APPLICATION OF FLOER HOMOLOGY OF LANGRANGIAN SUBMANIFOLDS TO SYMPLECTIC TOPOLOGY

KENJI FUKAYA

Contents

1. Introduction 1 2. Lagrangian submanifold of ℂ^{n} 2 3. Perturbing Cauchy-Riemann equation 4 4. Maslov index of Lagrangian submanifold with vanishing second Betti

number. 8 5. Floer homology and a spectre sequence. 11 6. Homology of loop space and Chas-Sullivan bracket. 14 7. Iterated integral and Gerstenhaber bracket. 17 8. A_{∞} deformation of de Rham complex. 20 9. S^{1} equivariant homology of Loop space and cyclic A_{∞} algebra. 23 10. L_{∞} structure on H(S^{1} × S^{n}; ℚ). 25 11. Lagrangian submanifold of ℂ^{3}. 29 12. Aspherical Lagrangian submanifold. 30 13. Lagrangian submanifold homotopy equivalent to S^{1} × S^{2m}. 34 14. Lagrangian submanifold of ℂP^{n}. 35 References 36

1. Introduction

The purpose of this article is two fold. One is to review theory of Floer homology of Lagrangian submanifolds devel-

opped in [15] jointly with Oh, Ohta, Ono. The construction of present article is a bit different from one in [15]. Namely in this article we use de Rham theory while in [15] we used a version of singular homology. We also emphasise application to symplectic topology. There are other survey articles [12, 13, 29] of [15], where more emphasise is put on its application to mirror symmetry. We also include some results which were not in the 2000 preprint version of [15] but will be included in its revised version.

The other purpose of this article is to describe a technique to use topology of loop space combined with the moduli space of pseudoholomorphic disk, together with some applications to symplectic topology. Especiall we explain a classification of prime 3 manifolds which are embeded to ℂ^{3} as a Lagrangian submanifold.

The author would like to thank the organizers of the school “Morse theoretic methods in non-linear analysis and symplectic topology” to give him an opportunity to write this article. He would also like to thank his collaborators Y-G. Oh, H. Ohta, K. Ono and acknowledge that many of the results discussed in this article are our joint work. The author would also like to thank D.Sullivan the discussion with whom was quite useful for the author to develop the idea how Floer theory and loop space homology are relted. The author also thanks P. Biran who pointed out some errors in the earier version of the manuscript.

2. Lagrangian submanifold of ℂ^{n}

In this section we extract some arguments of the Floer theory and use it to study Lagrangian submanifolds of ℂ^{n}. We do not introduce Floer homology in this section yet. Let x_{i}+\sqrt{−1}y_{i}, i = 1, ⋯, n be the canonical coordinate of ℂ^{n}. We put the standard symplectic structure ω = ∑ dx_{i} ∧ dy_{i} to ℂ^{n}. Let J be the standard complex structure J(∂/∂x_{i}) = ∂/∂y_{i}, J(∂/∂y_{i}) = −∂/∂x_{i}.

Definition 2.1. An n dimensional submanifold L of ℂ^{n} is said to be a Lagrangian submanifold if ω|_{L} = 0.

In this article we always assume that our Lagrangian submanifold is compact. For a Lagrangian submanifold L ⊂ ℂ^{n} we define an energy function E : π_{1}(L) →

ℝ by

(2.1)  E_{L}(γ) = ∑_{i} ∫_{S^{1}} γ^{∗}x_{i}dy_{i} = ∫_{D^{2}} u^{∗}ω.

Here u : D^{2} → ℂ^{n} is a map with u(∂D^{2}) ⊆ L and [u(∂D^{2})] = γ.

Remark 2.1. For a general Lagrangian submanifold L ⊂ M, we can define E_{L} : H_{2}(M, L) → ℝ in the same way.

The following result due to Gromov is one of the most basic results of symplectic topology.

Theorem 2.1 (Gromov [20]). E_{L} ≠ 0 for any embedded Lagrangian submanifold L ⊂ ℂ^{n}.

Theorem 2.1 in particular implies H_{1}(L; ℚ) ≠ 0 for any Lagrangian submanifold L of ℂ^{n}. To state the next result we need to define Maslov index of Lagrangian submanifold.

Definition 2.2. The set of all real n dimensional ℝ linear subspaces V of ℂ^{n} with ω|_{V} = 0 is called Lagrangian Grassmannian. We write it as GLag_{n}.

Let Gr_{n} be the set of all real n dimensional ℝ linear subspaces V of ℂ^{n}, that is the Grassmanian manifold.

Lemma 2.2. Let n ≥ 2. π_{1}(GLag_{n}) = ℤ. The map GLag_{n} → Gr_{n} induces a surjective homomorphism ℤ = π_{1}(GLag_{n}) → π_{1}(Gr_{n}) = ℤ_{2}.

See for example [1] for its proof.

In case L ⊂ ℂ^{n} is a Lagrangian submanifold, we define a map T i_{L} : L → GLag_{n} by

(2.2)  T i_{L}(p) = T_{p}L ∈ GLag_{n}.

Definition 2.3. The Malsov index η_{L} : π_{1}(L) → ℤ = π_{1}(GLag_{n}) is a homomorphism induced by T i_{L}.

Remark 2.2. In case of general Lagrangian submanifold L ⊂ M, we can define η_{L} : π_{2}(M, L) → ℤ in a similar way. The composition of π_{2}(M) → π_{2}(M, L) and η_{L} coincides with β ↦ 2β ∩ c^{1}(M), where c^{1}(M) is the first chern class.

Lemm 2.2 imnplies that, if L is oriented, then image of η_{L} is in 2ℤ. We recall that a spin manifold is an oriented manifold whose second Stiefel-Whiteney class vanishes.

Theorem 2.3 ([15]). Let L ⊆ ℂ^{n} be a Lagrangian submanifold. If H^{2}(L; ℚ) = 0 and if L is spin, then η_{L} ≠ 0.

Remark 2.3. In case H^{2}(L; ℤ_{2}) = 0, we have η_{L} ≠ 0 without assuming L is neither spin or nor oriented. See [15].

Let us consider the case L = S^{1} × S^{n} for example. The assumption of Theorem 2.3 is satisfied if n > 2.

For n = 1, L = S^{1} × S^{1}, it was proved by Viterbo [38] and Polterovich [31], that there exists γ ∈ π_{1}(L) such that E(γ) > 0 and η_{L}(γ) = 2. For general n we can prove the following.

Proposition 2.4 ([28, 15]). Let L = S^{1} × S^{n} ⊂ ℂ^{n+1} be a Lagrangian submanifold with n ≥ 2. We choose the generator γ ∈ π_{1}(L) = ℤ, such that E(γ) > 0.

If n is odd then there exists a positive integer ℓ such that η_{L}(ℓγ) = n + 1.

If n is even then either η_{L}(γ) = 2 or exists a positive integer ℓ such that η_{L}(ℓγ) = 2 − n.

Remark 2.4. In case η_{L}(γ) > 0, Proposition 2.4 follows from [28]. Proposition 2.4 follows from the results of [15] directly as we will see in §5, though it was not written in its 2000 version explicitely.

Actually, in the case n is even, we can show η_{L}(γ) = 2. Namely we can exclude the second case. See §13.

Before proving the results stated above we mention some of the constructions of Lagrangian submanifolds. See [2] for detail.

We say i : L → ℂ^{n} is a Lagrangian immersion if i is an immersion, dim L = n and if i^{∗}ω = 0. The map T i_{L} : L → ℂ^{n} is defined in the same way.

Two Lagrangian immersions i_{0}, i_{1} are said to be regular homotopic to each other if there exists a faimily of Lagrangian immersions i_{s} s ∈ [0, 1] joining them.

Theorem 2.5 (Gromov[21]-Lees[25]). (1) There exists a Lagrangian immersion i : L → ℂ^{n} if and only if T L ⊗ ℂ is a trivial bundle.

(2) We assume that T L ⊗ ℂ is trivial. Then i ↦ [i_{L}] induces a bijection from the set of regular homotopy classes of Lagrangian immersions i : L → ℂ^{n} to the set of the homotopy classes of maps L → GLag_{n}.

The proof is based on Gromov’s h-principle. See [10].

Theorem 2.5 implies that every element of π_{n}(GLag_{n}) is realized by a Lagrangian immersion i : S^{n} → ℂ^{n}. (We remark that Theorem 2.1 implies that none of them is realized by a Lagrangian embedding.)

Theorem 2.6 (Audin-Lalonde-Polterovich[2]). If i_{L} : L → ℂ^{n} is a Lagrangian immersion and i_{L′} : L′ → ℂ^{m} is a Lagrangian embedding then there exists a Lagrangian embedding i : L × L′ → ℂ^{n+m} such that T i : L × L′ → GLag_{n+m} is homoltopic to sum ∘ (T i_{L} ⊗ T i_{L′}). Where sum : GLag_{n} × GLag_{m} → GLag_{n+m} is given by (V, W) ↦ V ⊕ W.

Proof: Let us take f : L → ℝ^{m} ⊂ ℂ^{m} such that I = (i_{L}, f) : L → ℂ^{n+m} is an embedding. Clearly I^{∗}ω = 0. Then Weinstein’s theorem (see [1, 10]) implies that there exists a neighborhood U of I(L) in ℂ^{n+m} and a symplectic embedding I′ : U ⊂ T^{∗}L × ℂ^{m} such that I′ ∘ I is an embedding q ↦ ((q, 0), 0). Then, for small ϵ, the map I_{+} : L × L′ → T^{∗}L × ℂ^{m}, I_{+}(q, Q) = ((q, 0), ϵi_{L′}(Q)) sends L × L′ to I′(U). We put i_{L,L′} = I′ ∘ I_{+} : L × L′ → ℂ^{n+m}. □

We take L = S^{n} and L′ = S^{1}. We remark that there is an obvious Lagrangian embedding S^{1} → ℂ. Maslov index of it is given by η_{S^{1}}[S^{1}] = 2. Hence we have an embedding S^{1} × S^{n} → ℂ^{n+1} such that η_{S^{1}×S^{n}}([S^{1}]) = 2. Note regular homotopy class of Lagrangian immersion of S^{1} × S^{n} is identifided with π_{1}(GLag_{n}) × π_{n}(GLag_{n}). The construction above realize any elements of the form (2, α) ∈ π_{1}(GLag_{n}) × π_{n}(GLag_{n}) as a Lagrangian embedding.

In [31] Polterovich constructed a Lagrangian embedding S^{1} × S^{2n−1} ⊂ ℂ^{2n} such that η_{L}(γ) = 2n, as follows^{1}. (γ ∈ π_{1}(L) is a generator with E(γ) > 0.) Let S^{2n−1} = {x⃗ ∈ ℝ^{2n}||x⃗| = 1}. We then put L = {θx⃗ ∈ ℂ^{2n}|x⃗ ∈ S^{2n−1}, θ ∈ ℂ, |θ| = 1}. We can show that L is diffeomorphic to S^{1} × S^{2n−1} and is a Lagrangian submanifold. We can also prove η_{L}(γ) = 2n.

In case k = 2, L = S^{1} × S^{3}, then η_{L}(γ) is 2 or 4 according to Proposition 2.4. They both actually occur. In case k = 3, L = S^{1} × S^{5}, Proposition 2.4 implies η_{L}(γ) is 2 or 6. They again both occur. In case k = 4, L = S^{1} ×S^{7}, Proposition 2.4 implies η_{L}(γ) is either 2, 4 or 8. 2 and 8 occur. However example with η_{L}(γ) = 4 does not seem to be known.

Problem 2.1. Are there any Lagrangian submanifold L ⊂ ℂ^{2n} diffeomorphic to S^{1} × S^{2n−1} such that η_{L}(γ) ≠ 2, 2n.

As for the the other factor π_{2n−1}(GLag_{2n}), there is no restriction in the case η_{S^{1}×S^{2n−1}}(γ) = 2. However, in other case, especially in the case η_{S^{1}×S^{2n−1}}(γ) = 2n the following seems to be open.

Problem 2.2. For which homotopy class α ∈ π_{2n−1}(GLag_{2n}), the class (2n, α) ∈ π_{1}(GLag_{2n}) × π_{2n−1}(GLag_{2n}) is realized by a Lagrangian embedding ?

The first case to study seems to be the case of S^{1} ×S^{3}. We remark Theorem 13.1 determines when the regular homotopy class of Lagrangian immersion S^{1} × S^{2m} → ℂ^{2m+1} is realized by a Lagrangian embedding. We will discuss more results on Lagrangian submanifolds in ℂ^{n} in §§11,12,13.

3. Perturbing Cauchy-Riemann equation

Let us start the proof of Theorems 2.1,2.3. We use the moduli space of holomorphic disks which bound L. We define it below. Let β ∈ π_{2}(ℂ^{n}, L) ≅ π_{1}(L).

Definition 3.1. We consider a map φ : D^{2} → ℂ^{n} with the following properties.

(1) φ is holomorphic.

(2) φ(∂D^{2}) ⊂ L.

(3) The homotopy type of φ is β.

We denote by Int M̃(L; β) the set of all such maps. We can define a topology on it (see [15]).

1The author thanks Polterovich who pointed out his example and corrected an error author made during author’s lecture at Montreal.

We consider the group P SL(2; ℝ). It can be identified with the group of biholomorphic maps D^{2} → D^{2}. We denote {g ∈ P SL(2; ℝ)|g(1) = 1} by Aut(D^{2}, 1). (Here we identify D^{2} = {z ∈ ℂ||z| < 1}, P SL(2; ℝ) = Aut(D^{2}). Aut denotes the group of biholomorphic automorphisms.) Then we put

Int M̂(L; β) = \frac{Int M̃(L; β)}{Aut(D^{2}, 1)}, Int M(L; β) = \frac{Int M̃(L; β)}{P SL(2; ℝ)}.

The moduli spaces Int M̂(L; β), Int M(L; β) can be compactified by including stable maps (from open Riemann surface of genus 0). (See [15] for its precise definition.) We denote them by M̂(L; β), M(L; β) respectively. These moduli spaces are used to define and study Floer homology of Lagrangian submanifold. We will discuss it in §5. In this section we use it more directly to prove Theorems 2.1,2.3.

For this purpose, we use a similar but a different moduli space using perturbation of the Cauchy-Riemann equation (which is the equation that φ is holomorphic) by Hamiltonian vector field. We need some notations to describe it.

Definition 3.2. For each positive R we consider a smooth function χ_{R} : ℝ → [0, 1] such that

(1) χ_{R}(t) = 1, if |t| > R,

(2) χ_{R}(t) = 0, if |t| < R − 1,

(3) The C^{k} norm of χ_{R} is bounded uniformly on R.

We next take a smooth function H : ℂ^{n} → ℝ of compact support and let

(3.1)  X_{H} = J grad H

be the Hamiltonian vector field generated by it.

Definition 3.3. We consider a map φ = φ(τ, t) : ℝ × [0, 1] → ℂ with the following properties.

\frac{∂φ}{∂τ}(τ, t) = J (\frac{∂φ}{∂t}(τ, t) − χ_{R}(τ)X_{H}(φ(τ, t))) , (3.2a)

φ(τ, 0), φ(τ, 1) ∈ L, (3.2b)

∫_{ℝ×[0,1]} φ^{∗}ω < ∞. (3.2c)

We denote by N (R) the set of all such φ.

Note (3.2a) implies that φ is holomorphic outside [−R, R] × [0, 1]. Therefore (3.2c) and removable singularity theorem of Gromov [20] and Oh [26] imply the following.

Lemma 3.1. There exists p_{+∞}, p_{−∞} ∈ L such that lim_{τ→±∞} φ(τ, t) = p_{±∞}.

We remark that D^{2} \ {1, −1} is biholomorphic to ℝ × [0, 1]. Hence element φ of N (R) may be regarded as a map \overline{φ} : (D^{2}, ∂D^{2}) → (ℂ^{n}, L).

