.14 and Theorem A4.26.

Lemma A4.51. Let μ_{i,p} and ℓ_{p} be the constants for C^{p} given as in (A4.15). If p and q satisfy qλ_{0} > μ_{i,p} for each i, then we have

Avdim E_{∞}^{p,q}(C) = ℓ_{p}.

We also recall Definition 8.3 and then we get from (A4.31) that

rank HF ((L, b), (L, b); Λ_{nov}) = ∑_{ℓ=0}^{2n} ℓ_{p}.

Therefore Theorem F will follow from the following

Main Lemma A4.52. Let A and B be the constants defined in Theorem F, i.e.,

A = ∑ rank H^{p}(L; ℚ),
B = ∑ rank Ker (i_{∗} : H_{n−p}(L; ℚ) → H_{n−p}(M; ℚ)) .

Then we have

lim_{q→∞} Avdim E_{r}^{p,q}(C) ≥ A − 2B.

Proof. Let us consider the subgroup Ker (i_{∗} : H_{n−p}(L; ℚ) → H_{n−p}(M; ℚ)) of H_{n−p}(L; ℚ) and its Poincaré dual. Recalling Lemma A4.23, we put

W_{2}^{p,q}(Ĉ) = {𝔵 ∈ Z_{2}^{p,q}(Ĉ)|[𝔵] ∈ P D (Ker (i_{∗} : H_{n−p}(L; ℚ) → H_{n−p}(M; ℚ))) ⊗ Λ(λ_{0})}

where [𝔵] is the class in E_{2}^{p,q}(Ĉ) ≅ H^{p}(\overline{C}, \overline{δ}) ⊗_{ℚ} Λ(λ_{0}). it follows that W_{2}^{p,q}(Ĉ) is pure since Z_{2}^{p,q}(Ĉ) is pure. We define

\begin{aligned}W_{r}^{p,q}(Ĉ) & = W_{2}^{p,q}(Ĉ) ∩ Z_{r}^{p,q}(Ĉ) \\ J_{r}^{p,q}(Ĉ) & = \frac{W_{r}^{p,q}(Ĉ)}{B_{r}^{p,q}(Ĉ) ∩ W_{r}^{p,q}(Ĉ)}. \end{aligned}

W_{r}^{p,q}(Ĉ) is also pure since W_{2}^{p,q}(Ĉ) and Z_{r}^{p,q}(Ĉ) are pure. Similarly J_{r}^{p,q}(Ĉ) is also pure.

Lemma A4.53. We have

δ(𝔉^{q−r+2}Ĉ^{p−1}) ∩ 𝔉^{q}Ĉ^{p} ⊆ W_{r}^{p,q}(Ĉ).

Proof. Let

φ : (C, δ) → (C(L; Λ_{0,nov}), 𝔪_{1}^{b})

be a chain map which induces the isomorphisms H(C, δ) ≅ H(C(L; Λ_{0,nov}), 𝔪_{1}^{b}) and H(\overline{C}, \overline{δ}) ≅ H(L; ℚ). It follows from Proposition 6.31 that there exists a map

b : C(L; Λ_{0,nov}) → C(M; Λ_{0,nov})

satisfying

(A4.54)  δ_{0}b = ±b𝔪_{1}^{b},

where δ_{0} is the (classical) boundary operator δ_{0} : C(M; Λ_{0,nov}) → C(M; Λ_{0,nov}). Moreover we have

(A4.55)  (id − b)(F^{λ}C(L; Λ_{0,nov})) ⊂ F^{λ+λ_{1}}C(M; Λ_{0,nov})

where λ_{1} is the infimum of symplectic area of the (nontrivial) pseudo-holomorphic discs bounding L. We may assume λ_{0} < λ_{1}.

Let δ𝔷 ∈ δ(𝔉^{q−r+2}Ĉ^{p−1}) ∩ 𝔉^{q}Ĉ^{p} i.e., let 𝔷 ∈ 𝔉^{q−r+2}Ĉ^{p−1} and δ(𝔷) ∈ 𝔉^{q}Ĉ^{p}. We have (id − b)φδ(𝔷) ∈ 𝔉^{(q+1)}C^{p} by (A4.55). Therefore

(A4.56)  \begin{aligned}φδ(𝔷) & = (id − b)φδ(𝔷) + bφδ(𝔷) \\ & = (id − b)φδ(𝔷) ± b𝔪_{1}^{b}φ(𝔷) \\ & ≡ ±δ_{0}bφ(𝔷) mod 𝔉^{(q+1)}C^{p}(M; Λ_{0,nov}). \end{aligned}

We also have

(A4.57)  δ_{0}δ(𝔷) ≡ 0 mod 𝔉^{(q+1)}C^{p}

because we have

0 = δδ(𝔷) = (δ_{0} + δ′)δ(𝔷) = δ_{0}δ(𝔷) + δ′δ(𝔷)

where we know that δ′δ(𝔷) is in 𝔉^{(q+1)}C^{p}. Here δ_{0} is the boundary operator induced by \overline{δ} : \overline{C} → \overline{C} and δ′ is the higher order term of δ. Since φ induces an isomorphism H(\overline{C}, δ) ≅ H(L; Λ_{0,nov}), (A4.56) and (A4.57) imply

δ(𝔷) ∈ W_{2}^{p,q}(Ĉ)

as required. More precisely, (A4.57) implies that δ(𝔷) defines a cocycle in C^{n}(L; ℚ)⊗ Λ(λ_{0}) and so defines a class in H^{n}(L; ℚ)⊗Λ(λ_{0}). On the other hand, (A4.56) implies that the image of φ of this class becomes zero. □

We remark that 𝔵 ↦ T^{λ_{0}}𝔵 defines an isomorphism

\begin{aligned}Z_{r}^{p,q}(Ĉ) & → Z_{r}^{p,q+1}(Ĉ)  any p, q, r, \\ B_{r}^{p,q}(Ĉ) & → B_{r}^{p,q+1}(Ĉ)  if q − r + 2 > 0. \end{aligned}

We take r > q − 2. Then

E_{r}^{p,q−r+1}(Ĉ) ⟶^{δ_{r}^{p,q−r+1}} E_{r}^{p,q}(Ĉ) ⟶^{δ_{r}^{p,q}} E_{r}^{p,q+r−1}(Ĉ)

can be identified to

(A4.58)  E_{r}^{p,q}(Ĉ) ⟶^{δ_{r}^{′p,q}} E_{r}^{p,q}(Ĉ) ⟶^{δ_{r}^{′p,q}} E_{r}^{p,q}(Ĉ)

so that

E_{r+1}^{p,q}(Ĉ) ≅ \frac{Ker(δ_{r}^{′p,q})}{Im(δ_{r}^{′p,q})}

Moreover we have

(A4.59.1)  Im(δ_{r}^{′p,q}) ⊆ J_{r}^{p,q}(Ĉ)
(A4.59.2)  J_{r+1}^{p,q}(Ĉ) ≅ \frac{Ker(δ_{r}^{′p,q}) ∩ J_{r}^{p,q}(Ĉ)}{Im(δ_{r}^{′p,q})}

by Lemma A4.53. As in Definition A4.49, we define the average dimension of the various modules appearing in the above formulae.

Definition A4.60. Let W_{1}, W_{2} be pure submodules of C. We assume 𝔉^{q}C ⊇ W_{1} ⊇ W_{2} ⊇ 𝔉^{q+1}C. Then we put

dim_{μ} \frac{W_{1}}{W_{2}} = dim_{μ} W_{1} − dim_{μ} W_{2},
Avdim \frac{W_{1}}{W_{2}} = \frac{1}{λ_{0}} ∫_{qλ_{0}}^{(q+1)λ_{0}} dim_{μ} \frac{W_{1}}{W_{2}} dμ.

Using the average dimension in place of the usual dimension, we will now refine the argument given in §8. For this purpose we need to state another lemma for which we need some more notations.

Let 𝔉^{q}C ⊇ W_{1} ⊇ W_{2} ⊇ 𝔉^{q+1}C and 𝔉^{q}C ⊇ W′_{1} ⊇ W′_{2} ⊇ 𝔉^{q+1}C. Denote by A an n × n matrix with rational entry. (Here n is the rank of C.) Let λ be any constant with 0 ≤ λ < λ_{0}. We put φ = T^{λ}A. We assume

φ(W_{1}) ⊆ W′_{1}, φ(W_{2}) ⊆ W′_{2}

in addition. We denote

W̃ = \frac{W_{1}}{W_{2}} ,  W̃′ = \frac{W′_{1}}{W′_{2}} .

Then φ induces a map φ̃: W̃ → W̃′ for which it is easy to see

Ker φ̃ = \frac{φ^{−1}W′_{2} ∩ W_{1}}{W_{2}} , Im φ̃ = \frac{φW_{1} + W′_{2}}{W′_{2}} .

Here Lemmas A4.35 implies that both φ^{−1}W′_{2}∩W_{1} and φW_{1}+W′_{2} are pure submodules of C. Hence dim_{μ} Ker φ̃, dim_{μ} Im φ, Avdim Ker φ̃ and Avdim Im φ̃ are defined by Definition A4.60.

Lemma A4.61. Let λ be any constant with 0 < λ ≤ λ_{0}. Then we have

(A4.62.1)  dim_{μ} Ker φ̃ + dim_{μ+λ} Im φ̃ = dim_{μ} W̃.

Here we put dim_{μ+λ} Im φ̃ = 0 if μ + λ ≥ (q + 1)λ_{0}. We also have

(A4.62.2)  Avdim Ker φ̃ + Avdim Im φ̃ = Avdim φW.

