FLOER HOMOLOGY AND MIRROR SYMMETRY II

Kenji FUKAYA

Abstract. This is the second part of a series of articles explaining applications of Floer homology to mirror symmetry and D-brane. This article is independent of part I [Fu9]. We will associate an A_{∞} category to a symplectic manifold. This is an improved version of previous ones [Fu1], [Fu4] in which there were some flaw. The correction is based on a book [FOOO] written jointly with Oh,Ohta,Ono. While correcting the flaw, we find various interesting new phenomena which are related to mirror symmetry.

We also discuss homological algebra of A_{∞} category in this article.

This article is a survey article. So most of the material written here are minor modifications of the results which already known to somebody. However it is rather hard to find a reference of them.

Table of contents

§0 Introduction

Chapter 1 : Floer homology and A_{∞} category.

§1 A_{∞} category and filtered A_{∞} category. §2 Floer homology and A_{∞} category I - the set of objects - §3 Floer homology and A_{∞} category II - the operator 𝔪_{k} - the transversal case- §4 Floer homology and A_{∞} category III - the operator 𝔪_{k} - the general case - §5 Unit and homotopy unit.

Chapter 2 : Homological algebra of A_{∞} category.

§6 Twisted complex and derived A_{∞} category. §7 A_{∞} functor and natural transformations. §8 Homotopy equivalence and A_{∞} analogue of J.H.C.Whitehead theorem. §9 An A_{∞} analogue of Yoneda’s lemma.

Key words and phrases. Symplectic geometry, Floer homology, mirror symmetry, D-brane, A_{∞} algebra.

Partially supported by Grant in-Aid for Scientific Research 13852001.

Typeset by AMS-TEX

0. Introduction.

This is the second part of a series of articles, describing a project in progress to study homological mirror symmetry [Ko1], [Ko2] and D-brane using Floer homology of Lagrangian submanifolds. See [Fu5], [Fu6], [Fu7], [FOOO], [Ko1], [Ko2], [KoS1], [Ot], [PZ], [Se2], [Se3], [ST], [SYZ] for related or different aspects of homological mirror symmetry. Though this is the second part, it is independent of the first part [Fu9]. In this second part, we focus the A_{∞} category constructed by using Floer homology of Lagrangian submanifolds, and homological algebra of A_{∞} categories. In this sense, this article is an updated version of author’s previous papers [Fu1], [Fu4]. Time has passed after [Fu1], [Fu4] were written. During those period, we made several progress, some of which are explained in this article.

Among the points where the contents of this article overlaps with [Fu4], there are three points where we improve the constructions there. One is that we removed an assumption in [Fu4], that all Lagrangian submanifolds are monotone and have minimum Maslov index ≥ 3. (See §2 for the definition of Maslov index.) We assumed it in [Fu4] because we used results by Oh [Oh1] to define Floer homology of Lagrangian submanifolds. This assumption is not extremely restrictive for the purpose of [Fu4], that is to study relative Floer homology of 3 manifolds with boundary. However, for the purpose of this article, that is to study mirror symmetry, this assumption is rather restrictive. To study mirror symmetry, the case of Lagrangian submanifolds in Calabi-Yau 3 fold with Maslov index 0 is the most important. Such Lagrangian submanifolds are monotone but do not satisfy the condition that minimal Maslov index ≥ 3.

To define Floer homology of non monotone Lagrangian submanifolds or to remove the condition that minimal Maslov index ≥ 3, we need the obstruction theory developed in [FOOO]. The summary of a part of it is included in this article. (See [Ot] for another summary of [FOOO].) We generalize the construction of A_{∞} category explained in [Fu1], [Fu4] to more general situation using the idea of [FOOO].

The second point is that we put precise sign in each formula. In the side of geometry, this requires us to describe orientations of the moduli spaces of pseudoholomorphic discs involved in the construction. Actually the argument we need to do so is already developed in detail in [FOOO] Chapter 6^{1}. There, the case when only one or two Lagrangian submanifolds appear, are studied. But the method there can be generalized easily to the present situation where more than two Lagrangian submanifolds appear at the same time.

We need to study sign also in the algebraic construction involved. Various constructions on homological (homotopical) algebra of A_{∞} category is developed in [Fu4] Part II. There we worked over ℤ_{2} coefficient. In this article, we give precise sign to the discussion there. The sign in the study of A_{∞} category is rather hard and is not at all a trivial matter. Actually the author was unable to work over ℤ coefficient in [Fu4] Part II, because he could not find correct sign when he was writing [Fu4]. To fix the sign we develop more systematic way to write the formulas in [Fu4] Part II. As a consequence, the description of this article is considerably simplified compared to one in [Fu4] Part II.