Definition 3.4. Let N (R; β) be the set of all φ ∈ N (R) such that the homotopy class of \overline{φ} is β.

For a fixed p_{0} ∈ L, let N (R; β; p_{0}) be the set of all element of φ ∈ N (R) such that lim_{τ→+∞} φ(τ, t) = p_{0}. We define N (R; p_{0}) in the same way.

Let exp_{t}^{X_{H}} : ℂ^{n} → ℂ^{n} be a one parameter group of transformations associated to X_{H}.

Assumption 3.1. exp_{1}^{X_{H}} (L) ∩ L = ∅.

Since L is compact we can always find H satisfying Assumption 3.1. Now the following three propositions are basic points of the proof of Theorems 2.1,2.3.

Proposition 3.2. There exists C depending only on H and L (and independent of R) such that if N (R; β) ≠ ∅ then

(3.3)  β ∩ ω > −C.

The proof is based on well established argument and is omitted. We remark that Definition 3.2 (3) is essential for the proof of Proposition 3.2.

Proposition 3.3. If Assumption 3.1 is satisfied then, there exists R_{0} depending only on H and L such that N (R) = ∅ for R > R_{0}.

Proof. (Sketch) If not there exists φ_{i} ∈ N (R_{i}) with R_{i} → ∞. We can use Proposition 3.2 and elliptic regurality to show the existence of τ_{i} ∈ [−R_{i} + 2, R_{i} − 2], δ > 0 such that the C^{∞} norm of φ_{i} on [τ_{i} − δ, τ_{i} + δ] × [0, 1] is bounded and that

(3.4)  lim_{i→∞} ∫_{[τ_{i}−δ,τ_{i}+δ]×[0,1]} φ_{i}^{∗}ω = 0.

(3.4), (3.2a) and elliptic regurality implies that

lim_{i→∞} sup_{t∈[0,1]} ∥\frac{∂φ_{i}}{∂t} (τ_{i}, t) − X_{H} ∥ = 0.

This implies

lim_{i→∞} dist (exp_{1}^{X_{H}} (φ_{i}(τ_{i}, 0)), φ_{i}(τ_{i}, 1)) = 0.

In view of (3.2b) this contradicts to Assumption 3.1. □

For the next proposition we need an assumption.

Assumption 3.2. If E(β) > 0 then η_{L}(β) ≤ 0.

Let β_{0} = 0 ∈ π_{2}(ℂ^{n}, L).

Proposition 3.4. Under Assumption 3.2, we may choose p_{0} such that N (0; β; p_{0}) = ∅, for any β ≠ β_{0}.

Proof. (Sketch) We have

(3.5)  dim M(L; β) = n + η_{L}(β) − 3,

(see [15]), for β ≠ β_{0}. Hence, by Assumption 3.2 we have

dim ⋃_{φ∈M(L;β)} φ(S^{1}) ≤ n + η_{L}(β) − 3 + 1 ≤ n − 2.

Therefore we may take p_{0} which is not contained in the union of ⋃_{φ∈M(L;β)} φ(S^{1}) for various β. We remark that the equation (3.2a) is the Cauchy-Riemann equation when R is zero. Proposition 3.4 holds. □

Lemma 3.5. N (0; β_{0}; p_{0}) is one point.

The lemma follows immedately from the fact that each element of N (0; β_{0}) is a constant map. We put

(3.6)  N (β; p_{0}) = ⋃_{R≥0} N (R; β; p_{0}) × {R}.

We can perturb our moduli spaces so that N (β; p_{0}) is a manifold with boundary. Now we are in the position to prove Theorem 2.1. The main point is as follows.

Lemma 3.6. If E : π_{2}(ℂ^{n}, L) → ℝ is zero then N (β; p_{0}) is compact.

Proof. (Sketch) We prove by contradiction. Let φ_{i} ∈ N (R_{i}; β_{0}; p_{0}) be a divergent sequence. We may assume by Proposition 3.3 that R_{i} converges to R. Then there exists p_{i} = (τ_{i}, t_{i}) ∈ ℝ × [0, 1] such that

(3.7)  |d_{p_{i}}φ_{i}| = sup{|d_{x}φ_{i}||x ∈ ℝ × [0, 1]}.

We put ϵ_{i} = 1/D_{i}. We consider the following three cases separately. Case 1 : |d_{p_{i}}φ_{i}| = D_{i} diverges. D_{i}dist(p_{i}, ∂(ℝ × [0, 1])) = C_{i} → ∞. Case 2 : |d_{p_{i}}φ_{i}| = D_{i} diverges. D_{i}dist(p_{i}, ∂(ℝ × [0, 1])) is bounded. Case 3 : |d_{p_{i}}φ_{i}| = D_{i} is bounded. |τ_{i}| diverges.

In Case 1, we define h_{i} : D^{2}(C_{i}) → ℂ^{n} by h_{i}(z) = φ_{i}(ϵ_{i}z + p_{i}) − φ_{i}(p_{i}). (3.7) implies that dh_{i} is uniformly bounded. Moreover h_{i}(0) = 0 and |d_{0}h_{i}| = 1. Hence, by elliptic regularity, h_{i} converges to h : ℂ → ℂ^{n}. Since |d_{0}h| = 1, h is not a constant. On the other hand, by (3.2a),

sup \{|\frac{∂h_{i}}{∂τ} − J \frac{∂h_{i}}{∂t} (z)| | z ∈ D^{2}(C_{i})\} < ϵ_{i}C

where C is a constant independent of i. Therefore h is holomorphic. Moreover by (3.2c) the integral ∫_{D^{2}(R_{i})} h_{i}^{∗}ω is uniformly bounded. If follows that ∫_{D^{2}(R_{i})} h^{∗}ω is finite. Hence the holomorphicity of h implies that h is a constant map. This is a contradiction.

In Case 2, we consider a similar limit and obtain a map h : 𝔥 = {z ∈ ℂ|Imz ≥ 0} → ℂ^{n} which is nonconstant and holomorphic. Moreover h(ℝ) ⊂ L and ∫_{𝔥} h^{∗}ω is finite. Therefore by removable singularity theorem of Gromov [20] and Oh [26], h can be extended to h^{+} : D^{2} → ℂ^{n} with h^{+}(∂D^{2}) ⊂ L. (Namely when we identify D^{2} \ {1} = 𝔥, h^{+} = h on 𝔥.) Since E = 0 by assumption, it follows that

∫_{𝔥} h^{∗}ω = ∫_{D^{2}} h^{+∗}ω = 0.

Therefore, since h is holomorphic, h is a constant map. This is a contradiction.

We next consider Case 3. We can show that D_{i} is bounded away from zero by using the fact φ_{i} is a divergent sequence. We define h_{i} : ℝ × [0, 1] → ℂ^{n} by h_{i}(z) = φ_{i}(z − (τ_{i}, 0)). Since |τ_{i}| diverges it follows that h_{i}, after taking a subsequence if necessary, converges to a holomomorphic map h : ℝ × [0, 1] → ℂ^{n} which is nonconstant. Moreover h(ℝ × {0, 1}) ⊂ L. Furthermore :

∫_{ℝ×[0,1]} h^{∗}ω ≤ lim sup ∫_{ℝ×[0,1]} φ_{i}^{∗}ω < ∞.

By Lemma 3.1 and E = 0, we find that ∫_{ℝ×[0,1]} h^{∗}ω = 0. Since h is holomorphic, this implies that h is a constant map. This is a contradiction.

The proof of Lemma 3.6 is now complete. □

Now the rest of the proof of Theorem 2.1 is as follows. We remark that

(3.8)  dim N (R; β) = n + η_{L}(β)

can be proved in the same way as (3.5). Then

(3.9)  dim N (R; β; p_{0}) = η_{L}(β)

and hence

(3.10)  dim N (β; p_{0}) = η_{L}(β) + 1.

In particular ∂CN (β_{0}; p_{0}) is a one dimensional manifold. It is compact by Lemma 3.6. Its boundary is one point by Proposition 3.4, Lemma 3.5. This is a contradiction.

Thus we proved Theorem 2.1. □

Remark 3.1. To make the above argument rigorous we need to find a perturbation by which various moduli spaces have fundamental chain. We omit the arguments on transversality in various places of this article. We explain one novel argument on transversality in §6, however. We do not discuss sign (orientation) either. Those points are discussed in detail in [15]. (The detail of the new argument on transversality in §6 will be in [14].)

4. Maslov index of Lagrangian submanifold with vanishing second Betti number.

In this section, we continue the argument of the previous section and sketch the proof of Theorem 2.3. We here assume η_{L} : π_{2}(ℂ^{n}, L) → ℤ is zero. It follows from (3.5), (3.9) and (3.10) that dim N (β; p_{0}) = 1, dim N (R; β; p_{0}) = 0, dim M(L; β) = n − 3.

We next need the following result :

Theorem 4.1 ([15, 35]). A spin structure of L determines an orientation of M(L; β), N (β; p_{0}) etc.

Remark 4.1. In case of (oriented) Lagrangian submanifold L in a symplectic manifold M, the orientationtability of the moduli space of pseudoholomorphic disks is a consequence of relative spin structure. Here L ⊂ M is said to be relatively spin if the second Stiefel-Whitney class w^{2}(L) is in the image of H^{2}(M; ℤ_{2}). We remark that spin manifold is always relatively spin.

Hereafter we assume that L is spin and use the orientation obtained by Theorem 4.1. Since we do not assume E = 0, Lemma 3.6 does not imply compactness of N (β; p_{0}). We are going to study its compactification. We use our assumption η_{L} = 0 in the next lemma.

Lemma 4.2. If η_{L} = 0, we may choose p_{0} such that, for any φ ∈ M(L; β), we have p_{0} ∉ φ(S^{1}).

Proof. Since dim M(L; β) = n − 3, it follows that the union of all φ(S^{1}) for φ ∈ M(L; β) is n − 2 dimensional. The lemma follows. □

Using Lemma 4.2, we can describe compactification of N (β; p_{0}) and of N (R; β; p_{0}). We put A = (ℝ × {0, 1}) ∪ {−∞} in this section. We define

ev : N (R; β; p_{0}) × A → L

by

(4.1)  ev(φ, (τ, ±1)) = φ(τ, ±1),  ev(φ, −∞) = lim_{τ→−∞} φ(τ, t).

We also define M̂(L; β) → L by

(4.2)  ev([φ]) = lim_{τ→+∞} φ(τ, t).

Now we have

Proposition 4.3. We assume η_{L} = 0. Then N (R; β; p_{0}), N (R; β; p_{0}) has compactifications CN (R; β; p_{0}), CN (R; β; p_{0}) which has Kuranishi structure with boundary. The boundary is identified as

\begin{aligned}∂CN (R; β; p_{0}) & = ⋃_{β_{1}+β_{2}=β} M̂(L; β_{1}) ×_{L} (N (R; β_{2}; p_{0}) × A), β ≠ β_{0} \\ ∂CN (β_{0}; p_{0}) \ N (0; β_{0}; p_{0}) & = ⋃_{β_{1}+β_{2}=β_{0}} M̂(L; β_{1}) ×_{L} (N (β_{2}; p_{0}) × A), β_{0} = 0. \end{aligned}

Remark 4.2. We do not explain the definition of “Kuranishi structure with boundary” here. See [16]. If the reader is not familier with it, he may read the statement as “there is a perturbation so that the virtual fundamental chain satisfies the equality … ”.

Proof. (Sketch) The proof of Proposition 4.3 goes in a similar way as the proof of Lemma 3.6. We consider a divergent sequence φ_{i} ∈ N (R_{i}; β_{0}; p_{0}). R_{i} is bounded by Proposition 3.3 and hence we may assume R_{i} → R. We choose p_{i} = (τ_{i}, t_{i}) ∈ ℝ × [0, 1] satisfying (3.7). Then we consider Cases 1,2 and 3 in the proof of Lemma 3.6. Case 1 does not occur by the same reason as the proof of Lemma 3.6. So we need to consider Cases 2 and 3 only.

If lim τ_{i} = τ ∈ ℝ and lim t_{i} = t_{∞} ∈ {0, 1}, the limit correspond to an element

([h^{+}], (φ_{∞}, (τ, t_{∞}))) ∈ M̂(L; β_{1}) ×_{L} (N (R; β_{2}; p_{0}) × A).

Here h^{+} : D^{2} → ℂ^{n} is a bubble at (τ, t_{∞}).

Figure 1

If τ_{i} → −∞ we have an element

([h^{+}], (φ_{∞}, (−∞))) ∈ M̂(L; β_{1}) ×_{L} (N (R; β_{2}; p_{0}) × A).

Using Lemma 4.2 we may assume τ_{i} → +∞ does not occur. (See Figure 2.)

Figure 2

We next consider Case 3. τ_{i} → +∞ does not occur by the same reason. In the case τ_{i} → −∞, we again get an element

([h^{+}], (φ_{∞}, (−∞))) ∈ M̂(L; β_{1}) ×_{L} (N (R; β_{2}; p_{0}) × A).

Figure 3

This implies Proposition 4.3. □

The next lemma is a consequence of Proposition 3.2 and Gromov compactness.

Lemma 4.4. There exists only a finitely many β ∈ π_{1}(ℂ^{n}, L) such that N (β; p_{0}) ≠ ∅ and E(β) < 0.

First we consider the case when the following is satisfied in addition.

Assumption 4.1. There exists a unique β ∈ π_{1}(ℂ^{n}, L) such that N (β; p_{0}) ≠ ∅ and E(β) < 0.

Moreover if β′ ∈ π_{1}(ℂ^{n}, L) with E(β′) > 0, −E(β) > E(β′) > 0, then M̂(L; β′) = ∅.

This assumption is rather restrictive. We will explain the argument to remove it later in the next section.

Now we assume η_{L} = 0, L is spin and Assumption 4.1. We consider the boundary ∂CN (β_{0}; p_{0}) by using Lemma 4.3. We have

(4.3)  ∂CN (β_{0}; p_{0}) \ N (0; β_{0}; p_{0}) = ⋃_{β_{1}+β_{2}=β_{0}} M̂(L; β_{1}) ×_{L} (N (β_{2}; p_{0}) × A).

Since E(β_{1}) > 0 if M̂(L; β_{1}) ≠ ∅ and since E(β_{0}) = 0 it follows from (4.3) and Assumption 4.1 that

(4.4)  ∂CN (β_{0}; p_{0}) \ N (0; β_{0}; p_{0}) = M̂(L; −β) ×_{L} (N (β; p_{0}) × A).

Note that CN (β_{0}; p_{0}) is a one dimensional oriented chain and N (0; β_{0}; p_{0}) is a one point. Hence the right hand side of (4.4) is an oriented zero dimensional manifold whose order counted with sign is 1.

By the second half of Assumption 4.1, we can show that M̂(L; −β) = Int M̂(L; −β) and defines a cycle. Its dimension is n − 2. Hence

ev_{∗}([M̂(L; −β)]) ∈ H_{n−2}(L; ℤ).

Similarly we find

ev_{∗}([N (0; β; p_{0})]) ∈ H_{2}(L; ℤ).

We now have

ev_{∗}([M̂(L; −β)]) · ev_{∗}([N (0; β; p_{0})]) = 1,

since it is equal to the right hand side of (4.4). (Here · is the intersection pairing.) This implies H^{2}(L; ℚ) ≠ 0. Theorem 2.3 is thus proved under additional hypothesis Assumption 4.1.

5. Floer homology and a spectre sequence.

We now introduce Floer cohomology of Lagrangian submanifold and explain how it can be used to study Lagrangian submanifold of, say ℂ^{n}.

Let L be a compact Lagrangian submanifold of a symplectic manifold M. (In case M is noncompact we assume that M is convex at infinity. See [9].) Let us define a universal Novikov ring Λ by

(5.1)  Λ = {∑ a_{i}T^{λ_{i}} | a_{i} ∈ ℝ, λ_{i} → +∞, λ_{i} ∈ ℝ} .