Proof. Let 𝔈_{μ}(\overline{W}_{i}) be the linear subspace of \overline{C} associated to the pure submodule W_{1} for each i = 1, 2 as in Lemma A4.33. We denote

E_{μ}(W̃) := 𝔈_{μ}(\overline{W}_{1})/𝔈_{μ}(\overline{W}_{2})

for W̃ and similarly for Ker φ̃ and Im(φ̃). Then we have an exact sequence

0 → E_{μ}(Ker φ̃) → E_{μ}(W̃) → E_{μ+λ}(Im φ̃) → 0.

(A4.62.1) follows immediately. (A4.62.2) then follows by integrating (A4.62.1) out. □

Now we apply Lemma A4.61 to δ_{r}^{′p,q}. An argument similar to the proof of Lemma A4.45 implies that δ_{r}^{′p,q} is induced by a pure homomorphism T^{λ_{i}}A_{i} : C → C. Hence we can apply Lemma A4.61 to φ = δ_{r}^{′p,q}. We then obtain

Avdim(Ker(δ_{r}^{′p,q})) + Avdim(Im(δ_{r}^{′p,q})) = Avdim(Z_{r}^{p,q})

Since

Z_{r+1}^{p,q} ≅ \frac{Ker(δ_{r}^{′p,q})}{Im(δ_{r}^{′p,q})},

it easily follows that

Avdim(Ker(δ_{r}^{′p,q})) − Avdim(Im(δ_{r}^{′p,q})) = Avdim(Z_{r+1}^{′p,q}).

Hence

Avdim(Z_{r+1}^{′p,q}) = Avdim(Z_{r}^{′p,q}) − 2 Avdim(Im(δ_{r}^{′p,q})).

On the other hand, we have

Avdim(J_{r+1}^{p,q}) ≤ Avdim(J_{r}^{p,q}) − Avdim(Im(δ_{r}^{′p,q})).

from (A4.59.2). Therefore

Avdim(Z_{r+1}^{p,q}) − 2 Avdim(J_{r+1}^{p,q}) ≥ Avdim(Z_{r}^{p,q}) − 2 Avdim(J_{r}^{p,q}).

Then by induction on r, we have

Avdim(Z_{∞}^{p,q}) − 2 Avdim(J_{∞}^{p,q}) ≥ Avdim(Z_{2}^{p,q}) − 2 Avdim(J_{2}^{p,q}) = A − 2B.

The proof of Main Lemma A4.52 is now complete. □

It now remains to prove Theorem A4.28, which has been postponed before. For this purpose, we use another version of Floer homology HF (L, L) which was proposed by the second named author in [Oh5] based on the idea of the first named author presented in [Fu8] (A similar picture in the case of holomorphic spheres was described by [PSS]). Let f be a Morse function on L. We put

Cr(f) = {x ∈ M | df(x) = 0}.

We consider the chain complex

C^{p}(L, f; ℚ) = ⨁_{x∈Cr(f),η(x)=p} ℚ[x].

Here η is the Morse index. Let grad f be the gradient vector field of f. Using the integral curves of grad f, we define a boundary operator

δ_{0} : C^{p}(L, f; ℚ) → C^{p+1}(L, f; ℚ).

Its (co)homology is known to be isomorphic to the cohomology of L. We are going to deform δ_{0} by adding quantum effects. Let ψ^{t} : L → L be the one parameter subgroup of transformations generated by the vector field grad f.

Let β_{0} = 0, β_{1}, ⋯ as in (4.3). We consider the moduli space M_{k+2}(β_{i}) = M_{k+2}(M, J : β_{i}) together with the evaluation maps

ev = (ev_{0}, ev_{1}, ⋯, ev_{k+1}) : M_{k+2}(β_{i}) → L^{k+2}.

We fix a bounding chain

b = ∑ B(β_{i}) ⊗ [β_{i}].

Here we put [β_{i}] = T^{∫_{β_{i}} ω}e^{μ(β)/2}. We define

\begin{aligned}M′(β) = ∑_{k} \frac{1}{(k − 1)!} ∑_{β_{i_{2}} +···+β_{i_{k+1}} +β_{i_{1}} =β} & M_{k+2}(β_{i_{1}}) \\ & _{(ev_{2},··· ,ev_{k+1})} × (B(β_{i_{2}}) × ⋯ × B(β_{i_{k+1}})). \end{aligned}

Figure A4.1

Together with the evaluation map (ev_{0}, ev_{1}), it realizes a current on L × L. For each t_{1}, ⋯, t_{ℓ} ≥ 0 and β, we define

\begin{aligned}M′(β; t_{1}, ⋯, t_{ℓ}) = ∑_{β_{i_{1}} +···+β_{i_{ℓ+1}=β}} M′(β_{i_{1}})_{ψ^{t_{1}} ∘ev_{1}} & ×_{ev_{0}} M′(β_{i_{2}})_{ψ^{t_{2}} ∘ev_{1}} \\ & ×_{ev_{0}} ⋯_{ψ^{t_{ℓ}} ∘ev_{1}} ×_{ev_{0}} M′(β_{i_{ℓ+1}}). \end{aligned}

The remaining evaluations maps ev_{0} : M′(β_{i_{1}}) → L and ev_{1} : M′(β_{i_{ℓ}}) → L induce the map

ev = (ev_{0}, ev_{1}) : M′(β; t_{1}, ⋯, t_{ℓ}) → L^{2}.

We also put

M′(β : ℓ) = ⋃ ({(t_{1}, ⋯, t_{ℓ})} × M′(β; t_{1}, ⋯, t_{ℓ})) .

For a critical point p of grad f we define its stable and unstable manifolds by

\begin{aligned}𝔖_{p} & = \{x ∈ L| lim_{τ→−∞} ψ^{τ} (x) = p\} \\ 𝔘_{p} & = \{x ∈ L| lim_{τ→+∞} ψ^{τ} (x) = p\} \end{aligned}

they are manifolds of dimension n − η(p) and η(p) respectively. We now have :

Lemma A4.63. M′(β : ℓ) has a Kuranishi structure of dimension is n + μ(β) − 1. Here n is the dimension of L and μ is the Maslov index.

Proof. We first recall that M′(β) has a Kuranishi structure of dimension n+μ(β)−1. Therefore M′(β : 0) has a Kuranishi structure of dimension n + μ(β) − 1. Since M′(β; t_{1}, ⋯, t_{ℓ}) are a fiber product of M′(β_{i} : 0)’s for i = 1, ⋯, ℓ + 1, it has a Kuranishi structure of dimension

∑_{i=1}^{ℓ+1}(n + μ(β_{i}) − 1) − ℓn = n + ∑_{i=1}^{ℓ} μ(β_{i}) − ℓ − 1 = n + μ(β) − ℓ − 1.

Therefore M′(β : ℓ) has a Kuranishi structure of dimension is n + μ(β) − 1. □

Finally we define the analog of the Floer moduli space

Definition A4.64 For each p, q ∈ Cr(f) and β ∈ π_{2}(M, L), we define

M(p, q : β) = ∑_{ℓ} M′(β : ℓ) _{(ev_{0},ev_{1})} × (𝔖_{p} × 𝔘_{q}).

The proof of the following lemma is immediate from Lemma A4.63.

Lemma A4.65. M(p, q : β) has a Kuranishi structure of dimension η(q) − η(p) + μ(β) − 1.

Now we define

Definition A4.66. We define δ_{β} : C^{p}(L, f; ℚ) → C^{p+1−μ(β)}(L, f; ℚ)

δ_{β}([p]) = ∑_{η(q)=η(p)+1−μ(β)} ♯(M(p, q : β))[q],

where ♯ means the order counted with sign. Note if β_{0} ∈ π_{2}(M, L) is zero, then δ_{β_{0}} is the usual boundary operator of the Morse homology. We then put

C(L, f; Λ_{0,nov}) := C^{p}(L, f; ℚ) ⊗ Λ_{0,nov},

and define δ : C^{p}(L, f; Λ_{0,nov}) → C^{p+1}(L, f; Λ_{0,nov}) by

δ([p]) = ∑_{β} δ_{β}([p]) ⊗ [β].

With this definition, we have

Proposition A4.67. δ ∘ δ = 0.

Proof. Let η(q) = η(p)+2−μ(β) for which case M(p, q; β) becomes one dimensional. We calculate the coefficient of [q]⊗[β] in δ∘δ[p]. To prove the proposition, it suffices to show that the order of the boundary of M(p, q : β), counted with sign, is precisely the coefficient of [q] ⊗ [β] in δ ∘ δ[p].

To see this, we study the limit of a sequence

(A4.68)  X_{i}((Σ_{i,1}, φ_{i,1}), ⋯, (Σ_{i,m}, φ_{i,m}); t_{i,1}, ⋯, t_{i,ℓ})

for i = 1, ⋯, in M(p, q : β). Here Σ_{i,j} is a genus zero open Riemann surface together with k_{j} + 2 marked points on the boundary and φ_{i,j} : (Σ_{i,j}, ∂Σ_{i,j}) → (M, L) is a pseudo-holomorphic map. We assume that its value at each marked point lies on some B(β_{∗}). Then (Σ_{i,j}, φ_{i,j}) gives an element of M′(β_{i,j}) for some β_{i,j} = [φ_{i,j}] ∈ π_{2}(M, L). We assume that ((Σ_{i,1}, φ_{i,1}), ⋯, (Σ_{i,m}, φ_{i,m})) gives an element of the fiber product M′(β; t_{i,1}, ⋯, t_{i,ℓ}) and satisfies

(A4.69.1)  ev_{0}((Σ_{i,1}, φ_{i,1}), ⋯, (Σ_{i,m}, φ_{i,m})) ∈ 𝔖_{p},
(A4.69.2)  ev_{1}((Σ_{i,1}, φ_{i,1}), ⋯, (Σ_{i,m}, φ_{i,m})) ∈ 𝔘_{q},

and hence gives an element of M(p, q : β). Suppose that the sequence (A4.68) diverges in M(p, q : β). Then one of the following must occur:

(A4.70.1) lim_{i→∞} t_{i,j} = ∞.