Actually Λ is a field. We also put

Λ_{0} = {∑ a_{i}T^{λ_{i}} ∈ Λ| λ_{i} ≥ 0} , Λ_{+} = {∑ a_{i}T^{λ_{i}} ∈ Λ| λ_{i} > 0} .

Λ_{0} is a local ring and Λ_{+} is its maximal ideal.

Let η_{L} : π_{2}(M, L) → ℤ be the Maslov index and E : π_{2}(M, L) → ℝ is defined by integrating the symplectic form. We say that L is relatively spin if there exists st ∈ H^{2}(M; ℤ_{2}) which is sent to the second Stiefel-Whiteney class of L. (If L is spin then it is relatively spin.) We say that L ⊂ M is it rational if the image of E : π_{2}(M, L) → ℝ is in λℚ for some λ.

Theorem 5.1 ([15]). We assume L is relatively spin. Then there exists a series of elements β_{i} ∈ π_{2}(M, L) with E(β_{i}) > 0, E(β_{i}) ≤ E(β_{i+1}) and cohomology classes

o_{β_{i}} ∈ H^{2−η_{L}(β_{i})}(L; ℚ),

such that o_{β} = 0 if 2−η_{L}(β) = 0 or n, and o_{β} = 0 if E(β) ≤ 0. It has the following properties.

If o_{β_{i}} = 0 for all β_{i} then there exists a Λ module HF (L, L) and a spectral sequence E_{∗}, which satisfies (1), (2), (4), (5), (6) below. In the case when (M, L) is rational it satisfies (3) also.

(1) E_{2}^{p,q} ≅ H(L; ℚ) ⊗ T^{qλ}Λ_{0}/T^{(q+1)λ}Λ_{0}.

(2) There exists 𝔉^{q}HF^{p}(L, L) such that E_{∞}^{p,q} ≅ 𝔉^{q}HF^{p}(L, L)/𝔉^{q+1}HF^{p}(L, L).

(3) The differential d^{i} = T^{E(β_{i})}d^{β_{i}} is a Λ module homomorphism induced by d^{β_{i}} : H^{k}(L; ℚ) → H^{k+1−η_{L}(β_{i})}(L; ℚ).

(4) d_{β}([1]) = 0. Here [1] ∈ H^{0}(L; ℚ) is a generator.

(5) The fundamental cocycle [L] ∈ H^{n}(L; ℚ) is not contained in the image of d_{β}.

(6) If Φ : M → M is a Hamiltonian diffeomorphism and L is transversal to Φ(L), then

dim_{Λ} HF (L, L) ≤ ♯(L ∩ Φ(L)).

The statement of Theorem 5.1 looks rather complicated. However it seems impossible to find a simpler way to describe Floer homology of Lagrangian submanifolds in general, by the reason we will explain below.

An ideal statement one might expect for Floer cohomology HF (L, L′) might be as follows.

(A) The Floer cohomology HF (L, L′) between two Lagrangian submanifolds L, L′ is always well-defined. (B) We have HF (L, L) ≅ H(L), where the right hand side is a usual cohomology group. (C) Floer cohomology HF (L, L′) is invariant of Hamiltonian isotopies of L and of L′. (D) dim HF (L, L′) ≤ ♯(L ∩ L′) holds for any transversal pair (L, L′) of Lagrangian submanifolds.

Floer [11] constructed such HF (L, L′) under the assumption that π_{2}(M, L) = π_{2}(M, L′) = 0. Under this assumption (A), (B), (C), (D) above holds. Oh [27] generalized it under milder assumption.

However it is too much to expect that we have Floer cohomology which has all of the above properties (A), (B), (C), (D). In fact, (B), (C), (D) would imply that L should always intersect with Φ(L) where Φ is a Hamiltonian isotopy. But this is not the case, especially in the case of Lagrangian submanifold in ℂ^{n}. So we need to modify (A), (B), (C), (D) above to have a correct Floer theory in the general situation.

The series of obstruction classes o_{β} = 0 in Theorem 5.1 precisely describes when (A) holds. (2), (3), (4), (5) of Theorem 5.1 above is related to (B) and shows how H(L) is deformed to HF (L, L). (6) of Theorem 5.1 is a modified version of (D).

Remark 5.1. In Theorem 5.1 we considered only one Lagrangian submanifold L and HF (L) = HF (L, L). There is a version for two Lagrangian submanifold L, L′ and the Floer cohomology HF (L, L′) between them. See [15].

Remark 5.2. Floer cohomology HF (L, L) is not an invariant of symplectic diffeomorphism type of (M, L) but depends on an element of a moduli space M(L) of bounding chains b. We introduced the notion of bounding chain and denote its moduli space by M(L) in [15]. We do no discuss bounding chain here, since we use only the properties stated Theorem 5.1 in this article. (Namely any choice of bounding chain is good for the applications which appear in this article.)

For other applications, especially for applications to mirror symmetry, the space M(L) itself plays a crucial role.

Remark 5.3. The boundary operator of the spectral sequence in Theorem 5.1 does not preserve the degree. Namely its degree is 1 − η_{L}(β_{i}) and is not 1 in general. In [15] we add one extra generator (denoted by e there) to Λ to avoid this strange phenomenon. The the difference between the way to present the result in [15] and one in Theorem 5.1 is not essentail at all.

We also mention that the Floer homology in Theorem 5.1 is the case of ‘weakly unobstructed after infinitesimal deformation’ in the terminology of the revised version of [15]. (For the purpose of the present article the reader does not need to know what it mean. We remark it here for the reader who want to know the relation of this article and [15].

A sketch of the proof of Theorem 5.1 is given later, (just after Definition 8.3). Though Theorem 5.1 looks rather complicated, we can apply it to various pur-

poses as we see below. We first show

Lemma 5.2. Theorem 5.1 implies Theorem 2.3.

Proof. Let L ⊂ ℂ^{n} be a spin Lagrangian submanifold. We assume H^{2}(L; ℚ) = 0 and η_{L} = 0 and deduce a contradiction.

Since o_{β} ∈ H^{2−η(β)}(L; ℚ) = H^{2}(L; ℚ) it follows that we can define HF (L, L). By (6), we have HF (L, L) = 0. Hence by (1)(2), [L] ∈ H^{n}(L) does not survive until E_{∞}. Since [L] is not in the image of d_{β} by (5), it follows that d_{β}([L]) ≠ 0 for some β. However this is impossible since d_{β}([L]) ∈ H^{n+1+η_{L}(β)}(L; ℚ) = 0. □

We next apply Theorem 5.1 to the case of (M, L) = (ℂ^{n+1}, S^{1} × S^{n}) and prove Proposition 2.4. Namely we prove the following two lemmas.

Lemma 5.3 (Oh [28]). Let L = S^{1} × S^{n} be a Lagrangian submanifold of ℂ^{n} and n is odd. We choose the generator β ∈ π_{2}(ℂ^{n+1}, L) so that E(β) > 0. Then η_{L}(β) is positive and divides n + 1.

Proof. We first remark that E ≠ 0 by Theorem 2.1. Hence there is a unique generator β with E(β) > 0.

Since η(β) is even, it follows that o_{β} ∈ H^{even}(L; ℚ). Since n if odd, the cohomology group H^{even}(L; ℚ) is nonzero for H^{0}(L) and H^{n+1}(L) only. In that case o_{β} = 0 by Theorem 5.1. Therefore HF (L, L) is well defined. Since L ⊆ ℂ^{n+1}, it follows that HF (L, L) ≅ 0. By (4) and (5) there exists k, k′ such that d_{kβ}([1]) ≠ 0, and [L] = d_{k′β}u for some u ∈ H(L). Since d_{kβ} is odd degree either d_{kβ}([1]) = c[S^{1}] or d_{kβ} = c[S^{n}]. (c ∈ ℚ.) (Since E(β) > 0, it follows that k > 0.) In the case d_{kβ}([1]) = [S^{1}], we have η_{L}(kβ) = 2. Hence k = 1 and η_{L}(β) = 2.

In the case d_{kβ}([1]) = [S^{n}], we have η_{L}(kβ) = n + 1. Hence η_{L}(β) divides n + 1. □

Lemma 5.4 ([15, 28]). Let L = S^{1} × S^{n} be a Lagrangian submanifold of ℂ^{n+1} and n is even. We choose the generator β ∈ π_{2}(ℂ^{n+1}, L) so that E(β) > 0. Then either η_{L}(β) = 2 or it is nonpositive and divides 2 − n.

Proof. If o_{kβ} ≠ 0 then, since deg o_{kβ} is nonzero and even, it follows that o_{kβ} ∈ H^{n}(L; ℚ). It implies 2 − η_{L}(kβ) = n. Hence η_{L}(β) is negative and divides 2 − n.

If o_{kβ} are all zero, then Floer cohomology HF (L, L) is well defined and is zero. Hence d_{kβ}([1] + 𝔠) ≠ 0 for some k. Since [L] is not in the image of d_{kβ}, it follows that d_{kβ}([1] + 𝔠) = c[S^{1}]. Hence η_{L}(kβ) = 2. Herefore k = 1 and η_{L}(β) = 2. □

Remark 5.4. The second possibility η_{L}(β) < 0 in Lemma 5.4 will be eliminated in §13. (Theorem 13.1.)

6. Homology of loop space and Chas-Sullivan bracket.

In §§6,7,8, we will explain a construction of a filtered A_{∞} structure on the cohomology group H(L) of Lagrangian submanifold L. We take a bit different way from one in [15] and uses de Rham cohomology.

Remark 6.1. In [15] a variant of singular chain complex was used. The way taken in [15] has an advantage that we can work over ℤ coefficient at least in case L is semipositive^{2}. The way taken here has an advantage that we can then keep more symmetry. Especially, cyclic symmetry is established in §9.

Let L be a compact smooth manifold. We denote its free loop space by L(L). Namely

(6.1)  L(L) = {ℓ : S^{1} → L | ℓ is piecewise smooth.}

M.Chas and D.Sullivan [6] introduced a structure of graded Lie algebra on the homology of L(L). Let us recall it here. We identifty S^{1} = ℝ/ℤ. Let f_{i} : P_{i} → L(L) be cycles. (We write P̂_{i} = (P_{i}, f_{i}).) We put

(6.2)  P_{1} ∗ P_{2} = {(x, y, t) ∈ P_{1} × P_{2} × S^{1} | (f_{1}(x))(0) = (f_{2}(y))(t)}

We define f_{1} ∗ f_{2} : P_{1} ∗ P_{2} → L(L) by the following formula.

(6.3)  ((f_{1} ∗ f_{2})(x, y, t))(s) = \begin{cases}(f_{2}(y))(2s) & 2s ≤ t, \\ (f_{1}(x))(2s − t) & t ≤ 2s ≤ t + 1, \\ (f_{2}(x))(2s − 1) & t + 1 ≤ 2s. \end{cases}

Figure 4

We then put

(6.4)  P̂_{1} ∗ P̂_{2} = (P_{1} ∗ P_{2}, f_{1} ∗ f_{2}).

If P̂_{1}, P̂_{2} are cycles, of dimension k_{1},k_{2} respectively, then, under an appropriate transversality condition, P̂_{1} ∗ P̂_{2} is a cycle of dimension k_{1} + k_{2} − n + 1. Here n = dim L. Therefore ∗ defines a map

(6.5)  ∗ : H_{k_{1}}(L(L)) ⊗ H_{k_{2}}(L(L)) → H_{k_{1}+k_{2}−n+1}(L(L)).

2Actually we can do it in general using the method of [17].

Definition 6.1 ([6]). We define loop bracket {·, ·} by

{[P̂_{1}], [P̂_{2}]} = [P̂_{1} ∗ P̂_{2}] + (−1)^{(deg P_{1}+1)(deg P_{2}+1)}[P̂_{2} ∗ P̂_{1}].

Theorem 6.1 ([6]). Loop bracket satisfies Jacobi identity. Namely it defines a structure of graded Lie algebra on H_{∗}(L(L)).

Actually we can work in chain level and construct an L_{∞} algebra, (that is a homotopy version of graded Lie algebra).

There are various ways to work out transversality problem to justify Chas-Sullivan’s construction. Here we use the following one which works best with our construction of virtual fundamental chain of the moduli space of pseudoholomorphic disks.

Definition 6.2. (P, f, ω) is said to be an approximate de Rham chain of L(L) if the following holds.

(1) P is an oriented smooth manifold of finite dimension (with or withour boundary). ω is a smooth diferential form on P of compact support.

(2) f : P → L(L) is a smooth map.

(3) The map ev : P → L defined by ev(x) = (f(x))(0) is a submersion. In case ∂P is nonempty we assume that ev : ∂P → L is also a submersion.

We say (P, f, ω) is an approximate de Rham cycle if P does not have a boundary and dω = 0.

We define the degree of approximate de Rham chain by

deg(P, f, ω) = dim P − deg ω.

We put

(6.6)  ∂(P, f, ω) = (∂P, f, ω) + (−1)^{deg P} (P, f, dω)

An approximate de Rham cycle (P, f, ω) of degree k determines an element H_{k}(L(L); ℝ) as follows. Let H_{c}^{∗}(P; ℝ) be the de Rham cohomology group of compact support and P D : H_{c}^{dim P −k}(P; ℝ) → H_{k}(P; ℝ) be the Poincaré duality. Then we associate f_{∗}(P D([ω])) ∈ H_{k}(L(L); ℝ) to (P, f, ω). It is easy to see that any element of H_{∗}(L(L); ℝ) can be realized by an approximate de Rham cycle.

An approximate de Rham chain (P, f, ω) determines a smooth differential form ev^{L}(P, f, ω) on L of degree k = deg(P, f, ω) as follows.

(6.7)  ev^{L}(P, f, ω) = ev!(ω) ∈ Ω^{n−k}(L).

Here ev! is integration along fiber. It is well defined since ω is of compact support and ev : P → L is a submersion.

We can easily check

P D([ev^{L}(P, f, ω)]) = f_{∗}(P D([ω]))

if (P, f, ω) is an approximate de Rham cycle.

The following lemma follows easily from Stokes’ theorem.

Lemma 6.2. d(ev^{L}(P, f, ω)) = ev^{L}(∂(P, f, ω))

Now we go back to the loop bracket. Let (P_{i}, f_{i}, ω_{i}) be approximate de Rham chains of degree k_{i} for i = 1, 2.

Lemma 6.3. (P_{1} ∗ P_{2}, f_{1} ∗ f_{2}, ω_{1} × ω_{2}) is an approximate de Rham chain of degree k_{1} + k_{2} − n + 1.

Proof. We remark that the map

P_{1} × P_{2} × S^{1} → L × L, (x, y, t) ↦ ((f_{1}(x))(0), (f_{2}(y))(t))

is transversal to diagonal by Condition (3) of Definition 6.2. The rest of the proof is straightforward. □

Thus we can define ∗ in the chain level using approximate de Rham chains. Hence we can define loop bracket also in the chain level. By the argument of [6], loop bracket satisfies graded Jacobi identity in the chain level modulo parametrization of the loop. Hence we can prove easily that it induces L_{∞} structure. (The argument we need to do so is the same as the argument to show that the (based) loop space is an A_{∞} space. See [36, 37]).

Now we want to use moduli space of pseudoholomorphic disks to construct a chain (approximate de Rham chain) in the loop space. Let M be a symplectic manifold and L be its Lagrangian submanifold. We assume that L is compact and relatively spin and M is convex at infinity (in case M is noncompact). For each β ∈ π_{2}(M; L) we define the spaces M̃(L; β), M̂(L; β), M(L; β) in the same way as Definition 3.1. Actually since M ≠ ℂ^{n} in general, we need to consider sphere buble to compactify them. But the argument to handle sphere bubble of M̂(L; β) is the same as the case of the moduli space of pseudoholomorphic map from Riemann surface without boundary (see [16]) since the sphere bubble is a codimension 2 phenomenon. So we do not discuss it. As for the compactification M(L; β) there exists one new point to discuss which we mention briefly in §14 (see Theorem 14.2 and its proof) and will be discussed in detail in the revised version of [15].

Now we consider M̂(L; β). Recall

M̂(L; β) = \frac{M̃(L; β)}{Aut(D^{2}, 1)}

and the group Aut(D^{2}, 1) is contractible. Hence we take a lift

Lift_{β} : M̂(L; β) → M̃(L; β)

and fix it. We define a map ev : M̃(L; β) → L(L). by ev(φ) = φ|_{∂D^{2}}, and consider a map ev ∘ Lift_{β} : M̂(L; β) → L(L). We want to use the chain (M̂(L; β), ev ∘ Lift_{β}) to construct Floer cohomology etc.

For the purpose of transversality, we want to replace (M̂(L; β), ev ∘ Lift_{β}) by an appropriate approximate de Rham chain. To describe it precisely we need to use the notion of Kuranishi structure more systematically. We will do it in [14]. Here we sketch the argument by simplifying the situation. Let us consider a Banach manifold B together with a Banach bundle E → B and a section s : B → E which is not necessary transversal to zero. We assume that the differential of s is Fredholm. Let f̃: B → L(L) be a smooth map. (In our application B = Map((D^{2}, ∂D^{2}); (M, L)), and s = 0 gives the equation φ ∈ B to be pseudoholomorphic.) We assume s^{−1}(0) is compact. Then we can find a finite dimensional space W and a family of perturbations s_{w} of s parametrized by w ∈ W which has the following properties.

(1) We put P = {(x, w) ∈ B × W |s_{w}(x) = 0}. Then P is a smooth manifold.

(2) The projection π_{W} : P → W is smooth and proper.

(3) If we put f(x, w) = f̃(x) then f : P → L(L) is a smooth map.

(4) The composition ev ∘ f : P → L is a submersion.

(4) becomes possible by taking W of very big dimension.

Now let ω_{W} be a smooth form on W of top dimension. We assume that it is of compact support and ∫_{W} ω_{W} = 1. We put ω = π_{W}^{∗} ω_{W}. Then (P, f, ω) is an approximate de Rham cycle.

In our actual application, we have locally an orbibundle E_{α} → U_{α} on an orbifold U_{α} together with its sections s_{α} such that ∪_{α}s_{α}^{−1}(0) = M̂(L; β). Moreover (E_{α}, U_{α}, s_{α}) are glued in an appropriate sense. More precisely we have a Kuranishi structure on M̂(L; β). We can use a smooth family of multisections (see [16]) in a similar way as above to obtain an approximate de Rham chain for each M̂(L; β). We write it by the same symbol M̂(L; β) for simplicity.

Theorem 6.4 ([14]). We can choose Lift_{β} and approximate de Rham chain M̂(L; β) such that

(6.8)  ∂M̂(L; β) + \frac{1}{2} ∑_{β=β_{1}+β_{2}}\{M̂(L; β_{1}), M̂(L; β_{2})\} = 0.

The proof is similar to the proof of Proposition 4.3 and will be given in detail in [14].

Remark 6.2. The method to realize virtual fundamental chain using approximate de Rham chain is somewhat similar to Ruan’s approach [33].

7. Iterated integral and Gerstenhaber bracket.

In §6, we studyed homology of the loop space L(L). The relation of homology of L and of L(L) is classical. Especially there is a construction by Chen [7], which we review here.

Let L be a smooth manifold and (Ω(L), d, ∧) be its de Rham complex. We put

Ω(L)[1]^{m} = Ω^{m+1}(L),
B_{k}(Ω(L)[1]) = \underbrace{Ω(L)[1] × ⋯ × Ω(L)[1]}_{kcopy}

B(Ω(L)[1]) = ⨁_{k} B_{k}(Ω(L)[1])

and define

\overline{𝔪}_{k} : B_{k}(Ω(L)[1]) → Ω(L)[1]

of degree +1 by

(7.1)  \overline{𝔪}_{1}(u) = (−1)^{deg u}du, \overline{𝔪}_{2}(u, v) = (−1)^{deg u(deg v+1)}u ∧ v

and \overline{𝔪}_{k} = 0 for k ≠ 1, 2. It will define a structure of A_{∞} algebra (which will be defined later). We now define

d̂: B(Ω(L)[1]) → B(Ω(L)[1]),

by

(7.2)  \begin{aligned}d̂(u_{1} ⊗ ⋯ ⊗ u_{k}) = & ∑_{i}(−1)^{∗_{i}}u_{1} ⊗ ⋯ ⊗ \overline{𝔪}_{1}(u_{i}) ⊗ ⋯ ⊗ u_{k}+ \\ & ∑_{i}(−1)^{∗_{i}}u_{1} ⊗ ⋯ ⊗ \overline{𝔪}_{2}(u_{i}, u_{i+1}) ⊗ ⋯ ⊗ u_{k} \end{aligned}

where ∗_{i} = deg u_{1} + ⋯ + deg u_{i−1} + i − 1, here deg is the degree as a differential form. It is easy to see that d̂^{2} = 0 and hence (B(Ω(L)[1]), d̂) is a cochain complex.

To define iterated integral we modify it a bit. We fix a base point p_{0} of L and define Ω_{0}(L) as follows.

\begin{aligned}Ω_{0}^{k}(L) & = Ω^{k}(L), k ≠ 0, \\ Ω_{0}^{0}(L) & = \{f ∈ C^{∞}(L)|f(p_{0}) = 0.\} \end{aligned}

We define B(Ω_{0}(L)[1]) in a similar way.

It is easy to see that d̂ preserves B(Ω_{0}(L)[1]) and hence (B(Ω_{0}(L)[1]), d̂) is a cochain complex also.

We denote by L_{0}(L) the based loop space. Namely :

L_{0}(L) = {ℓ : S^{1} → L|ℓ(0) = p_{0}}.

Chen [7] defined a cochain homomorphism

(7.3)  Ich : (B(Ω_{0}(L)[1]), d̂) → Ω(L_{0}(L), d).

Since L_{0}(L) is infinite dimensional we need to be careful to define de Rham complex Ω(L_{0}(L), d). We do not discuss this point. See [7].

Instead we take a smooth chain (P, f) of L_{0}(L) and define an integration of Ich(u_{1} ⊗ ⋯ ⊗ u_{k}) over (P, f) as follows.

We put

(7.4)  C_{k} = {(t_{1}, ⋯, t_{k}) ∈ [0, 1]^{k}|0 ≤ t_{1} ≤ ⋯ ≤ t_{k} ≤ 1}.

We define a map ev : L_{0}(L) × C_{k} → L^{k}, by

(7.5)  ev(ℓ, (t_{1}, ⋯, t_{k})) = (ℓ(t_{1}), ⋯, ℓ(t_{k})).

We now define

(7.6)  ∫_{P} f^{∗}(Ich(u_{1} ⊗ ⋯ ⊗ u_{k})) = ∫_{P ×C_{k}}(ev ∘ (f × id))^{∗}(u_{1} ∧ ⋯ ∧ u_{k}).

Theorem 7.1 (Chen [7]). Ich is a cochain homomorphism.

Proof. It suffices to show :

(7.7)  ∫_{∂P} f^{∗}(Ich(u_{1} ⊗ ⋯ ⊗ u_{k})) = ∫_{P} f^{∗}(Ich(d̂(u_{1} ⊗ ⋯ ⊗ u_{k}))).

by studying the boundarry of P × C_{k}. (We omit the detail.) □

We define a free loop space version of the homomorphism of Theorem 7.1. We consider

Hom(B(Ω(L)[1]), Ω(L)[1]) = ∏_{k} Hom(B_{k}(Ω(L)[1]), Ω(L)[1]),

and define a boundary operator δ on it as follows. Let φ ∈ Hom(B(Ω(L)[1]), Ω(L)[1]) then

(7.8) (δφ)(a_{1} ⊗ ⋯ ⊗ a_{k}) = d(φ(a_{1} ⊗ ⋯ ⊗ a_{k}))
−(−1)^{deg φ}(φ ∘ d̂)(a_{1} ⊗ ⋯ ⊗ a_{k})
+(−1)^{(deg a_{1}+1) deg φ}\overline{𝔪}_{2}(a_{1}, φ(a_{2} ⊗ ⋯ ⊗ a_{k}))
+\overline{𝔪}_{2}(φ(a_{1} ⊗ ⋯ ⊗ a_{k−1}), a_{k})

It is easy to check that δ^{2} = 0.

We denote by S^{D}(L(L)) the set of all approximate de Rham chains on L(L). It is a chain compex. We next define a chain homomorphism

Ich^{∗} : S^{D}(L(L)) → Hom(B(Ω(L)[1]), Ω(L)[1])

We first define a map

ev_{+} = (ev, ev_{0}) : L(L) × C_{k} → L^{k+1}

by

ev_{+}(ℓ, (t_{1}, ⋯, t_{k}))) = (ℓ(t_{1}), ⋯, ℓ(t_{k}), ℓ(0)).

Then, we put

(7.9)  \begin{matrix}Ich^{∗}((P, f, ω)) & (u_{1} ⊗ ⋯ ⊗ u_{k})) \\ = (ev_{0} ∘ π)! (ω ∧ (ev ∘ (f × id))^{∗}(u_{1} ∧ ⋯ ∧ u_{k})) . \end{matrix}

Here ev_{0} ∘ π : P × C_{k} → L is the composition of P × C_{k} → P, f : P → L(L) and ev_{0} : L(L) → L. (ev_{0} ∘ π)! is the integration along fiber.

Proposition 7.2. Ich^{∗} is a chain homomorphism

The proof is straightforward.

We next recall Gerstenhaber bracket, which was introduced by [18] to study deformation theory of associative algebra.

We restrict ourselves to the case of Ω(L). We define a structure of differential graded Lie algebra on Hom(B(Ω(L)[1]), Ω(L)[1]) as follows. Let

φ_{i} ∈ Hom(B_{k_{i}}(Ω(L)[1]), Ω(L)[1]).

We put

(φ_{1} ∘ φ_{2})(u_{1} ⊗ ⋯ ⊗ u_{k_{1}+k_{2}−1}) (7.10)

= ∑_{i}(−1)^{∗_{i}}φ_{1}(u_{1} ⊗ ⋯ φ_{2}(u_{i} ⊗ ⋯ ⊗ u_{i+k_{2}−1}) ⋯ ⊗ u_{k_{1}+k_{2}−1}).

where ∗_{i} = (deg φ_{2})(deg u_{1} + ⋯ + deg u_{i−1} + i − 1). We then define :

(7.11)  {φ_{1}, φ_{2}} = φ_{1} ∘ φ_{2} − (−1)^{deg φ_{1} deg φ_{2}}φ_{2} ∘ φ_{1}.

We call {·, ·} the Gerstenhaber bracket.

Theorem 7.3 ([18]). (Hom(B(Ω(L)[1]), Ω(L)[1]), δ, {·, ·}) is a differential graded Lie algebra.

Now we have :

Proposition 7.4. We have δ(Ich^{∗}((P, f, ω))) = Ich^{∗}(∂(P, f, ω)) and

Ich^{∗}({(P_{1}, f_{1}, ω_{1}), (P_{2}, f_{2}, ω_{2})}) = {Ich^{∗}((P_{1}, f_{1}, ω_{1})), Ich^{∗}((P_{2}, f_{2}, ω_{2}))}

where {·, ·} in the left hand side is Loop bracket and {·, ·} in the right hand side is Gerstenhaber bracket.

The proof is straight forward calculation and is omitted. The proposition implies that Ich^{∗} is a homomrphism of differential graded Lie algebra.

Remark 7.1. Chas-Sullivan in [6] already mentioned that their construction is an analogy of Gerstenhaber bracket.

Remark 7.2. Precisely speaking the loop bracket defines an L_{∞} structure since the Jacobi identity holds only modulo homotopy. However Jacobi identity on S^{D}(𝔏(L)) fails only because of parametrization. The difference of parametrization is killed by Ich^{∗}.

8. A_{∞} deformation of de Rham complex.

We now use the result of §§6,7 to define an A_{∞} deformation of the de Rham complex. We first recall the definition of filtered A_{∞} algebra. Let C_{ℝ} be a graded ℝ vector space. We recall

Λ_{0} = {∑ a_{i}T^{λ_{i}} ∈ Λ| λ_{i} ≥ 0} , Λ_{+} = {∑ a_{i}T^{λ_{i}} ∈ Λ| λ_{i} > 0} .

Λ_{0} has a filtration F^{λ}Λ_{0} = T^{λ}Λ_{0} which defines a (non Archimedian) norm on it. We put C^{k} ≅ C_{ℝ}^{k} ˆ⊗Λ_{0} here and hereafter ˆ⊗ means that we are taking an approariate completion with respect to the (non Archimedian) norm on Λ_{0}. (We omit the detail and refer [15].) C[1]^{k} = C^{k+1}. We put

B_{k}(C[1]) = \underbrace{C[1] ˆ⊗_{Λ_{0}} ⋯ ˆ⊗_{Λ_{0}}C[1]}_{k}, B(C[1]) = ⨁_{k} B_{k}(C[1])

and consider a series of Λ_{0} module homomorphisms

𝔪_{k} : B_{k}(C[1]) → C[1]

of odd degree. We assume that it is written as

𝔪_{k} = ∑_{i} T^{λ_{i}}𝔪_{k,i}

where λ_{i} ≥ 0, λ_{i} → +∞ and 𝔪_{k,i} is induced by ℝ linear map

\underbrace{C_{ℝ} ⊗_{ℝ} ⋯ ⊗_{ℝ} C_{ℝ}}_{k} → C_{ℝ},

of degree d_{i} + 1. (Here d_{i} are even numbers. We assume that λ_{i} and d_{i} are independent of k.) We assume also λ_{0} = 0 and λ_{i} > 0 for i > 0.

Definition 8.1. We say that (C, 𝔪_{k}), k = 0, 1, ⋯ is a filtered A_{∞} algebra if

∑_{k+ℓ=n+1} ∑_{i}(−1)^{∗}𝔪_{k}(x_{1} ⊗ ⋯ 𝔪_{ℓ}(x_{i} ⊗ ⋯ ⊗ x_{i+ℓ−1}) ⋯ ⊗ x_{n}) = 0

where ∗ = deg x_{1} + ⋯ + deg x_{i−1} + i − 1 (deg is a degree before shift) and if 𝔪_{0} ≡ 0 mod Λ_{+}.

We take C_{ℝ}^{k} = Ω^{k}(L) the de Rham complex. We are going to define 𝔪_{k}. We take 𝔪_{k,0} = \overline{𝔪}_{k}. We recall that \overline{𝔪}_{1} = ±d, \overline{𝔪}_{2} = ±∧ and other \overline{𝔪}_{k} are zero. We put

(8.1)  φ_{k} = ∑_{i>0} 𝔪_{k,i} ∈ Hom(B_{k}C[1], C[1]), φ = (φ_{0}, φ_{1}, ⋯).

The next proposition is in principle due to Gerstenhaber.

Proposition 8.1. 𝔪_{k} is an filtered A_{∞} algebra if and only if

δφ + \frac{1}{2}\{φ, φ\} = 0.

Here δ is as in (7.8).

We remark we can generalize Gersenhaber blacket {·, ·} in an obvious way to ∏_{k} Hom(B_{k}C[1], C[1]).

Now let L be a Lagrangian submanifold of M. We assume that it is relativelly spin and is compact.

Definition 8.2.

α(L) = ∑_{i} T^{E(β)}M̂(L; β_{i}).

α(L) is a Λ_{0} valued approximate de Rham chain of L(L).