(A4.70.2) lim_{i→∞} t_{i,j} = 0. (A4.70.3) Σ_{i,j} splits into two components as i → ∞. 0th and 1st marked points are on the same components. (A4.70.4) Σ_{i,j} splits into two components as i → ∞. 0th and 1st marked points are on the different components. (A4.70.5) An image of φ_{i,j} at a marked point goes to the boundary of B(β_{∗}). (A4.70.6) A value of (A4.69.1) goes to the boundary of 𝔖_{p}. (A4.70.7) A value of (A4.69.2) goes to the boundary of 𝔘_{q}.

It is easy to see that (A4.70.2) cancels with (A4.70.4), and (A4.70.3) cancels with (A4.70.5). On the other hand, (A4.70.1) gives the coefficient of [q] in the sum

∑_{β_{1}+β_{2}=β; β_{1},β_{2}≠0} δ_{β_{1}} ∘ δ_{β_{2}}([p]).

(4.70.6), (4.70.7) give the coefficients of [q] in

δ_{β} ∘ δ_{0}([p]) and δ_{0} ∘ δ_{β}([p]),

respectively. The proof of Proposition A4.67 is now complete. □

We note that the complex (C(L, f; Λ_{0,nov}) is obviously finite, which is the reason why we consider this particular complex. We will now construct a gapped chain map

ψ : (C(L; Λ_{0,nov}), 𝔪_{1}^{b}) → (C(L, f; Λ_{0,nov}), δ)

which induces isomorphisms HF (C(L; Λ_{0,nov}), 𝔪_{1}^{b}) ≅ H(C(L, f; Λ_{0,nov}), δ) and H(L; ℚ) ≅ H(L, f; ℚ). We define the space

M(·, q : β) = ∑_{ℓ}(M′(β : ℓ)_{ev_{1}} × 𝔘_{q})

and consider the evaluation map ev_{0} : M(·, q : β) → L. We recall that we have fixed countably many chains on L to define the chain complex C(L; Λ_{0,nov}). Therefore by Baire’s category theorem, we may choose a generic f and assume that M(·, q : β) is transverse to any such chain. We then define

(A4.71)  ψ([P, f]) = ∑_{β,q}(M(·, q : β)_{ev_{0}} ×_{f} P) [q] ⊗ [β]

where the sum is taken over the cases where the (virtual) dimension of M(·, q : β)_{ev} ×_{f} P is zero.

Lemma A4.72. ψ is a gapped chain map.

Proof. The proof of the chain property is similar to that of Lemma A4.67 and hence omitted. The gapped property also follows from its definition (A4.71). □

Finally we check the remaining properties of ψ required in Definition A4.25.

Lemma A4.73. The chain map

\overline{ψ} : C(L; ℚ) → C(L, f; ℚ)

induced by ψ induces an isomorphism on homology.

Proof. It is easy to see, considering β = 0, that

\overline{ψ}([P, f]) = ∑_{q} P_{f} × 𝔘_{q}[q].

The lemma then is standard. □

At this stage, if we could apply Proposition A4.27, it would imply that ψ induces an isomorphism on homology, the statement we need to complete the proof of Theorem A4.28. Unfortunately to apply Proposition A4.27, we need the weakly finiteness of C(L; Λ_{0,nov}) which is exactly what we want to prove here. so we will need to prove Proposition A4.27 directly in our case. We start with

Lemma A4.74.

ψ_{∗} : H(C(L; Λ_{0,nov}), 𝔪_{1}^{b}) → H(C(L, f; Λ_{0,nov}), δ)

is injective.

Proof. By Lemma A4.73 which will imply that ψ induces an isomorphism in the E_{2} term, we find that ψ_{∗} induces an isomorphism

E_{∞}^{p,q}(C(L; Λ_{0,nov}), 𝔪_{1}^{b}) ≅ E_{∞}^{p,q}(C(L, f; Λ_{0,nov}), δ).

Then it follows from Theorem A4.26

𝔉^{q}H^{p}(C(L, f; Λ_{0,nov}), δ)/𝔉^{q+1}H^{p}(C(L, f; Λ_{0,nov}), δ) ≅ E_{∞}^{p,q}(C(L, f; Λ_{0,nov}), δ)

because (C(L, f; Λ_{0,nov}), δ) is finite. We deduce from Addendum A4.26 bis that there exists an injective map

𝔉^{q}H^{p}(C(L; Λ_{0,nov}), 𝔪_{1}^{b})/𝔉^{q+1}H^{p}(C(L; Λ_{0,nov}), 𝔪_{1}^{b}) → E_{∞}^{p,q}(C(L; Λ_{0,nov}), 𝔪_{1}^{b}).

These three maps commute and ψ_{∗} provides a commutative diagram (The commutativity is obvious from construction.)

𝔉^{q}H^{p}(C(L; Λ_{0,nov}), 𝔪_{1}^{b})/𝔉^{q+1}H^{p}(C(L; Λ_{0,nov}), 𝔪_{1}^{b}) ⟶ E_{∞}^{p,q}(C(L; Λ_{0,nov}), 𝔪_{1}^{b})
↓  ≅↓
𝔉^{q}H^{p}(C(L, f; Λ_{0,nov}), δ)/𝔉^{q+1}H^{p}(C(L, f; Λ_{0,nov}), δ) ⟶^{≅} E_{∞}^{p,q}(C(L, f; Λ_{0,nov}), δ)

Diagram A4.1

Lemma A4.74 now easily follows from an easy diagram chase. □

Finally we prove the surjectivity of ψ_{∗}. Let 𝔷 = ∑ z_{i,j}[p_{j}]T^{μ_{i}} be an arbitrary element of C(L, f; Λ_{0,nov}) such that δ(𝔷) = 0. Here z_{i,j} ∈ ℚ[e, e^{−1}]. We put

G_{0} = {∫_{β} ω| β ∈ π_{2}(M, L), M(β) ≠ ∅} ∪ {μ_{i}|i = 1, 2, ⋯ }.

Let G be a monoid generated by G_{0}. Note that G is discrete. We consider the subring of Λ_{0,nov}^{(0)} generated by T^{λ}, λ ∈ G, and e, e^{−1}. Let Λ(G) ⊂ Λ_{0,nov}^{(0)} be its closure. We obtain the submodules

C(L, f; Λ(G)) ⊂ C(L, f; Λ_{0,nov}), C(L; Λ(G)) ⊂ C(L; Λ_{0,nov}).

by restricting coefficients to Λ(G) ⊂ Λ_{0,nov}. By definition, they are preserved by the differentials δ and 𝔪_{1}^{b} respectively. Moreover ψ induces a chain map

ψ : C(L; Λ(G)) → C(L, f; Λ(G)).

Note that 𝔷 ∈ C(L, f; Λ(G)). It suffices to show that [𝔷] ∈ H(L, f; Λ(G)) is in the image of

φ_{∗} : H(C(L; Λ(G)), 𝔪_{1}^{b}) → H(C(L, f; Λ(G)), δ).

We put

G = {λ_{1}, λ_{2}, ⋯, }

with λ_{i} < λ_{i+1}. We now prove the following lemma by the induction on k.

Lemma A4.75.k. There exists 𝔵_{k} ∈ C(L; Λ(G)), 𝔶_{k} ∈ C(L, f; Λ(G)) such that

(A4.76.k.1) 𝔪_{1}^{b}(𝔵_{k}) ≡ 0 mod T^{λ_{k+1}}. (A4.76.k.2) ψ(𝔵_{k}) + δ(𝔶_{k}) ≡ 𝔷 mod T^{λ_{k+1}}. (A4.76.k.3) 𝔵_{k−1} ≡ 𝔵_{k} mod T^{λ_{k}}. (A4.76.k.4) 𝔶_{k−1} ≡ 𝔶_{k} mod T^{λ_{k}}.

Proof. We may write 𝔷 = ∑ z_{i,j}[p_{j}]T^{λ_{i}}. We prove the lemma by the induction on k. In case k = 0, δ_{0}(∑ z_{0,j}[p_{j}]) = 0. Therefore it follows from Lemma A4.73 that there exists 𝔵_{0} ∈ C(L, f; Λ(G)) such that ψ(𝔵_{0}) is homologous to 𝔷 modulo T^{λ_{1}}.

We assume A4.75.k-1 and put

ψ(𝔵_{k−1}) + δ(𝔶_{k−1}) − 𝔷 ≡ w_{λ_{k},j}T^{λ_{k}} mod T^{λ_{k+1}}.

By applying δ to this and using the identity ψ𝔪_{1}^{b} = ±δψ, we prove

δ_{0}(w_{λ_{k},j}) = 0.

Hence, by Lemma A4.73, there exists x_{λ_{k},j} ∈ C(L; ℚ[e, e^{−1}]) and y_{λ_{k},j} ∈ C(L, f; ℚ[e, e^{−1}]) such that

\overline{ψ}(x_{λ_{k},j}) + δ_{0}(y_{λ_{k},j}) = w_{λ_{k},j}.

We put

\begin{aligned}𝔵_{k} & = 𝔵_{k−1} ∓ T^{λ_{k}}x_{λ_{k},j} \\ 𝔶_{k} & = 𝔶_{k−1} ∓ T^{λ_{k}}y_{λ_{k},j}. \end{aligned}

The properties (A4.76.k.3) and (A4.76.k.4) are self-evident. Easy calculation then also shows that they satisfy (A4.76.k.1) and (A4.76.k.2). This finishes the proof. □

Now we finishes off the proof of surjectiveness. It follows from (A4.76.k.3) and (A4.76.k.4) that both lim_{k→∞} 𝔵_{k} and lim_{k→∞} 𝔶_{k} converge. Let 𝔵 and 𝔶 be their limits. Then we conclude from (A4.76.k.1) and (A4.76.k.2)

𝔪_{1}^{b}(𝔵) = 0, ψ_{∗}([𝔵]) = [𝔷].