Here β_{i} ∈ π_{2}(M, L) such that 0 = E(β_{0}) < E(β_{1}) ≤ · · ·. We define

(8.2)  φ = (φ_{0}, φ_{1}, ⋯) = Ich^{∗}(α),

where Ich^{∗} is as in (7.9) We use φ_{k} to define 𝔪_{k} by (8.1).

Theorem 8.2. The operator 𝔪_{k} above defines a structure of filtered A_{∞} algebra on Ω(L) ˆ⊗Λ_{0}.

Proof. Theorem 8.2 follows immediately from Theorem 6.4, Propositions 7.4, 8.1. □

To define a structure of filtered A_{∞} algebra on the cohomology group H^{∗}(L; ℝ) we use the following theorem. Let (C, 𝔪_{k}) be a filtered A_{∞} algebra. We remark that 𝔪_{1,0} is induced from an ℝ linear map C_{ℝ}^{k} → C_{ℝ}^{k+1}, which we write \overline{𝔪}_{1}. Using 𝔪_{0,0} = 0, (which follows from 𝔪_{0} ≡ 0 mod Λ_{+}), we can prove \overline{𝔪}_{1} ∘ \overline{𝔪}_{1} = 0. Let H^{∗}(C_{ℝ}) be the cohomology group H^{∗}(C; \overline{𝔪}_{1}) that is nothing but the de Rham cohomology group in our main example. We consider H^{∗}(C; Λ_{0}) = H^{∗}(C_{ℝ}) ⊗_{ℝ} Λ_{0}.

Theorem 8.3 ([15]). There exists a structure of filtered A_{∞} algebra on H^{∗}(C; Λ_{0}) such that it is homotopy equivalent to (C, 𝔪).

Theorem 8.3 is a filtered version of a classical result of homotopical algebra (see Kadeishvili [23] etc.) and is proved in the revised version of [15] (see also [13]). We also refer [15] (and [13]) for the definition of homotopy equivalence of filtered A_{∞} algebra.

Theorems 8.2 and 8.3 imply that we have a structure of filtered A_{∞} algebra on H^{∗}(L; Λ_{0}) for a compact relatively spin Lagrangian submanifold L. We now show how the structures of Theorem 5.1 is deduced from it. First we consider

𝔪_{0} : B_{0}H(L; Λ_{0}) = Λ_{0} → H(L; Λ_{0}).

We put

o_{β_{i}} = 𝔪_{0,i}(1) ∈ H^{2+d_{i}}(L; ℝ).

By Definition 8.2 we can show that d_{i} = −η_{L}(β_{i}). Hence o_{β_{i}} ∈ H^{2−η_{L}(β_{i})}(L; ℝ) as required.

We next assume that o_{β} are all zero. Then

(𝔪_{1} ∘ 𝔪_{1})(x) = ±𝔪_{2}(𝔪_{0}(1), x) ± 𝔪_{2}(x, 𝔪_{0}(1)) = 0.

We define

Definition 8.3.

HF (L, L; Λ_{0}) = \frac{Ker𝔪_{1}}{Im𝔪_{1}} .

We remark that H^{∗}(L; Λ_{0}) has a filtration induced by the filtration F^{λ}Λ_{0} = T^{λ}Λ_{0}. (λ ∈ ℝ_{≥0}). We can use it to construct a spectral sequence in Theorem 5.1. Namely we take F^{(k)}H(L; Λ_{0}) = H(L; ℚ) ⊗ T^{kλ}Λ_{0} (k = 0, 1, 2, ⋯) and use it to construct a spectral sequence. In the case when E(π_{2}(M, L)) = λℤ, we use this particular λ to define F^{(k)}H(L). Then the formula

𝔪_{1} = ∑_{i≥0} T^{λ_{i}}𝔪_{1,i}

implies that the differential of the spectral sequence is induced by 𝔪_{1,i}. We put d_{β_{i}} = 𝔪_{1,i}.

Remark 8.1. In the case when L is not rational (that is E(π_{2}(M; L)) ⊂ ℝ_{≥0} is not finitely generated), only the natural filtartion we have is one F^{λ}H(L; Λ_{0}) = H(L; ℚ)⊗T^{λ}Λ_{0} which is parametrized by λ ∈ ℝ_{≥0}. It is rather unusual to use such a filtration to construct a spectral sequence. Usually spectral sequence is induced by a filtration parametrized by integer or by natural number. Moreover the ring Λ_{0} is not Noetherian. This causes serious trouble to construct spectral sequence and prove its convergence. This problem is resolved in [15].

Remark 8.2. In the above argument, we use Λ_{0} in place of Λ. Off course Floer cohomology over Λ_{0} induces Floer homology over Λ. The reason we need to work over Λ is that (6) of Theorem 5.1 (or more generally the invariance of Floer cohomology under Hamiltonian deformation of Lagrangian submanifold) is not true over Λ_{0} and can be proved only over Λ coefficient. See [15].

We need more argument to establish the properties asserted in Theorem 5.1. (Especially (4), (5), (6) of it.) We do not explain it here and refer [15].

We mention one application of the construction of this article.

Theorem 8.4. Let L be a compact relatively spin Lagrangian submanifold of M. We assume that L admits a metric of negative sectional curvature and dim L is even. Then o_{β} = 0. Moreover HF (L; L) ≅ H(L; Λ).

Proof. (Sketch) We remark that dim M̂(L; β) = n−2+η_{L}(β) is even, since n is even and η_{L}(β) is even. Let us decompose L(L) into connected components ∪L_{[γ]}(L), where γ ∈ π_{1}(L) and [γ] be its conjugacy class. Since L has negative curvature H_{i}(L_{[γ]}(L)) ≅ H_{i}(S^{1}) for γ ≠ 1. We remark that if M̂(L; β) is nonempty then it is at least one dimensional. (In fact if M̂(L; β) is nonempty then M(L; β) is nonempty.) This implies that M̂(L; β) is homologous to zero if ∂β ≠ 1. Hence 𝔪_{k,i} is nonzero only for β_{i} with ∂β_{i} = 1. However since L is aspherical L_{[1]}(L) is homotopy equivalent to L. We can use it to show that Ich^{∗} is trivial on L_{[1]}(L). Thus the A_{∞} algebra (Ω(L) ˆ⊗Λ_{0}, 𝔪_{k}) is homotopy equivalent to the de Rham complex. Theorem 8.4 follows. □

Remark 8.3. Actually there is one point where the argument above is not sufficient. Namely since M̂(L; β) is not a cycle in general (see Theorem 6.4), it is not clear how to use vanishing of cohomology of L_{[∂β]}(L) to modify M̂(L; β) to zero, without changing the homotopy type of filtered A_{∞} algebra induced by it on Ω(L) ˆ⊗Λ_{0}.

We can overcome this point in the following way.

First there is a notion of gauge equivelence between elements satisfing Maurer-Cartan equation (that is the conclusion of Theorem 6.4), such that gauge equivalent M̂(L; β) induces a homotopy equivalent A_{∞} structure on Ω(L) ˆ⊗Λ_{0}. See [13].

Second we find that the set of homotopy equivalence class of elements satisfing Maurer-Cartan equation, is independt of the homotopy type of differential graded Lie algebra (or more generally of L_{∞} algebra).

Third we can show that any differential graded Lie algebra C is homotoyp equivalent to an L_{∞} algebra defined on cohomology group of C. This fact is an L_{∞} analogue of Theorem 8.3 and is proved by various people including [23].

Since our solution M̂(L; β) of Maurer-Cartan equation has degree where cohomology group vanish, it follows that it is gauge equivalent to zero. Summing up we find that the A_{∞} structure induced by M̂(L; β) is homotopy equivalent to one induced by zero. Theorem 8.4 follows.

Remark 8.4. Theorem 8.4 implies that negatively curved spin manifold of even dimension is not embeded to ℂ×M (for any symplectic manifold M which is convex at infinity) as a Lagrangian submanifold. This fact was established in stronger form by Viterbo. Namely negatively curved manifold is not embeded as a Lagrangian submanifold to ℂ × M or ℂP^{n}. (He does not need to assume neigher L is spin nor that L is of even dimension.) We will discuss related problems in §14.

Remark 8.5. The idea to use Chan-Sullivan’s String topology to study open string theory is also applied by [5]^{3}. Their interest is in its application to Physics. We here emphasise its application to sympletic topology. The application of our approach to mirror symmetry will be discussed elsewhere.

The idea to use homology of loop space in Floer theory is independently proposed in [3], where Barraud and Cornea applied it in the case when there exist no pseudoholomorphic disk and Floer homology is isomorphic to the usual homology^{4}. F.Lalonde informed the author that together with authors of [3] he is now trying to apply it in more general situation.

9. S^{1} equivariant homology of Loop space and cyclic A_{∞} algebra.

The loop space L(L) has a canonical S^{1} action defined by (s · ℓ)(t) = ℓ(t + s). Chas-Sullivan [6] also defined a blacket (which they call string blacket) on the S^{1} equivariant homology H_{∗}^{S^{1}}(L(L)). Namley they define

3The author would like to thank A. Schwarz who pointed out the relation of [5] to the authors idea to use loop space homology and Chas-Sullivan bracket to study Floer homology.

4Our main application in §§11,12,13,14 is in the case when Floer homology may not be well defined in the sense of [15].

(9.1)  {·, ·} : H_{k}^{S^{1}}(L(L)) ⊗ H_{ℓ}^{S^{1}}(L(L)) → H_{k+ℓ−n+2}^{S^{1}}(L(L)),

by

(9.2)  {x, y} = I_{∗}{I^{∗}(x), I^{∗}(y)}

where I_{∗} : H_{k}(L(L)) → H_{k}^{S^{1}}(L(L)) and I^{∗} : H_{k}^{S^{1}}(L(L)) → H_{k+1}(L(L)) are obvious maps and {} in the right hand side is the loop bracket. In a similar way, we can construct it in the chain level using S^{1} equivariant approximation de Rham chain of L(L). Here S^{1} equivariant approximation de Rham chain of L(L) is an approximate de Rham chain (P, f, ω) such that S^{1} acts on P and that f,ω are S^{1} equivariant.

We next consider M(L; β). We may regard it as an S^{1} equivariant approximate de Rham chain of L(L) of degree n − 3 + η_{L}(β). Then we have

(9.3)  ∂M(L; β) + \frac{1}{2} ∑_{β=β_{1}+β_{2}}\{M(L; β_{1}), M(L; β_{2})\} ∈ S^{D}(L).

Here we embed L to L(L) as the set of trivial loops.

We next define a cyclic Bar complex B_{k}^{cyc}(C[1]) deviding B_{k}(C[1]) by the equivalence relation generated by

x_{1} ⊗ ⋯ ⊗ x_{k} ∼ (−1)^{(deg x_{k}+1)(deg x_{1}+··· deg x_{k−1}+k−1)}x_{k} ⊗ x_{1} ⊗ ⋯ ⊗ x_{k−1}.

We can define a Gerstenhaver bracket on ∏_{k≥1} Hom(B_{k}^{cyc}(C[1]), ℝ) in a similar way. We have homomorphism

Ich : ⨁_{k} B_{k}^{cyc}(C[1]) → Ω_{S^{1}}^{∗}(L(L))

here the right hand side is the set of S^{1} equivarent forms. (See [19].) Its dual Ich^{∗} is a chain homomorphism and sends string blacket to Gerstenhaber bracket.

Now we put

\overline{α}(L) = ∑_{i} T^{E(β)}M(L; β_{i}).

which is an S^{1} equivalent approximate de Rham chain of L(L). Hence pulling it back by Ich^{∗}, we have an element Ich^{∗}(\overline{α}(L)) of ∏_{k≥1} Hom(B_{k}^{cyc}(Ω(L)[1]), Λ_{+}) Then, using the fact that S^{D}(L) is killed by iterated integral, we have

δ(Ich^{∗}(\overline{α}(L))) + \frac{1}{2}\{Ich^{∗}(\overline{α}(L)), Ich^{∗}(\overline{α}(L))\} = 0.

It defines a family of operations

𝔪_{k}^{+} : B_{k}^{cyc}(Ω(L)[1]) → Λ_{0}.

It is related to the operations 𝔪_{k} in the last section by the fomula

(9.4)  ⟨𝔪_{k}(u_{1}, ⋯, u_{k}), u_{k+1}⟩ = 𝔪_{k+1}^{+}(u_{1}, ⋯, u_{k}, u_{k+1})

if we take perturbation appropriately. Here ⟨·, ·⟩ : Ω^{k}(L) ⊗ Ω^{n−k}(L) → ℝ is defined by

⟨u, v⟩ = ∫_{L} u ∧ v.

(9.4) implies that 𝔪_{k} satisfies the following cyclic symmetry.

(9.5)  ⟨𝔪_{k}(u_{1}, ⋯, u_{k}), u_{k+1}⟩ = (−1)^{∗}⟨𝔪_{k}(u_{k+1}, u_{1} ⋯, u_{k−1}), u_{k}⟩

where ∗ = (deg x_{k+1} + 1)(deg x_{1} + ⋯ deg x_{k} + k). We will discuss the contents of this section in more detail in [14].

Remark 9.1. We constructed filtered cyclic A_{∞} algebra in this section using de Rham cohomology. Hence it is defined over ℝ. The author does not know how to do it keeping cyclic symmetry over ℤ even in semipositive case. (Actually he does not know how to do it for classical cohomology either.) Compare Remark 6.1.

Remark 9.2. The author would like to thank P. Seidel a discussion with whom let the author realize the importance of cyclic symmetry of Floer cohomology.

10. L_{∞} structure on H(S^{1} × S^{n}; ℚ).

In this section we consider the case of S^{1} × S^{n} in ℂ^{n+1} and study the leading term of the symmetrization of A_{∞} structure of it.

We first discuss symmetrization of (filtered) A_{∞} algebra briefly. Let B_{k}(C[1]) is as in §8. We divide it by the equivalence relation generated by

x_{1} ⊗ ⋯ ⊗ x_{i} ⊗ x_{i+1} ⊗ ⋯ ⊗ x_{k} (10.1)

∼ (−1)^{(deg x_{i}+1)(deg x_{i+1}+1)}x_{1} ⊗ ⋯ ⊗ x_{i+1} ⊗ x_{i} ⊗ ⋯ ⊗ x_{k},

and denote it by E_{k}(C[1]). 𝔪_{k} induces 𝔩_{k} : E_{k}(C[1]) → C[1] by

(10.2)  𝔩_{k}([x_{1} ⊗ ⋯ ⊗ x_{k}]) = \frac{1}{k!} ∑_{σ∈𝔖_{k}}(−1)^{∗_{σ}}𝔪_{k}(x_{σ(1)} ⊗ ⋯ ⊗ x_{σ(k)})

here ∗_{σ} = ∑_{i<j;σ(i)>σ(j)}(deg x_{i} + 1)(deg x_{j} + 1). In case 𝔪_{k} = \overline{𝔪}_{k} is induced by the structure of (graded commutative) differential graded algebra by (7.1) then we can check by an easy calculation that the induced operations 𝔩_{k} becomes zero. Hene the part of 𝔩_{k} on E_{k}(Ω(L)[1]) induced by wedge product of differential forms vanishes.

𝔩_{k} defines an L_{∞} structure. Here we define :

Definition 10.1. (C, 𝔩_{k}) is said to be a filtered L_{∞} algebra if the following holds :

∑_{k+ℓ=n} ∑_{I,J}(−1)^{∗}𝔩_{ℓ+1}(𝔩_{k}(x_{i_{1}} ⊗ ⋯ x_{i_{k}}) ⊗ x_{j_{1}} ⊗ ⋯ ⊗ x_{j_{ℓ}}) = 0

where the second sum is taken over all I = {i_{1}, ⋯, i_{k}}, J = {j_{1}, ⋯, j_{ℓ}} with i_{1} < ⋯ < i_{k}, j_{1} < ⋯ < j_{ℓ}, I ∩ J = ∅, I ∪ J = {1, ⋯, n} and

∗ = ∑_{a,b;i_{a}>j_{b}}(deg x_{i_{a}} + 1)(deg x_{j_{b}} + 1).

We thus obtained a filtered L_{∞} algebra (Ω(L), 𝔩_{k}) for a relatively spin compact Lagrangian submanifold L ⊂ M. We consider the case L = S^{1} × S^{n} ⊂ ℂ^{n+1} and calculate the leading term of 𝔩_{k}.

We put

(10.3)  ⟨𝔩_{k}(u_{1}, ⋯, u_{k}), u_{k+1}⟩ = 𝔩_{k+1}^{+}(u_{1}, ⋯, u_{k}, u_{k+1}).

Let us chooose a generator γ ∈ π_{1}(S^{1} × S^{n}) such that E(γ) = λ_{1} > 0. (Such γ exists by Theorem 2.1.) We consider the case η_{L}(γ) = 2, n + 1. They are the only cases we have an examples. Let a, b, c, e ∈ H^{∗}(S^{1} × S^{n}; ℤ) be generators of degree 1, n, n + 1, 0 respectively.

Theorem 10.1. If η_{L}(γ) = 2 then there exists an integer ℓ independent of k such that

𝔩_{k+1}^{+}(a, ⋯, a, c) ≡ ℓT^{λ_{1}} \frac{1}{k!} mod T^{2λ_{1}}.

All other operations 𝔩^{+} among generators vanish.

Theorem 10.2. If η_{L}(γ) = n + 1 (n is odd) then

𝔩_{k+2}^{+}(a, ⋯, a, b, c) ≡ ±T^{λ_{1}} \frac{1}{(k + 1)!} mod T^{2λ_{1}}.

All other operations 𝔩^{+} among generators vanish.

Proofs of Theorems 10.1, 10.2 will be discussed later in this section. Detail of them will be in the revised version of [15].

Remark 10.1. We remark that filtered A_{∞} algebra and filtered L_{∞} algebra associated to a Langrangian submanifold L (which is defined on the cohomology group of L) is well defined only up to filtered A_{∞} or filtered L_{∞} isomorphism. (Here filtered L_{∞} isomorphism is an L_{∞} homomorphism which has an inverse.) So precisely speaking we need to state Theorems 10.1, 10.2 such as : “We may take filtered L_{∞} structure so that the following holds … ”.

To prove Theorems 10.1,10.2, we first remark that Theorem 6.4 implies that M̂(S^{1} × S^{n}; γ) is a cycle in S^{D}(L(S^{1} × S^{n})). (This is because there is no γ′ ∈ π_{1}(S^{1} × S^{n}) with 0 < E(γ′) < E(γ).)

Actually we can prove more. Namely, since our Lagrangian submanifold is monotone we can use a result of [15] to find an appropriate (single valued) perturbation so that the fundamental chain of M̂(S^{1} × S^{n}; γ) is a cycle over ℤ.

Hence in case of Theorems 10.1, we have [M̂(S^{1}×S^{n}; γ)] ∈ H_{n+1}(L(S^{1}×S^{n}); ℤ), and in case of Theorems 10.2 we have [M̂(S^{1} × S^{n}; γ)] ∈ H_{2n}(L(S^{1} × S^{n}); ℤ). We are going to study them below. We first use the following :

Lemma 10.3. [M̂(S^{1} × S^{n}; γ)] is in the image of I^{∗} : H_{k}^{S^{1}}(L(S^{1} × S^{n}); ℚ) → H_{k+1}(L(S^{1} × S^{n}); ℚ).

This is a consequence of the §9.

We next remark that it is easy to show

(10.4)  H_{∗}(L(X × Y)) = H_{∗}(L(X)) ⊗ H_{∗}(L(Y)).

Let γ_{0} ∈ π_{1}(S^{1}) is a generator and L_{γ_{0}}(S^{1}) be the component containing γ_{0}. It is easy to see that H_{∗}(L_{γ_{0}}(S^{1})) ≅ H_{∗}(S^{1}), H_{∗}^{S^{1}}(L_{γ_{0}}(S^{1})) ≅ H_{∗}^{S^{1}}(S^{1}) ≅ ℤ.

Now we consider the component of L_{γ}(S^{1} × S^{n}). Then using a commutative diagram

L(S^{n})
⊂↓
L_{γ}(S^{1} × S^{n}) ⟶^{/S^{1}} L_{γ}(S^{1} × S^{n})/S^{1}
π↓  ^{π}↓
L_{γ_{0}}(S^{1}) ⟶^{/S^{1}}  one point

Diagram 1 where the left vertical maps are fibration, we find that

(10.5)  H_{k}^{S^{1}}(L_{γ}(S^{1} × S^{n})) ≅ H_{k}(L(S^{n}))

and the map I^{∗} : H_{k}^{S^{1}}(L_{γ}(S^{1} × S^{n})) → H_{k+1}(L_{γ}(S^{1} × S^{n})) is identifided with x ↦ [S^{1}] ⊗ x where we use the identification (10.4) and H_{∗}(L_{γ_{0}}(S^{1})) ≅ H_{∗}(S^{1}).

We now recall the calculation of homology group of loop space of S^{n}. (See [32] for detail.)

Let E(a, b, ⋯, c) be the free graded commutative graded algebra generated by a, b, ⋯, c. (Namely if all of a, b, ⋯, c are of even degree then E(a, b, ⋯, c) are polynomial algebra and if all of them are of odd degree then E(a, b, ⋯, c) is an exterior algebra.) We recall the following classical result. Let L_{0}(X) be the based loop space.

Lemma 10.4. If n is odd then H^{∗}(L_{0}(S^{n}); ℚ) ≅ E(x) with deg x = n − 1. If n is even then H^{∗}(L_{0}(S^{n}); ℚ) ≅ E(x, y) with deg x = n − 1, deg y = 2n − 2.

To caclucate the cohomology of free loop space L(S^{n}) we use the Leray-Serre spectral sequence associated to the fiberation L_{0}(S^{n}) → L(S^{n}) → S^{n}. Let [S^{n}] be the fundametal cohomology class of S^{n}. Then the E^{2} term of the spectral sequence is (ℚ[S^{n}] ⊗ ℚ[pt]) ⊗ H^{∗}(L_{0}(S^{n}); ℚ).

Lemma 10.5. The boundary operator of the spectral sequence (ℚ[S^{n}] ⊗ ℚ[pt]) ⊗ H^{∗}(L_{0}(S^{n}); ℚ) ⇒ H^{∗}(L(S^{n}); ℚ) is zero if n is odd and is given by

d(x ⊗ [pt]) = 0, d(y ⊗ [pt]) = 2x ⊗ [S^{n}].

if n is even.

Now we consider first the case when η_{L}(γ) = 2. Then [M̂(S^{1} × S^{n}; γ)] ∈ H_{n+1}(L(S^{1} × S^{n}); ℤ) corresponds to an element of H_{n}(L(S^{n}); ℚ). We can use Lemmas 10.4,10.5 to see that it corresponds to ℓ [pt] ⊗ [S^{n}] for some ℓ ∈ ℤ. We remark that the operations 𝔩_{k} depends only on the homology class of M̂(S^{1}×S^{n}; γ). Also it is sufficient to consider the case ℓ = 1. Hence we may assume that M̂(S^{1} × S^{n}; γ) = S^{1} × S^{n} and elements (s, x) ∈ S^{1} × S^{n} corresponds to the curve t ↦ (s + t, x) in S^{1} × S^{n}. Let dt be the one form on S^{1} and Ω be the volume form of S^{n}. Then a, b, c are the de Rham cohomology classes represented by dt, Ω and dt ∧ Ω respectively. Let us write 𝔪_{k} = ∑_{i=0}^{∞} T^{iλ_{1}}𝔪_{k,iγ}. Then, by definition, we have

⟨𝔪_{k,γ}(a,…, a), c⟩
= ± ∫_{x∈S^{n}} ∫_{t_{1}=0}^{1} ∫_{t_{2}=0}^{t_{1}} … ∫_{t_{k}=0}^{t_{k−1}} d(s + t_{1}) ∧… ∧ d(s + t_{k}) ∧ ds ∧ Ω
= ±1/k!.

This implies Theorem 10.1. □

We next consider the situation of Theorem 10.2. Then [M̂(S^{1} × S^{n}; γ)] ∈ H_{2n}(L(S^{1} × S^{n}); ℤ) corresponds to an element of H_{2n−1}(L(S^{n}); ℤ). We can use Lemmas 10.4,10.5 to see that it corresponds to ℓ x ⊗ [S^{n}] for some ℓ ∈ ℤ.

Lemma 10.6. ℓ = ±1.

Postponing the proof of Lemma 10.6 for a while let us complete the proof of Theorem 10.2.

Let P → L(S^{n}) be the cycle representing x. For z ∈ P, let μ_{z} : S^{1} → S^{n} be the curve represented by it. We consider the map ev : P × S^{1} → S^{n} × S^{n} defined by ev(z, t) = (μ_{z}(0), μ_{z}(t)). By the definition of the class x, we find easily that ev : P × S^{1} → S^{n} × S^{n} is of degree one.

Now by Lemma 10.6 [M̂(S^{1} × S^{n}; γ)] is homologous to the cycle represented by f : P × S^{1} → L(S^{1} × S^{n}) given by f(z, s)(t) = (s + t, μ_{z}(t))

Then, by definition, we have

∑_{i=0}^{k}⟨𝔪_{k+1,γ}(\overbrace{a,…, a}^{i}, b, \overbrace{a,…, a}^{k−i}), c⟩

= ± ∫_{(x,s)∈P ×S^{1}} ∫_{t=0}^{1} ∫_{t_{1}=0}^{1} ∫_{t_{2}=0}^{t_{1}} … ∫_{t_{k}=0}^{t_{k−1}}

d(s + t_{1}) ∧… ∧ d(s + t_{k}) ∧ dt ∧ ev^{∗}(Ω ∧ Ω)

= ±1/k!.

Theorem 10.2 follows. □

Proof. Let us prove Lemma 10.6. We can show (without using Lemma 10.6) that ⟨𝔪_{k,γ}(a,…, a, b), c⟩ = ±ℓ/k!. In particular 𝔪_{1,γ}(b) = ±ℓe. Since e should be on the image of 𝔪_{1} by Theorem 5.1, it follows that ℓ = ±1 as required. (We remark that since our Lagrangian submanifold is monotone we can work over integer. (See [15].) Hence we can prove not only ℓ ≠ 0 but also ℓ = ±1.) □

Remark 10.2. In Lemma 10.6 we proved ℓ = ±1. However the author does not know whether ℓ = 1 in case η_{L}(γ) = 2. The difference is as follows. Let η_{L}(γ) = 2. Then Theorem 10.1 implies that 𝔪_{1}(a) = ±ℓe mod T^{2λ_{1}}. We remark that e must be killed (over integer) by differentials of the spectral sequence. However this is not enough to show ℓ = ±1. If ℓ ≠ ±1 then e is not yet killed over integer by the differential H^{n−1}(L) → H^{n}(L). But there may be other differentials later on which kills e.

In the situation η_{L}(γ) = n+1, by degree reason there can be no other differential which kills e other than first one.

See revised version of [15] and [14] for more detail.

Remark 10.3. We remark that we still did not yet present the arugment to elliminate the possibility η_{L}(γ) = 2 − n when n is even. (We will do it in §14.)

Let us consider the situation of L ⊂ M such that η_{L}(γ) = 2 − n, L ≅ S^{1} × S^{n}. (It seems very likely that this actually happens in case M is a Calabi-Yau 3 fold and n = 2.)

In this case we have : [M̂(S^{1} × S^{n}; γ)] ∈ H_{1}(L_{γ}(S^{1} × S^{n})) ≅ ℤ. (Here γ ∈ π_{2}(M, L). We assume M̂(S^{1} × S^{n}; γ′) = ∅ for γ′ with 0 < E(γ′) < E(γ).) Let us suppose moreover that [M̂(S^{1} × S^{n}; γ)] is a generator. It follows that, to calculate 𝔩_{k}, we may assume that [M̂(S^{1} × S^{n}; γ)] ≅ S^{1} such that s ∈ S^{1} corresponds to the loop t ↦ (s + t, x_{0}) for some fixed x_{0}. Then we can calculate in a similar way as above and obtain

𝔩_{k+1}^{+}(a, ⋯, a, a) ≡ ±T^{λ_{1}} \frac{1}{k!} mod T^{2λ_{1}}.

and all other operations are zero modulo T^{2λ_{1}}. Let us suppose 𝔩_{k+1}^{+}(a, ⋯, a, a) = T^{λ_{1}}/k! for simplicity. Then 𝔩_{k}(a, ⋯, a) = T^{λ_{1}}b/k!.

We remark that 𝔩_{k} (k = 0, 1, ⋯) induces a coderivation d̂on E(H(S^{1}×S^{m})[1]) = E(a, b, c, e) in the same way as 𝔪_{k} induces a coderivation on B(H(L)). We then have, in our case,

(10.6)  d̂= ∑ T^{λ_{1}} \frac{1}{k!}b\frac{∂^{k}}{∂a^{k}}

In other words, d̂ is f(a, b, c, e) ↦ T^{λ_{1}}bf(a + 1, b, c, e). (Here we identify element of E(a, b, c, e) as a (Λ coefficient) polynomial of a, c tensored with an element of ℚ[1, b] ⊗ ℚ[1, e] ≅ E[b, e]. Note that a, c has even degree after shifted and b, e has odd degree after shifted.) From this calculation, we find easily that the cohomology of (E(H(S^{1} × S^{m})[1]), d̂) vanishes.

We remark that the calculation above means that we can not use the calculation of d̂ cohomolgy of E(H(S^{1} × S^{m})) to eliminate the possibility η_{L}(γ) = 2 − n, in the case L ⊂ ℂ^{n+1}. (Note in case L ⊂ ℂ^{n+1}, the d̂ cohomology of Lagrangian submanifold of ℂ^{n} should vanish. ) Actually we are going to use H(L(S^{1} × S^{m})) (which is closely related to B(H(S^{1} × S^{m}))) to eliminate the possibility η_{L}(γ) = 2 − n.

11. Lagrangian submanifold of ℂ^{3}.

We first state our main result on Lagrangian submanifold of ℂ^{3}. We recal that a 3 dimensional manifold L is said to be prime if L ≅ L_{1}♯L_{2} implies L_{1} ≅ S^{3} or L_{2} ≅ S^{3}. Here ♯ stands for connected sum and ≅ means diffeomorphism. Two Lagrangian immersions i_{0} : L → M and i_{1} : L → M are said to be exactly regular homotopic to each other if there exists a smooth family of Lagrangian immersions i_{t} : L → M connecting them such that E : π_{2}(M, i_{t}(L)) → ℝ is independent of t. (Here π_{2}(M, i_{t}(L)) ≅ π_{1}(L) is the set of homotopy class of pair of maps (f, g), f : S^{1} → L, g : D^{2} → M such that i_{t} ∘ f = g|_{∂D^{2}}.)

Theorem 11.1 ([14]). An oriented and connected prime 3 dimensional manifold L can be embeded to ℂ^{3} as a Lagranaian submanifold if and only if L is diffeomorphic to S^{1} × Σ_{g} where Σ_{g} is an oriented 2 dimensional manifold.

Moreover a Lagrangian immersion i : L = S^{1} × Σ_{g} → ℂ^{3} is exactly regular homotopic to an Lagrangian embedding if and only if there exists γ ∈ π_{1}(L) such that E(γ) > 0, η(γ) = 2.