The proof of Theorem A4.28 is now complete. □

§A5. Independence of the choice of countably generated sub-complex.

This section is a continuation of §15 and §19. We complete the proof of Theorem 15.19 and of 19.8. Especially we clarify the dependence of the choice of countably generated sub-complex C(L; ℚ) of \overline{S}(L; ℚ).

We start with the proof of Theorem 19.8. We first define a notion of chain homotopy between two A_{∞} maps. Let (C_{1}, 𝔪), (C_{2}, 𝔪) be A_{∞} algebras (with 𝔪_{0} = 0), and let 𝔣_{k}, 𝔤_{k} : B_{k}(C_{1}[1]) → C_{2}[1] (k = 1, 2, ⋯) define A_{∞} homomorphisms. (We assume 𝔣_{0} = 𝔤_{0} = 0). Let φ̂, ψ̂ : (B(C_{1}[1]), d̂) → (B(C_{2}[1]), d̂) be the chain maps induced by them, respectively. (See (15.8) for the definitions of φ̂ and ψ̂.) In this appendix, we will just write φ = φ̂ and ψ = ψ̂.

We consider a family T = {T_{k}} of operators T_{k} : B_{k}(C_{1}[1]) → C_{2}[1] of degree −1 (after shifted) and extend each T_{k} to T̂_{k} : B(C_{1}[1]) → B(C_{2}[1]) as follows:

Let x ∈ B(C_{1}[1]). We put

ΔΔx = ∑_{a} x′_{a} ⊗ x′′_{a} ⊗ x′′′_{a},

and

(A5.1)  T̂_{k}(x) = ∑_{a}(−1)^{deg′ x{′ \atop a}}φ(x′_{a}) ⊗ T_{k}(x′′_{a}) ⊗ ψ(x′′′_{a}).

Here deg′ x′_{a} is the degree after shifted, and T_{k}(x′′_{a}) is zero unless x′′_{a} ∈ B_{k}(C[1]). We remark that T̂_{k} is a left φ and right ψ bi-coderivation. Namely

Δ ∘ T̂_{k} = (φˆ⊗T̂_{k} + T̂_{k} ˆ⊗ψ) ∘ Δ,

where ˆ⊗ is defined by

(Aˆ⊗B)(x ⊗ y) = (−1)^{deg′ x deg′ B}A(x) ⊗ B(y).

We put

T̂ = ∑ T̂_{k}.

Definition A5.2. A family T = {T_{k}} is called a chain homotopy from 𝔣 = {𝔣_{k}} to 𝔤 = {𝔤_{k}} if

(A5.3)  𝔪 ∘ T̂ + T ∘ d̂= 𝔣_{k} − 𝔤_{k}.

on each B_{k}(C_{1}[1]). (Here 𝔪 and T is defined as follows. Let π_{1} : B(C[1]) → B_{1}(C[1]) = C[1] be the projection. Then 𝔪 = π_{1} ∘ d̂, T = π_{1} ∘ T̂.)

Remark A5.4. (1) (A5.3) is an analog of the definition of a natural transformation between two A_{∞} functors, which is given in [Fu5] Definitions 10.1, 10.3.

(2) In case k = 1, (A5.3) becomes

𝔪_{1} ∘ T_{1} + T_{1} ∘ 𝔪_{1} = 𝔣_{1} − 𝔤_{1}.

This means that T_{1} is a chain homotopy between 𝔣_{1} and 𝔤_{1} in the standard sense.

Lemma A5.5. If T is a chain homotopy from 𝔣 to 𝔤, then T̂ is a chain homotopy from φ to ψ. Namely

d̂∘ T̂ + T̂ ∘ d̂= φ − ψ.

Proof. We put

Δ^{m−1}x = ∑_{a} x_{a}^{(1)} ⊗ ⋯ ⊗ x_{a}^{(m)}.

Then

(d̂∘ T̂)(x) = ∑_{a}(−1)^{deg′ x{(2) \atop a}}φ(x_{a}^{(1)}) ⊗ 𝔪(φ(x_{a}^{(2)}) ⊗ T (x_{a}^{(3)}) ⊗ ψ(x_{a}^{(4)})) ⊗ ψ(x_{a}^{(5)})

+ ∑_{a}(−1)^{deg′ x{(1) \atop a}}(d̂∘ φ)(x_{a}^{(1)}) ⊗ T (x_{a}^{(2)}) ⊗ ψ(x_{a}^{(3)})

+ ∑_{a}(−1)^{deg′ x_{a}^{(2)}+1}φ(x_{a}^{(1)}) ⊗ T (x_{a}^{(2)}) ⊗ (d̂∘ ψ)(x_{a}^{(3)}).

Using the fact that φ, ψ are d̂ chain maps, the second and third terms cancel the corresponding terms in T̂ ∘ d̂ and we have

(d̂∘ T̂ + T̂ ∘ d̂)(x) = ∑_{a}(−1)^{deg′ x{(2) \atop a}}φ(x_{a}^{(1)}) ⊗ 𝔪(φ(x_{a}^{(2)}) ⊗ T (x_{a}^{(3)}) ⊗ ψ(x_{a}^{(4)})) ⊗ ψ(x_{a}^{(5)})

+ ∑_{a} φ(x_{a}^{(1)}) ⊗ (T ∘ d̂)(x_{a}^{(2)}) ⊗ ψ(x_{a}^{(3)})

= ∑_{a} φ(x_{a}^{(1)}) ⊗ (𝔣 − 𝔤)(x_{a}^{(2)}) ⊗ ψ(x_{a}^{(3)}). (A5.6)

We use (A5.3) in the last identity. Most of the terms in (A5.6) cancel one another. What will be left is the following two terms: One is

φ(x_{a}^{(1)}) ⊗ −𝔤(x_{a}^{(2)}) ⊗ ψ(x_{a}^{(3)})

with x_{a}^{(1)} = 1 ∈ B_{0}(C_{2}[1]). This gives −ψ(x). The other is

φ(x_{a}^{(1)}) ⊗ 𝔣(x_{a}^{(2)}) ⊗ ψ(x_{a}^{(3)})

with x_{a}^{(3)} = 1 ∈ B_{0}(C_{2}[1]). This gives φ(x). The proof of Lemma A5.5 is now complete. □

We also need the following Corollary A5.7 and A5.8, which immediately follow from the proof of Lemma A5.5. We put

B_{k,··· ,ℓ}(C[1]) = \frac{⨁_{i=0}^{ℓ} B_{i}(C[1])}{⨁_{i=0}^{k−1} B_{i}(C[1])}.

B_{k,··· ,ℓ}(C[1]) naturally becomes a chain complex by d̂. We remark that φ, ψ and T̂ induce homomorphisms

B_{k,··· ,ℓ}(C_{1}[1]) → B_{k,··· ,ℓ}(C_{2}[1]),

which we write as φ_{k,··· ,ℓ}, ψ_{k,··· ,ℓ} and T̂_{k,··· ,ℓ}. In Addison, φ_{k,··· ,ℓ}, ψ_{k,··· ,ℓ} become chain homomorphisms.

Hereafter we sometimes identify B_{k,··· ,ℓ}(C[1]) with ⨁_{i=k}^{ℓ} B_{i}(C[1]). When we regard φ_{k,··· ,ℓ}, ψ_{k,··· ,ℓ} and etc as homomorphisms on ⨁_{i=k}^{ℓ} B_{i}(C[1]) rather than on B_{k,··· ,ℓ}(C[1]), to avoid confusion, we write them as φ̃_{k,··· ,ℓ}, ψ̃_{k,··· ,ℓ} etc.

The following corollary can be proved in the same way as Lemma A5.5.

Corollary A5.7.

d̂_{k,··· ,ℓ}T̂_{k,··· ,ℓ} + T̂_{k,··· ,ℓ}d̂_{k,··· ,ℓ} = φ_{k,··· ,ℓ} − ψ_{k,··· ,ℓ}.

The second half of the following corollary is also immediate from the proof of Lemma A5.5. (The first half is obvious.)

Corollary A5.8. (A5.8.1) Let m = ℓ − k + 1. Then φ_{k,··· ,ℓ}, ψ_{k,··· ,ℓ}, and T̂_{k,··· ,ℓ} depend only on 𝔣_{1}, ⋯, 𝔣_{m}, 𝔤_{1}, ⋯, 𝔤_{m} and T_{1}, ⋯, T_{m} respectively. (A5.8.2) Suppose only 𝔣_{i}, 𝔤_{i} and T_{i} for i = 1, ⋯ m are given and assume that they satisfy (A5.3) and the condition in Definition 15.9 (i.e., 𝔣 and 𝔤 to be A_{∞} homomorphisms) only on B_{1,··· ,m}(C([1])). Under this weaker assumption, we still have the conclusion of Corollary A5.7.

Now we are ready to give the proof of Theorem 19.8.

Proof of Theorem 19.8. We will construct 𝔣′_{k} inductively over k. At the same time, we will construct the chain homotopy T = {T_{k}} from φ ∘ φ′ to the identity.

Suppose we have constructed 𝔣′_{1}, ⋯, 𝔣′_{k−1} and let

φ′_{i} : B_{1,··· ,k−1}(C_{2}[1]) → B_{1,··· ,k−1}(C_{1}[1])

i = 1, ⋯, k − 1 be the coalgebra homomorphisms extending 𝔣′_{1}, ⋯, 𝔣′_{k−1}. (See (15.8) for their definitions.) We put

φ′_{1,··· ,k−1} = ∑_{i=1}^{k−1} φ′_{i}.

on B_{1,··· ,k−1}(C_{2}[1]) and assume (as an induction hypothesis)

(A5.9.k − 1)  φ′_{1,··· ,k−1} ∘ d̂= d̂∘ φ′_{1,··· ,k−1}, on B_{1,··· ,k−1}(C_{2}[1]),

and that we have constructed T_{1}, ⋯, T_{k−1} such that

(A5.10.k − 1) The condition (A5.3) T being a homotopy from φ ∘ φ′ to the identity, holds on B_{1,··· ,k−1}(C_{2}[1]).

The first step of the induction is easy. Namely we can construct 𝔣′_{1} which is a 𝔪_{1} chain map and induces an inverse to 𝔣_{1} on homology. Then (A5.10.1) means that T_{1} is a chain homotopy from 𝔣_{1} ∘ 𝔣′_{1} to the identity. We can construct such T_{1} in a standard way.

We next assume the induction hypotheses (A5.9.i) and (A5.10.i) for i = 1, ⋯, k− 1. We will construct 𝔣′_{k}, T_{k} that satisfy (A.5.9.k) and (A.5.10.k). It follows from Corollary A5.8 that the maps 𝔣′_{i}, i = 1, ⋯, k − 1 determine a map

φ′_{2,··· ,k} : B_{2,··· ,k}(C_{2}[1]) → B_{2,··· ,k}(C_{1}[1]).

The induced map φ̃′_{2,··· ,k} will become the ⨁_{i=2}^{k} B_{i}(C_{1}[1])-component of φ′_{1,··· ,k} when 𝔣′_{k} and φ′_{1,··· ,k} are defined. (Here we use the convention just before Corollary A5.6 in φ̃′_{2,··· ,k}.) It also follows from (A5.9.k − 1) that φ′_{2,··· ,k} is a chain map. Hence, we have

(φ̃′_{2,··· ,k} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(B_{k}(C_{2}[1])) ⊆ B_{1}(C_{1}[1]) = C_{1}[1].

We put

\begin{aligned}K_{k} & = \{x ∈ B_{k}(C_{2}[1]) |d̂x ∈ B_{1,··· ,k−1}(C_{2}[1])\} = \{x ∈ B_{k}(C_{2}[1]) |d_{1}x = 0\} , \\ B_{k} & = \{d_{1}x| x ∈ B_{k}(C_{2}[1])\} \end{aligned}

Note B_{k} ⊆ K_{k} since d_{1}d_{1} = 0. (We assumed 𝔪_{0} = 0.)

We choose a basis {v_{1}, ⋯, v_{a}} of B_{k} and extend it to a basis {v_{1}, ⋯, v_{a}, v_{a+1}, ⋯, v_{b}} of K_{k}. We put v_{i} = d_{1}w_{i}, i = 1, ⋯, a. Note that {v_{1}, ⋯, v_{b}, w_{1}, ⋯, w_{a}} forms a basis of B_{k}(C_{2}[1]).

We first define 𝔣′_{k} on K_{k}. Since d̂(v_{i}) ∈ B_{1,··· ,k−1}(C_{2}[1]) it follows that

(φ′_{1,··· ,k−1} ∘ d̂− φ̃′_{2,··· ,k} ∘ d̂)(v_{i}) ∈ C_{1}[1].

On the other hand, we have

(φ̃′_{2,··· ,k} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(v_{i}) ∈ C_{1}[1].

Therefore

(A5.11)  (φ′_{1,··· ,k−1} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(v_{i}) ∈ C_{1}[1].

Moreover we can prove

Lemma A5.12. (φ′_{1,··· ,k−1} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(v_{i}) ∈ Im 𝔪_{1}.

Proof. Since φ′_{1,··· ,k−1} is a chain map it follows that :

(A5.13)  d̂∘ (φ′_{1,··· ,k−1} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(v_{i}) = 0.

Since φ_{1} induces an isomorphism on homology, it suffices to show that

φ_{1} ∘ (φ′_{1,··· ,k−1} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(v_{i})

is in the image of 𝔪_{1}. We calculate (using φ = φ_{1} on C_{1})

(A5.14)  \begin{matrix}φ_{1}∘ & (φ′_{1,··· ,k−1} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(v_{i}) \\ = (φ ∘ φ′_{1,··· ,k−1} ∘ d̂− d̂∘ φ ∘ φ̃′_{2,··· ,k})(v_{i}) \\ = ((id + d̂∘ T̂ + T̂ ∘ d̂) ∘ d̂− d̂∘ (id + d̂∘ T̂ + T̂ ∘ d̂+ Ψ))(v_{i}). \end{matrix}

Here we use Corollary A5.8 and induction hypothesis (A5.10.k − 1) to show

φ_{2,··· ,k} ∘ φ′_{2,··· ,k} − id = d̂∘ T̂ + T̂ ∘ d̂∈ Hom(B_{2,··· ,k}(C_{2}[1]), B_{2,··· ,k}(C_{2}[1])).

Therefore

φ ∘ φ̃′_{2,··· ,k} − id = d̂∘ T̂ + T̂ ∘ d̂+ Ψ ∈ Hom(B_{1,··· ,k}(C_{2}[1]), B_{1,··· ,k}(C_{2}[1])).

and

Im(Ψ) ⊆ B_{1}(C_{2}[1]) = C_{2}[1].

Hence we have

(A5.15)  φ_{1} ∘ (φ′_{1,··· ,k−1} ∘ d̂− d̂∘ φ̃′_{2,··· ,k})(v_{i}) = −d̂(Ψ(v_{i})) = −𝔪_{1}(Ψ(v_{i})).

(A5.13) and (A5.15) imply Lemma A5.12. □

Lemma A5.12 implies that there exists y_{i} ∈ C_{1}[1] such that

(A5.16)  (φ′_{1,··· ,k−1} ∘ d̂)(v_{i}) = (d̂∘ φ̃′_{2,··· ,k})(v_{i}) + 𝔪_{1}y_{i}.

We then define 𝔣′_{k} on K_{k} by putting

(A5.17)  𝔣′_{k}(v_{i}) = y_{i}.

Then φ′_{1,··· ,k−1} is extended to a map φ′_{1,··· ,k} on B_{1,··· ,k−1}(C_{2}[1]) ⊕ K_{k}. (A5.16) implies that φ′_{1,··· ,k} satisfies (A5.9.k), on B_{1,··· ,k−1}(C_{2}[1]) ⊕ K_{k}.

Note that we may change the element y_{i} by an arbitrary element of Ker 𝔪_{1} and the condition (A5.17) will still be satisfied. We use this freedom in the next and the last steps.

We are next going to define 𝔣′_{k}(w_{i}). Since

d̂w_{i} − v_{i} ∈ B_{1,··· ,k−1}(C_{2}[1]),

it follows from (A5.17) that

(A5.18)  \begin{aligned}(d̂ & ∘ φ̃′_{2,··· ,k})(w_{i}) − (φ′_{1,··· ,k} ∘ d̂)(w_{i}) \\ & = (d̂∘ φ̃′_{2,··· ,k})(w_{i}) − y_{i} − ∑_{i=1}^{k−1}(𝔣′_{i} ∘ d̂)(w_{i}) − (φ̃′_{2,··· ,k} ∘ d̂)(w_{i}). \end{aligned}

(Note d̂(w_{i}) ∈ B_{1,··· ,k−1}(C_{2}[1]) ⊕ K_{k}. Hence (φ′_{1,··· ,k} ∘ d̂)(w_{i}) is already defined.) On the other hand, since φ′_{2,··· ,k} is a chain map it follows that

(d̂∘ φ̃′_{2,··· ,k})(w_{i}) − (φ̃′_{2,··· ,k} ∘ d̂)(w_{i}) ∈ C_{1}[1].

Thus

(A5.19)  (d̂∘ φ̃′_{2,··· ,k} − φ′_{1,··· ,k} ∘ d̂)(w_{i}) ∈ C_{1}[1].

Since d̂(w_{i}) ∈ B_{1,··· ,k−1}(C_{2}[1]) ⊕ K_{k} and since φ′_{1,··· ,k} is a chain map (where it is defined) it follows that

d̂(d̂∘ φ̃′_{2,··· ,k} − φ′_{1,··· ,k} ∘ d̂)(w_{i})) = 0.

Therefore, by (A5.19), we have

(A5.20)  𝔪_{1}((d̂∘ φ̃′_{2,··· ,k} − φ′_{1,··· ,k} ∘ d̂)(w_{i})) = 0.

Hence, by (A5.18) and (A5.20), we may change y_{i} = 𝔣′_{k}(v_{i}) by an element of Ker 𝔪_{1} and may assume

(A5.21)  (d̂∘ φ̃′_{2,··· ,k} − φ′_{1,··· ,k} ∘ d̂)(w_{i}) = 0.

We now put 𝔣′_{k}(w_{i}) = 0. Then (A5.21) implies (A5.9.k).

We remark that we have fixed the value y_{i} = 𝔣′_{k}(v_{i}) for i = 1, ⋯, a, at this stage. But we still have freedom to change the value y_{i} of 𝔣′_{k}(v_{i}) for i = a + 1, ⋯, b by elements of Ker 𝔪_{1}, which we will exploit in the last step.

Now we are going to construct T_{k} satisfying (A5.10.k). The condition (A5.10.k) can be written as :

(A5.22)  𝔪_{1} ∘ T_{k} + T_{k} ∘ d̂_{1} = π_{1} ∘ φ ∘ φ′ − (𝔪 ∘ ∑_{i=1}^{k−1} T̂_{i} + ∑_{i=1}^{k−1} T_{i} ∘ d̂)

on B_{k}(C_{2}[1]). Here π_{1} is a projection to C_{2}[1]. The left hand side is not defined yet but the right hand side is already defined. Let Ξ be the right hand side.

Lemma A5.23.

(A5.24)  Ξ = φ ∘ φ′ − id − d̂∘ ∑_{i=1}^{k−1} T̂_{i} − ∑_{i=1}^{k−1} T̂_{i} ∘ d̂,

on B_{k}(C_{2}[1]).