Remark 11.1. For any oriented 3 manifold L, we have T L⊗ℂ ≅ ℂ^{3}. Hence Theorem 2.5 implies that L has a Lagrangian immersion i : L → ℂ^{3}. We may assume that L is transversal to itself. Hence applying Lagrangian surgery (Lalonde-Sikorav [24], Polterovich [30]), we can prove that there exists k such that L♯k(S^{1} × S^{2}) can be embeded as a Lagrangian submanifold of ℂ^{3}. (Here k(S^{1} × S^{2}) is a conneted sum of k copies of S^{1} × S^{2}.)

The following seems to be open yet.

Problem 11.1. Let L_{i} be prime oriented 3 manifolds which are not diffeomorphic to ℚ homology sphere or S^{1} × S^{2}. Let L = L_{1}♯ ⋯ ♯L_{k}, k ≥ 2. Can any such L be embeded to ℂ^{3} as a Lagrangian submanifold?

The fact that S^{1} × Σ_{g} can be embeded to ℂ^{3} as a Lagrangian submanifold follows from Theorem 2.6. We can also prove that if E(S^{1}) > 0 and η(S^{1}) = 2 for an Lagrangian immersion i : S^{1} × Σ_{g} → ℂ^{3} then it is exaxcly regular homotopic to an embedding, by carefully examining the proof of Theorem 2.6. So the main new part of the proof of Theorem 11.1 is the proof of “only if” part.

Let L ⊂ ℂ^{3} be an embeded Lagrangian submanifold. We assume that L is oriented and prime. By Theorem 2.1 H_{1}(L; ℚ) ≠ 0. Hence by a well known result of 3 manifold topology (see for example [22]), L is diffeomorphic either to S^{1} × S^{2} or an aspherical manifold. Here a manifold L is said to be aspherical if π_{k}(L) = 0 for k ≥ 2. We can generalize Theorem 11.1 in both cases to higher dimensions. We will discuss it in next two sections together with a sketch of their proofs.

12. Aspherical Lagrangian submanifold.

In this section we consider an aspherical Lagrangian submanifold L of a symplectic manifold M of arbitrary dimension. We assume M is convex at infinity in case M is noncompact. To prove Theorem 11.1 and its generalization in the case of aspherical Lagrangian submanifold L we are going to use the moduli space N (R, β) introduced in §3. To study this moduli space we need to use an assumption similar to Assumption 3.1 in §3. We state it below. Let H : M × [0, 1] → ℝ be a smooth function of compact support. We put H_{t}(x) = H(x, t), then H_{t} : M → ℝ is a smooth function of compact support. We denote by X_{H_{t}} the Hamiltonian vector field generated by H_{t}. Namely dH_{t} = i_{X_{H_{t}}} (ω). Let exp_{t}^{X_{H}} : M → M be a one parameter group of transformations associated with X_{H}. Namely

\frac{∂}{∂t} exp_{t}^{X_{H}} (x)|_{t=t_{0}} = X_{H_{t_{0}}} (exp_{t_{0}}^{X_{H}} (x)).

exp_{t}^{X_{H}} is a symplectic diffeomorphism for each t.

Assumption 12.1. exp_{1}^{X_{H}} (L) ∩ L = ∅.

Theorem 12.1. Let L ⊂ M be a Lagrangian submanifold. We assume that L is relatively spin and aspherical. We also assume Assumption 12.1. Then there exists β ∈ π_{2}(M, L) with the following properties.

(1) E(β) > 0.

(2) η_{L}(β) = 2.

(3) ∂β ∈ π_{1}(L) is nonzero. Its centralizer Z_{∂β} = {γ ∈ π_{1}(L)|γ (∂β) = (∂β) γ} is of finite index in π_{1}(L).

Remark 12.1. In case L = T^{n} ⊂ ℂP^{n} existence of β ∈ π_{2}(ℂP^{n}, L) with η_{L}(β) = 2 was conjectured by M.Audin and is independently proved by Eliashberg-Givental-Hofer [8] (the result they claimed is actually more general) and by K. Cielieback also.

Remark 12.2. (A) Let us assume that c^{1}∩ : π_{1}(M) → ℤ is zero in Theorem 12.1. Then η_{L} induces a homomorphism η_{L} : π_{1}(L) → ℤ. We now have an exact sequence

(12.1)  0 → (Ker η_{L}) ∩ Z_{∂β} → Z_{∂β} ⟶^{η_{L}/2} ℤ → 0.

(We remark that the image of η_{L} is even since L is orientable.) Therefore Z_{∂β} ≅ ℤ × ((Ker η_{L}) ∩ Z_{∂β}) by (2). It follows that the finite covering space L̂ of L with π_{1}(L̂) = Z_{∂β} is homotopy equivalent to S^{1} × L′ for a K(Z_{∂β}/ℤ, 1) space L′.

(B) Under the assumption of Theorem 12.1, the finite covering space L̂ of L with π_{1}(L̂) = Z_{∂β} is homotopy equivalent to an S^{1} bundle over L′ for a K(Z_{∂β}/ℤ, 1) space L′. If the image of c^{1}∩ : π_{1}(M) → ℤ is mℤ, we can show that the Euler class of this S^{1} bundle is divisible by m in a similar way.

Let us consider the situation of Theorem 11.1. As we remarked before either L ≅ S^{1} × S^{2} or L is aspherical. We discuss the first case in the next section. So we may assume that L satisfies the assumption of Theorem 12.1. (We remark that any oriented 3 manifold is spin.) Moreover c^{1}(ℂ^{3}) = 0. Hence we have Z_{∂β} ≅ ℤ × ((Ker η_{L}) ∩ Z_{∂β}) as in Remark 12.2 (A). Using a standard result of 3 dimensional topology (see [22]), (we remark that L is sufficiently large in the sense of Waldhausen, since H_{1}(L; ℚ) ≠ 0), we can prove that L̂ is diffeomorphic to Σ_{g} × S^{1}. Let G = π_{1}(L)/π_{1}(L̂). It acts freely on L̂ and L̂/G = L. Since G = Ker η_{L}/((Ker η_{L}) ∩ Z_{∂β}) it follows that G acts freely on Σ_{g} and trivially on the S^{1} factor. Hence Σ_{g}/G is again a Rieman surface. We can use this fact and Aut(π_{1}(Σ_{g})) ⊂ P SL(2; ℝ) for g ≥ 2 to show that actually Z_{∂β} = π_{1}(L). Namely L is diffeomorphic to S^{1} × Σ_{g}. This proves Theorem 11.1 except the case L = S^{1} × S^{2}. □

We now sketch the proof of Theorem 12.1. We consider a map φ = φ(τ, t) : ℝ × [0, 1] → M with the following properties. Here χ_{R} is as in Definition 3.2.

\frac{∂φ}{∂τ}(τ, t) = J (\frac{∂φ}{∂t}(τ, t) − χ_{R}(τ)X_{H_{t}}(φ(τ, t))) , (12.2a)

φ(τ, 0), φ(τ, 1) ∈ L, (12.2b)

∫_{ℝ×[0,1]} φ^{∗}ω < ∞. (12.2c)

(Compare Definition 3.6.) We denote by N (R) the set of all such φ. In the same way as §3, we can extend φ to a map \overline{φ} : (D^{2}, ∂D^{2}) → (M, L). Let N (R; β) be the set of all φ ∈ N (R) such that the homotopy class of \overline{φ} is β. Element of N (R; β) may be regarded as a map (D^{2}, ∂D^{2}) → (M, L). We define a map ev : N (R; β) → L(L) by ev(φ) = φ|_{∂D^{2}}. We put

N (β) = ⋃_{R∈[0,∞)} N (R; β) × {R}

and define ev : N (β) → L(L) in an obvious way. We remark

(12.3)  dim N (β) = n + 1 + η_{L}(β).

Definition 12.1. We define 𝔅(L, H) by

𝔅(L, H) = ∑_{β} T^{β∩ω}ev_{∗}[N (β)].

Here we may regard ev_{∗}[N (β)] as an approximate de Rham chain of L(L) in a similar way to §6. We can prove an analogue of Lemma 4.4. Together with Gromov compactness it implies the following.

Lemma 12.2. There exists C such that if β ∩ ω < −C then N (β) is empty.

Moreove, for any C, there exists only a finite number of β such that β ∩ ω < C and N (β) ≠ ∅.

Lemma 12.2 implies that

𝔅(L, H) ∈ S^{D}(L) ˆ⊗Λ,

where ˆ⊗ means the completion of the algebraic tensor product. Now the main point of the proof of Theorem 12.1 is the following equality.

Theorem 12.3. Let L ⊂ M be a Lagrangian submanifold. We assume that L is relatively spin and Assumption 12.1. Then we have

(12.4)  ∂𝔅(L, H) + {α(L), 𝔅(L, H)} = [L].

Here we embed L → L(L) as the set of constant maps. Hence [L] ∈ S_{n}^{D}(L(L)). α(L) is defined in Definition 8.2.

Remark 12.3. We remark that we do not assume that L is aspherical in Theorem 12.3.

Proof. (Sketch) We are going to study the boundary of 𝔅(L, H). Let φ_{i} ∈ N (R_{i}; β) be a divergent sequence. In a way similar to the proof of Proposition 3.3, we can show that R_{i} is bounded. (This is the place we use Assumption 12.1.) There exists p_{i} = (τ_{i}, t_{i}) ∈ ℝ × [0, 1] such that

(12.5)  |d_{p_{i}}φ_{i}| = sup{|d_{x}φ_{i}||x ∈ ℝ × [0, 1]}.

We then consider three cases separately. (Compare the proof of Lemma 3.6.) Case 1: |d_{p_{i}}φ_{i}| = D_{i} diverges. D_{i}dist(p_{i}, ∂(ℝ × [0, 1])) = C_{i} → ∞. Case 2 : |d_{p_{i}}φ_{i}| = D_{i} diverges. D_{i}dist(p_{i}, ∂(ℝ × [0, 1])) is bounded. Case 3 : |d_{p_{i}}φ_{i}| = D_{i} is bounded. |τ_{i}| diverges. Case 4 : R_{i} → 0.

Case 1 is a sphere bubble. Hence it happens in codimension two. So it does not contribute to ∂𝔅(L, H). (We can make this argument rigorous in the same way as [16].)

Cases 2 and 3 correspond to a bubble at the boundary ∂D^{2}. We can show that this gives the term {α(L), 𝔅(L, H)}.

Let us consider Case 4. We remark that the equation (12.2a) becomes pseudoholomorphicity for R = 0. Therefore, the limit lim_{i→0} φ_{i} will give a pseudoholomorphic map φ : (D^{2}, ∂D^{2}) → (M, L). The moduli space of such maps has an extra symmetry {g ∈ P SL(2; ℝ)|g(1) = 1} = Aut(D^{1}, 1). The action of this group is nontrivial if β ≠ 0. So in that case the contribution of Case 4 is zero as a chain. In case β = 0 we have [L].

We thus obtain the fomula (12.4). The detail will be in [14]. □

To apply Theorem 12.3 to the proof of Theorem 12.1, we use a series of lemmas of purely topological nature. For γ ∈ π_{1}(L), let L_{[γ]}(L) ⊂ L(L) be the set of all loops in the free homotopy class of γ. (Here [γ] stands for the free homotopy class of γ.) Let Z_{γ} ⊂ π_{1}(L) be the centralizer. Let L̂_{γ} be the covering space of L corresponding to Z_{γ}.

Lemma 12.4. If γ ∈ π_{1}(L) is nonzero, then the natural projection induces a homeomorphism : π_{∗} : L_{[γ]}(L̂_{γ}) → L_{[γ]}(L)

Proof. Let ℓ : S^{1} → L be a loop in L_{[γ]}(L). Let p_{0} ∈ L be the base point. We can choose a path m : [0, 1] → L joining p_{0} to ℓ(1) such that m^{−1} ∘ ℓ ∘ m is homotopic to γ. It then lifts to a loop m̃^{−1} ∘ ℓ̃∘ m̃ in L_{[γ]}(L̂_{γ}). It is easy to see that π_{∗}(ℓ̃) = ℓ. Hence π_{∗} is surjective.

We next assume π_{∗}(ℓ̃_{1}) = π_{∗}(ℓ̃_{2}) = ℓ and ℓ̃_{1}, ℓ̃_{2} ∈ L_{[γ]}(L̂_{γ}). There exists g ∈ G = π_{1}(L)/Z_{γ} such that g · ℓ̃_{1} = ℓ̃_{2}. We remark that both ℓ̃_{1} and ℓ̃_{2} are in the free homotopy class of γ. Since γ is in the center of π_{1}(L_{[γ]}) it follows that g^{−1}γg = γ. Since g ∈ G = π_{1}(L)/Z_{γ}, it follows g = 1. Namely π_{∗} is injective. □

Lemma 12.5. Let L is a K(π, 1) space and γ ∈ π_{1}(L) = π, then ev : L_{[γ]}(L̂_{γ}) → L̂_{γ} is a homotopy equivalence.

Proof. If γ = 1, then L̂_{γ} = L̂_{1} = L. Let L̃ be the universal covering space of L. Then L_{[1]}(L) = L_{[1]}(L̃)/π. Since L̃ is contractible it follows that L_{[1]}(L̃) is contractible. Hence ev : L_{[1]}(L) → L is a homotopy equivalence.

Next we assume γ ≠ 1. We put g_{γ} = γ ∈ π, which acts on L̃. We put X = {ℓ : ℝ → L̃ | ℓ(t + 1) = gℓ(t)}. Z_{γ} acts on X freely and X/Z_{γ} = L_{[γ]}(L̂_{γ}). Hence it suffices to show that X is contractible.

We consider a map ev_{0} : X → L̃, ℓ ↦ ℓ(0). It is a fibration. Hence X is homotopy equivalent to the fiber of ev_{0}. The fiber can be identified to the space of path ℓ′ : [0, 1] → L̃ joining p̃_{0} with gp̃_{0} and hence is contractible. □

Lemma 12.6. Let L is an n dimensional aspherical manifold and γ ∈ π_{1}(L). Then H_{k}(L_{[γ]}(L); ℤ) = 0 for k ∉ {0, ⋯, n}. Moreover, if H_{n}(L_{[γ]}(L); ℤ) ≠ 0 then Z_{γ} is of finite index in π_{1}(L).

Proof. H_{k}(L_{[γ]}(L); ℤ) ≅ H_{k}(L̂_{γ}; ℤ) by Lemmas 12.4,12.5. Since L̂_{γ} is a covering space of n dimensional manifold L it follows that H_{k}(L̂_{γ}; ℤ) = 0 for k ∉ {0, ⋯, n}. If H_{n}(L̂_{γ}; ℤ) ≠ 0, it follows that L̂_{γ} is compact. Hence Z_{γ} is of finite index in π_{1}(L). □

Lemma 12.7. Let L be an n dimensional compact oriented aspherical manifold and [L] ∈ H_{n}(L_{0}(L); ℚ). Then for any [P] ∈ H_{∗}(L(L); ℚ) we have {[L], [P]} = 0.

Proof. For p ∈ L let ℓ_{p} be the constant loop at p. For x ∈ P let ℓ_{x} be the loop corresponding to it. Now [P] ∗ [L] is supported at

P ∗ L = {(p, x, t) ∈ L × P × S^{1}|ℓ_{p}(t) = ℓ_{x}(0)}.

If (p, x, t) ∈ P ∗ L then (p, x, t′) ∈ P ∗ L for any t′. Moreover ∗ : P ∗ L → L(L) sends (p, x, t) to the same loop as (p, x, t′). This implies [P] ∗ [L] = 0.

On the other hand, [L] ∗ [P] is supported at

L ∗ P = {(x, p, t) ∈ L × P × S^{1}|ℓ_{x}(t) = ℓ_{p}(0)}.

If (p, x, t) ∈ L ∗ P then (p, x, t′) ∈ L ∗ P for any t′. Moreover ∗(x, p, t) is different from ∗(x, p, t′) only by the parametrization. This implies [L] ∗ [P] = 0. □

Now we go back to the (sketch of the) proof of Theorem 12.1. By (12.4) we have β ∈ π_{2}(M) such that

{M(L; β), N (−β)} ≠ 0.