Proof. It suffices to show that the image of right hand side of (AS.24) on B_{k}(C_{2}[1]) is contained in B_{1}(C_{2}[1]) = C_{2}[1] . But this is an immediate consequence from the identity

∑_{i=1}^{k−1} T̂_{i} = T̃_{2,··· ,k}

on B_{k}(C_{2}[1]) ⊆ B_{2,··· ,k}(C_{2}[1]), the induction hypothesis and from Corollary A5.8. □

We denote by Ξ′ the right hand side of (A5.24). (It is also defined outside of B_{k}(C_{2}[1]).)

Lemma A5.25.

𝔪_{1} ∘ Ξ − Ξ ∘ d_{1} = 0.

Proof. It is easy to see

(A5.26)  d̂∘ Ξ′ = Ξ′ ∘ d̂.

Since the image of Ξ′ is in C_{2}[1] it follows that

(A5.27)  d̂∘ Ξ′ = 𝔪_{1} ∘ Ξ′.

We remark that Ξ′ vanishes on B_{1,··· ,k−1}(C_{2}[1]) by the induction hypothesis and Corollary A5.8. Therefore

(A5.28)  Ξ′ ∘ d̂= Ξ′ ∘ d_{1}.

Lemma A5.23, (A5.26), (A5.27) and (A5.28) imply Lemma A5.25. □

We have thus proved that Ξ is a d_{1}-cocycle on Hom(B_{k}(C_{2}[1]), C_{2}[1]). Therefore Ξ induces a chain map

(A5.29)  Ξ_{∗} : (B_{k}(C_{1}[1]), d_{1}) → (C_{2}[1], 𝔪_{1}).

To find T_{k} satisfying (A5.22), it suffices to show that Ξ is a coboundary which will follow from the following

Lemma A5.30. We may choose y_{i} = 𝔣′_{k}(v_{i}) for i = a + 1, ⋯, b so that the homomorphism

Ξ_{∗} : H(B_{k}(C_{1}[1]), d̂_{1}) → H(C_{2}[1], 𝔪_{1})

induced by the chain map (A5.29) in fact becomes zero.

Proof. We remark that v_{i}, i = a + 1, ⋯, b induces a basis on H(B_{k}(C_{1}[1]), d̂_{1}). In the definition of Ξ, the term 𝔣′_{k} is involved only in φ_{1} ∘ 𝔣′_{k}. We recall that φ_{1} induces isomorphism on 𝔪_{1}-homology and that we can freely modify y_{i} = 𝔣′_{k}(v_{i}) for i = a + 1, ⋯, b by arbitrary elements of Ker 𝔪_{1} in its definition. Lemma A5.30 follows. □

The proof of Theorem 19.8 is now complete. □

Remark A5.31. In the proof of Theorem 19.8 above we use existence of the basis {v_{i}} and {w_{i}} of modules over the commutative ring over which our A_{∞} algebras are defined. When we work over ℚ or ℤ_{2}, this does not cause any trouble. We however need to work over the universal Novikov ring Λ_{0,nov} sometimes. By reviewing our proof, we see that existence of desired bases {v_{i}}, {w_{i}} can be proved under the following two conditions.

(A5.32.1) C_{1}^{k}, C_{2}^{k} are free or completion of a free module over Λ_{0,nov}. (A5.32.2) The cohomology H(B_{k}(C_{2}[1]), d̂_{1}) is a free module over Λ_{0,nov}.

Note the condition (A5.32.2) is necessary to obtain basis {v_{i}} so that {v_{i}; i = 1, ⋯, a} is a basis of B_{k} and {v_{i}; i = a + 1, ⋯, b} is a basis of the complement of B_{k} in K_{k}.

It is easy to show that (A5.32.1) is satisfied in the cases we considered. By the Künneth formula, we have

(A5.33)  H(B_{k}(C_{2}[1]), d̂_{1}) ≅ H(C_{2}[1], 𝔪_{1})^{⊗k}.

For the case C(L; Λ_{0,nov}) which we are interested in, we have

H(C_{2}[1], 𝔪_{1}) ≅ H(L, ℚ) ⊗_{ℚ} Λ_{0,nov}

and hence (A5.32.2) is satisfied.

Proof of Theorem 19.9. Let X_{0} and X_{1} be two countable subsets of chains, and 𝔪_{k}^{0} and 𝔪_{k}^{1} be the A_{∞} structures on ℤX_{0} and ℤX_{1} respectively constructed by Theorem 19.6. We remark that a diffeomorphism F : L → L induces a map X → F (X) which we write as F_{∗}.

Main Lemma A5.34. There exists a diffeomorphism F : L → L arbitrarily close to the identity, X_{2} containing X_{0} ∪ F (X_{1}), an A_{∞} structure 𝔪_{k}^{2} on ℤX_{2} and, A_{∞} homomorphisms 𝔣_{k}^{02} and 𝔣_{k}^{12} from (ℤX_{0}, 𝔪_{k}^{0}) and (ℤX_{1}, 𝔪_{k}^{1}) to (ℤX_{2}, 𝔪_{k}^{2}) respectively such that

(A5.35.1) 𝔣_{0}^{02} = 𝔣_{0}^{12} = 0. (A5.35.2) 𝔣_{1}^{02} is induced from canonical inclusion X_{0} ⊆ X_{2}. (A5.35.3) 𝔣_{1}^{12} = F_{∗}. (A5.35.4) 𝔣_{1}^{02}, 𝔣_{1}^{12} induce isomorphisms in 𝔪_{1} homology. (A5.35.5) 𝔪_{k}^{2} coincides with F_{∗} ∘ 𝔪_{k}^{1} ∘ (F_{∗}^{−1})^{⊗k} on (ℚ · F (X_{1}))^{⊗k}. (A5.35.6) 𝔣_{k}^{12} = 0 for k ≥ 2.

It is easy to see that Main Lemma A5.34 and Theorem 19.8 imply Proposition 19.9.

Proof of Main Lemma A5.34. Let us take F to be generic. Then we may first assume that X_{0} ∩ F (X_{1}) = ∅. We apply Theorem 19.6 to X_{0} ∪ F (X_{1}). Then we obtain 𝔪_{k}^{2′} on ℤX′_{2}, where X′_{2} ⊃ X_{0} ∪ F (X_{1}). Furthermore using the genericity of F we may assume that X′_{2} satisfies (A5.35.5).

During the proof of Main Lemma A5.34, we are going to replace X′_{2} countably many times to obtain our X_{2}. To save notation we use the same symbol X_{2} for all of them. It suffices to show the statement for X_{0} since the argument for X_{1} is simpler as far as we take generic F and can choose our 𝔪_{k}^{2} satisfying (A5.35.5).

Let 𝔰^{0,k} and 𝔰^{2,k} be the maps (19.2) which are used to define 𝔪_{k}^{0} and 𝔪_{k}^{2} respectively. We consider a map

𝔰^{para,k} : [0, 1] × Δ_{k} × \overline{Gr}_{k+1} → N_{Δ_{k}}L^{k+1},

satisfying the following properties:

(A5.36.1) The restriction of 𝔰^{para,k} to \{j\}×Δ_{k} ×\overline{Gr}_{k+1} (j = 0, 1) coincides with 𝔰^{0,k}, 𝔰^{2,k} respectively.

To state (A5.36.2) we need more notations. Let s ∈ [0, 1]. Then the restriction 𝔰^{(s),k} of 𝔰^{para,k} to \{s\} × Δ_{k} × \overline{Gr}_{k+1} defines the map

𝔪_{k}^{(s)}(P_{1}, ⋯, P_{k}) = (Δ_{k} × \overline{Gr}_{k+1})_{𝔰^{(s),k}} × (P_{1} × ⋯ × P_{k}),

for generic P_{i} and s. (Note 𝔪_{k}^{(0)} = 𝔪_{k}^{1}, 𝔪_{k}^{(1)} = 𝔪_{k}^{3}.) We require :

(A5.36.2) 𝔪_{k}^{(s)} satisfies the A_{∞} relation for generic P_{i} and s.

We can rewrite (A5.36.2) in a similar way to (19.3). But we do not try to do it. We will also require various transversality properties later.

Lemma A5.37. There exists 𝔰^{para,k} satisfying (A5.36).

Proof. The proof is by induction on k. The case k = 1 being trivial. Suppose we have 𝔰^{para,1}, ⋯, 𝔰^{para,k−1}. Then (A5.36) required automatically determine the restriction of 𝔰^{para,k} on ∂([0, 1]×Δ_{k}×\overline{Gr}_{k+1}). The consistency of the maps determined on each component of ∂([0, 1]×Δ_{k} ×\overline{Gr}_{k+1}) follows from induction hypothesis. We remark that we can take 𝔰^{para,i} so that it is almost constant in the [0, 1] × \overline{Gr}_{k+1} direction. Using this fact we can check that the obstruction extending the map on ∂([0, 1]×Δ_{k}×\overline{Gr}_{k+1}) to [0, 1]×Δ_{k}×\overline{Gr}_{k+1} vanishes. Thus the lemma is proved. □

We will construct 𝔣_{k}^{02}, k = 2, 3, ⋯ using 𝔰^{para,k}. The proof here is quite similar to that of Theorem 15.19.

Consider the set of ribbon trees G_{k} introduced in (17.4). Let (T, i, v_{0}) ∈ G_{k+1}. Note v_{0} is the first exterior vertices according to the order we fixed. (We write v_{0} here. In (17.4) we wrote v_{1}.) We denote Γ = (T, i, v_{0}).

Let C_{0}(Γ) be the set of all vertices of Γ. We define a partial order of it as follows. Let v, w ∈ C_{0}(Γ). We say w ≤ v if the unique minimal path in T which joins v to v_{0} contains w. Note that this definition is similar to Definition 15.21. In fact it will coincide with Definition 15.21 through the homeomorphism \overline{Θ} : \overline{Gr}_{k} → M_{k}^{b,main} quoted in §17.

We define the set S(Γ) as follows. An element τ of S(Γ) is a map : C_{0,int}(Γ) ∪ {v_{0}} → [0, 1]; v ↦ τ(v) such that

(A5.38) If w ≤ v then τ(w) ≤ τ(v).

For τ ∈ S(Γ) and P_{i} we define the element

(A5.39)  𝔪_{Γ,τ} (P_{1}, ⋯, P_{k})

inductively on the order of C_{0,int}(Γ). In case Γ has no interior vertex we put

𝔪_{Γ,τ} (P_{1}, ⋯, P_{k}) = 𝔪_{k}^{(τ(v_{0}))}(P_{1}, ⋯, P_{k}).

Let v be an interior vertex of Γ. There exists a unique edge e which contains v and which is contained in the minimal path joining v to v_{0}. We cut Γ into two trees at a point w of e. Let Γ_{1} be the piece containing v_{0} and Γ_{2} be the piece containing v.

Figure A5.1

We let w be the first (0-th) marked point of Γ_{2}. (w becomes a vertex of both Γ_{1} and Γ_{1}.) Let w be the ℓ-th marked point of Γ_{1}. Suppose that Γ_{2} ∈ Gr_{m+1}. τ induces τ_{1} ∈ S(Γ_{1}), τ_{2} ∈ S(Γ_{2}) in an obvious way. We then put

(A5.40)

𝔪_{Γ,τ} (P_{1}, ⋯, P_{k}) =(−1)^{deg P_{1}+···+deg P_{ℓ−1}+ℓ−1}𝔪_{Γ_{1},τ_{1}}(P_{1}, ⋯, P_{ℓ−1},
𝔪_{Γ_{2},τ_{2}}(P_{ℓ}, ⋯, P_{ℓ+m−1}), P_{ℓ+m}, ⋯, P_{k}).

We now put

(A5.41)  𝔪_{Γ}(P_{1}, ⋯, P_{k}; para) = ⋃_{τ∈S(Γ)} 𝔪_{Γ,τ} (P_{1}, ⋯, P_{k}).

Now we put

(A5.42)  𝔣_{k}^{02}(P_{1}, ⋯, P_{k}) = ∑_{Γ∈G_{k+1}} 𝔪_{Γ}(P_{1}, ⋯, P_{k}; para).

We can use Baire’s category theorem to show the following

Lemma A5.43. There exists X_{2}, 𝔰^{2,k}, 𝔰^{para,k} such that the following holds.

(A5.44.1) If P_{i} ∈ X_{0} then the fiber products in (A5.40) and (A5.41) are all transversal. (A5.44.2) If P_{i} ∈ X_{0} then the right hand side of (A5.42) is in X_{2}.

Note we do not assert that fiber product in (A5.41) for each individual τ is transversal. (A5.44.1) means that the union of (A5.41) over all τ is transversal. (In fact requiring (A5.40) to be transversal for all individual τ means we put uncountably many conditions. So it can not be proved by Baire’s category theorem.) The proof of Theorem A5.43 is standard.

Now once Lemma A5.43 is established we can define 𝔣_{k}^{02} by (A5.42). The proof that it is an A_{∞} homomorphism is similar to that of Proposition 15.26. (A5.45.3) is also easy to see. The proof of Main Lemma A5.34 is now complete. □

Now we are in the position to complete the proof of Theorem 15.19. The point postponed in §15.2 is about how to choose countably generated sub-complex of \overline{S}(L). First we consider X_{0} and X_{1} used to define C_{(0)}(L; Λ_{0,nov}, 𝔪(J_{0}, Ξ_{0})), C_{(1)}(L; Λ_{0,nov}, 𝔪(J_{1}, Ξ_{1})). Note the construction of 𝔪 (or the choice of the multisection of the Kuranishi structure of various moduli spaces of pseudo-holomorphic discs) is carried out by the induction on the energy. Then the first step is to construct the A_{∞} structure by using the perturbations 𝔰^{0,k}, 𝔰^{1,k} which we have already discussed this point in Main Lemma A5.34 above. We obtain X_{2} and A_{∞} homomorphism 𝔣_{k}^{02}.

The rest of the proof is given in §15.2 and in Chapter 6. Namely we first take X_{2} as in Main Lemma A5.34 and the map there as the energy zero part of the A_{∞} maps C_{(0)}(L; Λ_{0,nov}, 𝔪(J_{0}, Ξ_{0})) → C_{(2)}(L; Λ_{0,nov}, 𝔪(J_{1}, Ξ_{1})). We remark that the moduli space

⋃_{Γ∈G_{k+1}} 𝔪_{Γ}(P_{1}, ⋯, P_{k}; para).

in (A5.42) can be identified with the perturbed moduli space of pseudo-holomorphic discs with energy zero. (See §19.) Hence we can start the construction of the A_{∞} maps using the energy filtration. Once the point how to choose a countably generated sub-complex is taken care of as above, the rest of the proof is already given in §15.2. □

§A6. Construction of the Kuranishi map.

In this section, we prove Theorem D stated in the introduction. We consider the countably generated sub-complex C(L; ℚ) of \overline{S}(L; ℚ) introduced in Chapter 5. We first need to define a combinatorial analogue of Green’s operator. We fix a subspace H^{k}(L; ℚ) ⊂ Ker ∂_{k} ⊂ C^{k}(L; ℚ), such that the restriction of the projection Ker ∂_{k} → H^{k}(L; ℚ) to H^{k}(L; ℚ) is an isomorphism. (Here ∂ denotes the classical boundary operator.) There exists a projection Ker ∂_{k} → H^{k}(L; ℚ) whose kernel coincides with Im ∂_{k−1}. We extend the projection to a map C^{k}(L; ℚ) → H^{k}(L; ℚ) in an arbitrary way, which we write by Π_{H,k}. We also denote

Π_{H} = ∑_{k} Π_{H,k}.

Lemma A6.1. There exists a sequence of maps 𝔊_{k} : C^{k}(L; ℚ) → C^{k−1}(L; ℚ) such that

1 − Π_{H,k} = (−1)^{n}(∂_{k} ∘ 𝔊_{k} + 𝔊_{k+1} ∘ ∂_{k}).

Proof. We can find a subspace F_{k} ⊂ C^{k}(L; ℚ) such that

F_{k} ⊕ Im δ_{k−1} = Ker Π_{H,k}.

It is easy to see that ∂_{k} induces an isomorphism : F_{k} ≅ Im ∂_{k}. Let 𝔊_{k+1} be the inverse of it on Im ∂_{k} multiplied by (−1)^{n}. We put it zero on F_{k} + H^{k}(L; ℚ). It is easy to check the lemma. (We put the sign (−1)^{n} since the contribution of the classical boundary operator ∂ to 𝔪_{1} is (−1)^{n}∂.) □

We extend 𝔊 and ∂ to C(L; ℚ[e, e^{−1}]) → C(L; ℚ[e, e^{−1}]) and denote them by the same symbols. Now let

𝔞 = ∑ a_{i}T^{λ′_{i}}

where

a_{i} ∈ C^{1}(L; ℚ[e, e^{−1}]); λ′_{i} ≥ 0, λ′_{i} → ∞.

To each such 𝔞, we will associate an element s(𝔞) ∈ H^{2}(L; ℚ[e, e^{−1}]). Let G(𝔞) be the monoid generated by G_{+}(L) and {λ′_{0}, ⋯, λ′_{i}, ⋯ } in ℝ_{≥0}. Noting that G(𝔞) is discrete, we put

G(𝔞) = {λ_{0}, ⋯, λ_{i}, ⋯ }

where λ_{i} < λ_{i+1}, λ_{0} = 0. Using this new sequence, we rewrite 𝔞 as

𝔞 = ∑_{i=1}^{∞} a_{i}T^{λ_{i}}.

We recall our A_{∞} map {𝔪_{k}}_{0≤k<∞} introduced in Chapter 4. We define 𝔪_{k,i} by

𝔪_{k} = ∑_{i=1}^{∞} 𝔪_{k,i}T^{λ_{i}},

where 𝔪_{k,i} does not involve the factor T. Note 𝔪_{0,0}.

We inductively define b_{j} and s̃_{j} over j as follows:

(A6.2.1)  b_{j} = a_{j} − ∑_{k} ∑_{{λ_{j} =λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔊(𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}})),

(A6.2.2)  s̃_{j} = ∑_{k} ∑_{λ_{j}=λ_{i_{0}} +···+λ_{i_{k}}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}).

Lemma A6.3. Let s̃_{j} be defined as above. Then we have

(A6.3.1) s̃_{j} ∈ Im 𝔊 + H^{2}(L; ℚ[e, e^{−1}]). (A6.3.2) If s̃_{0} = ⋯ = s̃_{j−1} = 0, then s̃_{j} ∈ H^{2}(L; ℚ[e, e^{−1}]).

Proof. Using (A6.2.1), 𝔪_{1,0} = (−1)^{n}δ and δa_{j} = 0 we compute

s̃_{j} = 𝔪_{1,0}(b_{j}) + ∑_{k} ∑_{{λ_{j} =λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}})

= 𝔊 ∘ 𝔪_{1,0} (∑_{k} ∑_{{λ_{j} =λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}))

+ Π_{H} (∑_{k} ∑_{{λ_{j} =λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}))

and so (A6.3.1) holds.