We remark that dim M(L; β) = n − 2 + η_{L}(β), N (−β) = n + 1 − η_{L}(β). Hence, by Lemma 12.6, n − 2 + η_{L}(β), n + 1 − η_{L}(β) ∈ {0, ⋯, n}. Since η_{L}(β) is even, this implies η_{L}(β) = 2. Then since dim M(L; β) = n and is nonzero, it follows from Lemma 12.6 that the centralizer Z_{γ} of γ = ∂β ∈ π_{1}(L) is of finite index in π_{1}(L). This implies Theorem 12.1. □

Remark 12.4. Actually there is one point we need a clearification in the proof above. Namely the chain N (−β) is not necessary a cycle. So we need to work in the chain level. So the way to apply Lemma 12.6 is not so clear. We can overcome this trouble in a way similar to Remark 8.3 by using a theorem that any L_{∞} algebra is homotopy equivalent to an L_{∞} algebra defined on its homology group. (We use also Lemma 12.7.) The detail will appear in [14].

Remark 12.5. Loop space homology was used in [39] in a related context. To find a relation of Floer homology to [39] was one of the motivations of the author to modify the construction of [15] to ones described in §§6,7,8,9. In [39] and in [38], Viterbo used closed geodesic. Closed geodesic appears also in the study of Lagrangian submanifold using contact homology and in the approach by Eliashberg-Givental-Hofer and Cielieback mentioned in Remark 12.1 using [8]. Closed geodesic is closely related to the homology of loop space. In a sense, our approach is a topological version of one using closed geodesic.

It seems possible to describe relation of Floer homology of Lagrangian submanifold to [39] and to [8], using the ideas developped in §§6,7,8,9. We will discuss it elsewhere.

13. Lagrangian submanifold homotopy equivalent to S^{1} × S^{2m}.

Theorem 13.1. Let L ⊂ M be a Lagrangian submanifold. We assume that L is homotopy equivalent to S^{1} × S^{2m}. We also assume Assumption 12.1. Then there exists β ∈ π_{2}(M, L) such that E(β) > 0, η_{L}(β) = 2, ∂β ∈ π_{1}(L) is a generator^{5}.

Proof. (Sketch) (We need to apply the same remark as Remark 12.4, to make the argument below precise.)

We put n = 2m. Lemma 5.4 (and its proof) implies that, if the theorem is false, then there exists β ∈ π_{2}(M, L) with M(L; β) ≠ 0, η_{L}(β) = 2 − n. Moreover by Theorem 12.3, we may assume

(13.1)  {M(L; β), N (−β)} = [L].

Then dim N (−β) = 2n. Let γ = ∂β. We remark that [N (−β)] ∈ H_{n}(L_{γ}(S^{1} ×S^{n})). Hence [N (−β)] is either of the form [pt] ⊗ a or [S^{1}] × a′ where H(L_{γ}(S^{1} × S^{n})) ≅ H(L_{γ}(S^{1})) ⊗ H(L(S^{2n})) ≅ H(S^{1}) ⊗ H(L(S^{n})).

By Lemma 10.5, x⊗[S^{n}] in the E_{2} term of the spectral sequence does not survive. (Note deg(x ⊗ [S^{n}]) = deg a′.) Using this fact, we can prove that [N (−β)] lies in the image of H(S^{1}) ⊗ H(L_{0}(S^{n})). Here L_{0}(S^{n}) denotes the based loop space.

We define ev_{0} : L_{γ}(S^{1} × S^{n}) → S^{1} × S^{n} by ev_{0}(ℓ) = ℓ(0). Then by definition ev_{0}({P, Q}) ⊆ ev_{0}(P) ∪ ev_{0}(Q). Moreover the image of ev_{0} of elements of H(S^{1}) ⊗ H(L_{0}(S^{n})) is on S^{1} × {p_{0}} where p_{0} ∈ S^{n} is the base point. Furthermore dim M(L; β) = 0.

Hence the support of ev({M(L; β), N (−β)}) is contained in a one dimensional space. On the other hand, ev_{0∗}[L] = [L]. This contradicts to (13.1). □

5The author thanks to Prof. A. Kono who provides informations on the homology of free loop space useful to prove this Theorem.

14. Lagrangian submanifold of ℂP^{n}.

As we mentioned in Remark 8.4 Viterbo, (Eliashberg-Givental-Hofer, Cieliback) proved that if L admits a metric of negative curvature it can not be embeded to ℂP^{n} as a Larangian submanifold. Theorem 12.1 does not imply this result even in spin case since Assumption 12.1 may not be satisfied. We however have an alternative argument which implies Viterbo’s result in spin case.

Theorem 14.1. Let L be a Lagrangian submanifold of ℂP^{n}. We assume that L is asperical and spin. Then there exists β ∈ π_{2}(ℂP^{n}, L) with the following properties.

(1) E(β) > 0.

(2) η_{L}(β) = 2.

(3) ∂β ∈ π_{1}(L) is nonzero. Its centralizer Z_{∂β} = {γ ∈ π_{1}(L)|γ (∂β) = (∂β) γ} is of finite index in π_{1}(L).

Remark 14.1. There are also results by Seidel [34], Biran [4] etc. on Lagrangian submanifold of ℂP^{n} etc. The Lagrangian submanifolds discussed in those works seems to be opposite extreme to one in Theorem 14.1. Namely in those works Lagrangian submanifolds with small fundamental group or homology group are mainly studyed, while main targent of Theorem 14.1 is one with huge fundamental group.

Proof. (Sketch) We are going to construct an S^{1} equivariant chain \overline{𝔅}(L) ∈ S^{D}(L(L)) which has a similar property as (12.4). We fix p_{0} ∈ ℂP^{n} \ L. For each β ∈ π_{2}(ℂP^{n}, L), we consider the moduli space of maps φ : D^{2} → ℂP^{n} with the following properties.

(1) φ is holomorphic.

(2) φ(∂D^{2}) ⊂ L.

(3) The homotopy type of φ is β.

(4) φ(0) = p_{0}.

(Compare Definition 3.1.) Let N′(L; β) be the space of all such φ. We consider Aut(D^{2}, 0) = {g ∈ P SL(2; ℝ)|g(0) = 0} ≅ S^{1}. It acts on N′(L; β) in an obvious way. φ ↦ φ|_{S^{1}} defines an S^{1} equivariant map ev : N′(L; β) → L(L).

Hereafter we denote by S_{S^{1}}^{D}(L(L)) the set of all S^{1} equivariant de Rham chains in L(L).

We can use an argument similar to §6 and may regard ev_{∗}[N′(L; β)] ∈ S_{S^{1}}^{D}(L(L)).

Definition 14.1. We define \overline{𝔅}(L) ∈ S_{S^{1}}^{D}(L(L)) ˆ⊗Λ by

\overline{𝔅}(L) = ∑_{β} T^{β∩ω}ev_{∗}[N′(β)].

In §9 we regarded M(L; β) as an S^{1} equivariant approximate de Rham chain of L(L) of degree n − 3 + η_{L}(β) and used it to define \overline{α}(L) = ∑_{i} T^{E(β)}M(L; β_{i}) ∈ S_{S^{1}}^{D}(L(L)).

Theorem 14.2. We normalize our symplectic form ω so that ω ∩ S^{2} = 1 for the generator [S^{2}] ∈ H_{2}(ℂP^{n}; ℤ). We then have

∂\overline{𝔅}(L) + \{\overline{α}(L), \overline{𝔅}(L)\} ≡ [L] mod T^{2}.

Proof. (Sketch) We consider a divergent series of elements φ_{i} of N′(L; β). Then, in the limit, one of the following occurs.

(1) A bubble occurs at the boundary.

(2) A bubble occurs at interior.

(1) gives the term \{\overline{α}(L), \overline{𝔅}(L)\}.

In general (2) is a phenomenon of codimension 2 and do not contribute to our formula. However there is an exception. Namely φ_{i} may converge to a union of trivial disk φ_{0} : D^{2} → ℂ^{2} that is a constant map to p ∈ L, and a pseudoholomorphic sphere φ : S^{2} → ℂP^{n} such that φ(∞) = p_{0} and φ(0) = p = φ_{0}(D^{2}).

The reason that this is codimension one phenomenon is that the map φ_{0} ∨ φ : D^{2} ∨ S^{2} → ℂP^{n} is not stable, since its group of automorphism is S^{1}. We can analyse the neighborhood of this map in the moduli space and can show that they correspond to the boundary point of the moduli space N′(L; β). (We remark that in order such phenomenon to occur, ∂β must be zero.) We are interested in the case when φ ∩ ω = 1. It is easy to see that, for each p, there exists exactly one such φ. (This is because there exists exactly one rational curve of degree one containing p and ∞.) Hence this gives [L]. Theorem 14.2 follows. (More detail of the proof will be in the revised version of [15] and in [14].) □

Now we can use Theorem 14.2 in place of Theorem 12.3 and prove Theorem 14.1 in the same way as Theorem 12.1. We also replace Lemma 12.6 by the following :

Lemma 14.3. Let L is an n dimensional aspherical manifold and γ ∈ π_{1}(L). If x ∈ H_{k}^{S^{1}}(L_{[γ]}(L); ℤ) for k ∉ {0, ⋯, n − 1}, then {x, y} = 0 for any y. Moreover, if x ∈ H_{n−1}^{S^{1}}(L_{[γ]}(L); ℤ) and {x, y} ≠ 0 for some y, then Z_{γ} is of finite index in π_{1}(L).

Lemma 14.3 follows immediately from Lemma 12.6 and (9.2). □

Remark 14.2. We remark that we do not assume L to be aspherical in Theorem 14.2. So it can be used to study Lagrangian submanifold L of ℂP^{n} for more general L. For example the case L = S^{1} × S^{n} can be studyed in a way similar to §13. The case when L is rational homology sphere is also of interest since Gromov’s theorem 2.1 is not generalized directly to a Lagrangian submanifold of ℂP^{n}.

One may also study Lagrangian submanifold of more general symplectic manifold M than ℂP^{n}. For example the case when M is uniruled may be handled in a similar way.

The author is planning to explore these points elsewhere.

References

[1] V.I. Arnol’d, and A.V. Givental. Symplectic Geometry. in “Dynamical Systems IV” Encyclopaedia of Mathematical Sciences, ed. by V.I. Arnold and S.P. Novikov, Springer-Verlag, 1985.

[2] M.Audin, F.Lalonde, and L.Polterovich. Symplectic rigidity: Lagrangian submanifolds. in “Holomorphic curves in symplectic geometry”, 271–321, Progr. Math., 117, Birkhäser, Basel, 1994.

[3] J.F. Barraud and O.Cornea. Lagrangian Intersections and the Serre Spectral Sequence. math.DG/0401094

[4] P. Biran. Lagrangian non-intersection SG/0412110

[5] A.S. Cattaneo J. Fröhlich, and P.B. Jürg. Topological field theory interpretation of string topology. Comm. Math. Phys. 240 (2003), no. 3, 397–421.

[6] M.Chas and D.Sullivan. String Topology. preprint, math.GT/9911159.

[7] K.-T.Chen. Iterated integrals of differential forms and loop space homology. Ann. of Math. (2) 97 (1973), 217–246.

[8] Y.Eliashberg. A.Givental, and H.Hofer. Introduction to symplectic field theory. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part II, 560–673

[9] Y.Eliashberg, and M.Gromov. Convex symplectic manifolds. in “Several complex variables and complex geometry, Part 2,” 135–162, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.

[10] Y.Eliashberg, and N.Mishachev. Introduction to the h-principle. Graduate Studies in Mathematics, 48. American Mathematical Society, Providence, RI, 2002.

[11] A. Floer Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.

[12] K. Fukaya. Floer homology and mirror symmetry I. in “Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds,” 15–43, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI, 2001.

[13] K. Fukaya. Deformation theory, homological algebra and mirror symmetry. in “Geometry and physics of branes,” Ser. High Energy Phys. Cosmol. Gravit., IOP, Bristol, 2003.

[14] K.Fukaya. in preparation.

[15] K.Fukaya, Y.G.Oh, H.Ohta, and K.Ono. Langrangian intersection Floer theory -anomaly and obstruction-. preprint, http://www.kusm.kyoto-u.ac.jp/∼fukaya/ fukaya.html, 2000.

[16] K.Fukaya and K.Ono. Arnold conjecture and Gromov-Witten invariants. Topology, 38:933 – 1048, 1999.

[17] K.Fukaya and K.Ono. Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds—summary. Taniguchi Conference on Mathematics Nara ’98, 75–91, Adv. Stud. Pure Math., 31, Math. Soc. Japan, Tokyo, 2001.

[18] M.Gerstenhaber. The cohomology structure of an associative ring. Ann. of Math. (2) 78 1963 267–288.

[19] T.G.Goodwillie. Cyclic homology, derivations, and the free loopspace. Topology 24 (1985), no. 2, 187–215.

[20] M.Gromov. Pseudoholomorphic curves in symplectic manifolds. Invent. Math., 82(2):307– 347, 1985.

[21] M.Gromov. Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. Springer-Verlag, Berlin, 1986. x+363 pp.

[22] J.Hempel 3-Manifolds. Ann. of Math. Studies, No. 86. Princeton University Press, Princeton, N. J., 1976

[23] T.V.Kadeishvili. The algebraic structure in the homology of an A(∞)-algebra. Soobshch. Akad. Nauk Gruzin. SSR, 108(2):249–252 (1983), 1982.

[24] F.Lalonde, and J.-C.Sikorav. Sous-vari 師市 lagrangiennes et lagrangiennes exactes des fibr 市 cotangents. Comment. Math. Helv. 66 (1991), no. 1, 18–33.

[25] J.A.Lees. On the classification of Lagrange immersions. Duke Math. J. 43 (1976), no. 2, 217–224.

[26] Y-G.Oh. Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions. Comm. Pure Appl. Math. 45 (1992), no. 1, 121–139.

[27] Y-G.Oh. Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. Comm. Pure Appl. Math. 46 (1993), no. 7, 949–993.

[28] Y-G.Oh. Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings. Internat. Math. Res. Notices 1996, no. 7, 305–346.

[29] H.Ohta. Obstraction to and deformation of Lagrangian intersection Floer cohomology. In “Symplectic Geometry and Mirror Symmetry”. World Sci. Press, 2001.

[30] L.Polterovich. The surgery of Lagrange submanifolds. Geom. Funct. Anal. 1 (1991), no. 2, 198–210.

[31] L.Polterovich. Monotone Lagrange submanifolds of linear spaces and the Maslov class in cotangent bundles. Math. Z. 207 (1991), no. 2, 217–222.

[32] M.Vigué-Poirrier and D.Sullivan The homology theory of the closed geodesic problem. J. Differential Geometry 11 (1976), no. 4, 633–644.

[33] Y.Ruan, Virtual neighborhoods and pseudo-holomorphic curves. Proceedings of 6th G 嗅 ova Geometry-Topology Conference. Turkish J. Math. 23 (1999), no. 1, 161–231.

[34] P.Seidel, Graded Lagrangian submanifolds. Bull. Soc. Math. France, 128 (2000), 103–149.

[35] V.D.Silva Products on symplectic Floer homology. Thesis, Oxford Uinv. 1997.

[36] J.Stasheff. Homotopy associativity of H-spaces. I, II. Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 1963 293–312.

[37] M.Sugawara. A condition that a space is group-like. Math. J. Okayama Univ. 7 1957 123–149.

[38] C.Viterbo. A new obstruction to embedding Lagrangian tori. Invent. Math. 100 (1990), no. 2, 301–320.

[39] C.Viterbo. Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces. J. Differential Geom. 47 (1997), 420–468.

Department of Math. Facult of Science, Kyoto University

   E-mail address: fukaya@math.kyoto-u.ac.jp