If s̃_{0} = ⋯ = s̃_{j−1} = 0, the A_{∞} formula implies

𝔪_{1,0} (∑_{k} ∑_{{λ_{j} =λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}))

= − ∑_{k,ℓ,m} ∑_{{λ_{j} =λ_{i_{−1}} +···+λ_{i_{k}} \atop (k−m+ℓ,i_{0})≠(1,0)}} 𝔪_{k−m+ℓ,i_{0}}(b_{i_{1}}, ⋯, 𝔪_{m−ℓ+1,i_{−1}}(b_{i_{ℓ}}, ⋯, b_{i_{m}}), ⋯, b_{i_{k}})

= − ∑_{k,m} ∑_{{λ_{j} =λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{m−1}}, s̃_{i_{m}}, b_{i_{m+1}}, ⋯, b_{i_{k}}) = 0.

This combined with (A6.3.1) implies (A6.3.2). □

Definition A6.4. s(𝔞) = ∑ Π_{H}(s̃_{j})T^{λ_{j}}.

It is easy to see that we can write s(𝔞) = ∑ s_{j}(𝔞)T^{λ_{j}} as in Theorem D. In fact from the inductive formula (A6.2.1) and (A6.2.2), we observe that ∑ Π_{H}(s̃_{j}) can be written as a sum over appropriate graphs (Feynman diagram) such as drawn in the figure below.

Figure A6.1

Now we put

M̂(L) = {𝔞 = ∑ a_{j}T^{λ_{j}} | s(𝔞) = 0}.

and define the map π : M̂(L) → M(L) by

π(𝔞) = [∑ b_{j}T^{λ_{j}}] ,

where b_{j} is defined as in (A6.2.1), and [ ] denotes the ∼ equivalence class defined in Definition 15.37. Note that Lemma A6.3 implies ∑ b_{j}T^{λ_{j}} ∈ M̃(L) if 𝔞 ∈ M̂(L). We are only to prove the following proposition to complete the proof of Theorem D.

Proposition A6.5. The map π : M̂(L) → M(L) is surjective.

Proof. Let

𝔟′ = ∑ b′_{j}T^{λ_{j}} ∈ M̃(L).

We are going to construct

𝔞 = ∑_{j} a_{j}T^{λ_{j}} ∈ M̂(L), 𝔔 = ∑_{j} Q_{j}T^{λ_{j}},

so that the following conditions (A6.6.j) hold which will finish the proof: We put

𝔞(j) = ∑_{i=0}^{j} a_{i}T^{λ_{i}}, 𝔔(j) = ∑_{i=0}^{j} Q_{i}T^{λ_{i}}

where they satisfy

(A6.6.j.1) Q_{j} is a chain on L × [0, 1]. (A6.6.j.2) d̂e^{𝔔(j)} ≡ 0 mod T^{λ_{j+1}}. (A6.6.j.3) 𝔔(j)(0) ≡ π(𝔞) mod T^{λ_{j+1}}. Here 𝔔(j)(0) is the restriction of 𝔔(j) to L × {0}. (A6.6.j.4) 𝔔(j)(1) ≡ 𝔟′ mod T^{λ_{j+1}}. Here 𝔔(j)(1) is the restriction of 𝔔(j) to L × {1}. (A6.6.j.5) s(𝔞) ≡ 0 mod T^{λ_{j+1}}.

We remark that π(𝔞) mod T^{λ_{j+1}} and s(𝔞) mod T^{λ_{j+1}} depend only on 𝔞(j). Hence (A6.6.j) is indeed a statement on 𝔞(j), 𝔔(j). We will inductively construct a_{j}, Q_{j} over j so that they satisfy (A6.6.j).

We start with the case j = 0. We have 𝔪_{1,0}(b′_{0}) = 0. Hence there exists a_{0} ∈ C^{1}(L; ℚ[e, e^{−1}]) such that [b′_{0}] = [a_{0}] in H^{1}(L; ℚ[e, e^{−1}]). Therefore we can choose Q_{0} so that (A6.6.0) are satisfied.

We now assume (A6.6.0), …, (A6.6.j). By (A6.6.j.2), we may write

(A.6.7)  d̂e^{𝔔(j)} ≡ R_{j+1}T^{λ_{j+1}} mod T^{λ_{j+2}}.

Noting 𝔪_{1,0}(b′_{0}) = 0 we have, by (A6.6.j.3) and (A6.7),

(A6.8)  R_{j+1}(0) = ∑_{k} ∑_{{λ_{j+1}=λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}).

Here

𝔟 = ∑ b_{j}T^{λ_{j}} = π(𝔞).

By (A6.6.j.4) and (A6.7), we have

(A6.9)  R_{j+1}(1) = ∑_{k} ∑_{{λ_{j+1}=λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b′_{i_{1}}, ⋯, b′_{i_{k}}).

Since d̂(e^{𝔟′}) = 0, the A_{∞} formula implies

(A6.10)  𝔪_{1,0}(b′_{j+1}) = − ∑_{k} ∑_{{λ_{j+1}=λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b′_{i_{1}}, ⋯, b′_{i_{k}}).

Therefore we have

(A6.11) ∑_{k} ∑_{{λ_{j+1}=λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}) = 𝔪_{1,0}((−1)^{n+1}R_{j+1} − b′_{j+1}).

Note that (−1)^{n+1}R_{j+1} − b′_{j+1} in the right hand side may not be contained in our countably generated sub-complex, since we do not assume that Q_{j} is contained in it. However the left hand side of (A.6.11) sits in and so a cycle in our countably generated sub-complex C(L; ℚ). We recall that the inclusion C(L; ℚ) → \overline{S}(L; ℚ) induces an isomorphism in homology. It follows that since the left hand side of (A.6.11) is a boundary (with respect to the classical boundary operator) in \overline{S}(L; ℚ), it must also be a boundary of a chain in z_{j+1} ∈ C(L; ℚ). We then have

(A6.12)  ∑_{k} ∑_{{λ_{j+1}=λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}) = 𝔪_{1,0}(z_{j+1}).

Now Proposition A6.1 implies

(A6.13)  𝔪_{1,0}(𝔊(𝔪_{1,0}(z_{j+1}))) = (id − Π_{H,∗})(𝔪_{1,0}(z_{j+1})) = 𝔪_{1,0}(z_{j+1}).

By (A6.2.1), (A6.12), we have

(A6.14)  b_{j+1} = a_{j+1} − 𝔊(𝔪_{1,0}(z_{j+1}))

where a_{j+1} will be determined later. Combining (A6.2.2), (A6.12), (A6.13) and (A6.14), we have

s̃_{j+1} = 𝔪_{1,0}(b_{j+1}) + ∑_{k} ∑_{{λ_{j+1}=λ_{i_{0}} +···+λ_{i_{k}} \atop (k,i_{0})≠(1,0)}} 𝔪_{k,i_{0}}(b_{i_{1}}, ⋯, b_{i_{k}}) = 0,

for any choice of a_{j+1} (such that 𝔪_{1,0}(a_{j+1}) = 0). (A6.6.j + 1.5) then follows. Now we consider

W = −𝔊(𝔪_{1,0}(z_{j+1})) + (−1)^{n+1}R_{j+1} − b′_{j+1}.

We regard W as a chain on L × [0, 1] by putting −𝔊(𝔪_{1,0}(z_{j+1})) on L × {0}, R_{j+1} on L × [0, 1], and b′_{j+1} on L × {1}. By (A6.11),(A6.12) and (A6.13), we have 𝔪_{1,0}(W) = 0. Therefore, we can choose a_{j+1} such that

a_{j+1} − 𝔊(𝔪_{1,0}(z_{j+1})) + (−1)^{n+1}R_{j+1} − b′_{j+1}

becomes a boundary. We choose Q_{j+1} such that

∂(Q_{j+1}) = −(a_{j+1} − 𝔊(𝔪_{1,0}(z_{j+1}))) + (−1)^{n}R_{j+1} + b′_{j+1}.

(Here ∂ = (−1)^{n}𝔪_{1,0} is the classical boundary operator.) It is now easy to check (A6.6.j + 1). The proof of Proposition A6.5 is now complete. □

Remark A6.15. Theorem D is in fact not in a satisfactory form yet in two reasons. First we did not prove that the Kuranishi map s is independent of the various choices involved, in an appropriate sense. Secondly we did not describe how the map π : M̂(L) → M(L) fails to be an isomorphism. For the first point, it appears that we need to find an appropriate notion of the “formal super scheme” and show that M(L) is well defined as such an object. For the second point, we need to define an appropriate notion of action of the “gauge group” on M̂(L) for which M(L) seems to become a “quotient” space by the action. We leave them to work out as a task for the future research. Remark A6.16. In Theorem D, we asserted that s_{i} in (1.6.1) is a formal power series, not a polynomial. More specifically, if we consider b = ∑_{i=1}^{∞} b_{i}T^{λ′_{i}}e^{n_{i}} with λ′_{i} ≥ 0 in place of λ′_{i} > 0, there may be infinitely many terms appearing in s(b) in a fixed energy level. (And hence s(b) may not be well defined.) Reviewing carefully the proof given in this section, we find that this is caused only by the contribution of the classical operators 𝔪_{k,β_{0}}. It might well be the case that contributions from the classical operators indeed vanish and s_{i} in Theorem D can be taken as polynomials. This is because the classical A_{∞} algebra is homotopy equivalent to a graded-commutative differential graded algebra and the Kuranishi map for a graded-commutative differential graded algebra is trivial. (Namely the equation da + a ∧ a = 0 is equivalent to da = 0.) However in §19, we needed to perturb our classical operator and to deform the classical differential graded algebra to a noncommutative A_{∞} algebra. Because of this, the proof given in this section is short of proving that s_{i} is a polynomial.

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Department of Mathematics, Faculty of Sciences, Kyoto University, Kyoto, Japan

Korea Institute for Advanced Study, Seoul, Korea & Department of Mathematics, University of Wisconsin, Madison, WI 53706 USA

Graduate School of Mathematics, Nagoya University, Nagoya, Japan

Department of Mathematics, Hokkaido University, Sapporo, Japan