1When we quote [FOOO] in this article, we refer its preprint version which was completed in 2000 December and can be obtained from the author’s home page http:www.kusm.kyotou.ac.jp/∼fukaya/fukaya.html at the time of writing this article. We are now adding more materials to it and there will be some change of the order of the chapters in the final version.

Thirdly we solved in [FOOO] the trouble related to the existence of the identity element in the A_{∞} category of Lagrangian submanifold, which was discussed in [Fu4] §13 in unsatisfactory way. In [FOOO] §20 we discussed the problem of identity in the A_{∞} algebra we associate to a Lagrangian submanifold. As a consequence of [FOOO] §20 and §5 of the present article, we have an A_{∞} category with identity. The identity element plays a central role in the proof of Yoneda’s lemma which we prove in §9 of present paper, refining the proof given in [Fu4] §12.

Now the outline of each sections are in order. In §1, we introduce the notions of A_{∞} category and filtered A_{∞} category. The other sections of Chapter 1 is devoted to the construction of its example. Namely we associate a filtered A_{∞} category to each symplectic manifold.

In §2, we describe an objects of this filtered A_{∞} category. The object is roughly speaking a pair of a Lagrangian submanifold and a flat U(1) bundle on it. But we need to add some additional topological data. The main point of this section is to describe precisely the additional data we add and explain the reason we need it. The additional data is relative spin structure (which is related to the orientation or sign), and the grading, (which is related to the degree). We follow [FOOO] Chapter 6 on the first point and follow Kontsevich, Seidel [Se1] on the second point.

§3,4 are devoted to the definition of the module of morphisms and operations to define our filtered A_{∞} category. To define an operator we need to consider two cases separately. The first case is when the Lagrangian submanifolds involved are mutually transversal. This case was discussed in [Fu1], [Fu4]. We discuss this case in §3.

In §4 we discuss the general case, namely the case when we do not assume transversality. Especially we need to study the case when two Lagrangian submanifolds involved coincide to each other. In a similar way, we can discuss more general case when they are of clean intersection. But we do not do it in this article. (See [Po] and [FOOO] §16.) The construction in §4 is a natural generalization of one in [FOOO] where we associate a filtered A_{∞} algebra to a Lagrangian submanifold.

In §5 we discuss the problem of unit. Namely we define a notion, homotopy unit (which was first introduced in [FOOO] in the case of A_{∞} algebra) and sketch the idea how to construct the homotopy unit in the case of the A_{∞} category of Lagrangian submanifolds.

In Chapter 2 we discuss homological algebra of A_{∞} category. After writing [Fu4], the author leaned that there are many works done in this direction, in the case of A_{∞} algebra or differential graded category, especially by Russian mathematicians. However the reference on it is rather scattered and it is rather hard to find a good reference where we find an appropriate description (and its proofs) of the results we need, especially in the case of A_{∞} category. So the author include it in this article. But he does not assert so much credit on it. Namely they are rather minor modification of the results already known to specialists in closely related situations, though some of the results in Chapter 2 are new in the sense they are not proved in the references. The author tries to quote appropriate reference in case he found it. However, since the author is far from being a specialist of homological algebra there should be many works which is closely related to Chapter 2 but is not known to the author.

In §6, we describe the notion of twisted complex and derived A_{∞} category. Twisted complex is a natural analogue of chain complex. Namely in the case of abelian category we study chain complex and use it to construct derived category. In a similar way we use twisted complex in the case of A_{∞} category. Twisted complex was introduced by Bondal-Kapranov [BoK] in the case of differential graded category. Kontsevitch [Ko1] proposed to use it in mirror symmetry. (It was also applied in [Fu7] to mirror symmetry.)

The author, in [Fu1], [Fu4], proposed to use the A_{∞} category of A_{∞} functors in the study of Floer homology of 3 manifolds with boundary. For each twisted complex of an A_{∞} category C, we can associate an A_{∞} functor from C to CH. Here CH is the A_{∞} category whose object is a chain complex. In this sense, A_{∞} functor to CH is a natural generalization of twisted complex. Also the idea of A_{∞} functor is important to understand Floer homology of family of Lagrangian submanifolds (see [Fu5]) and its application to mirror symmetry. Moreover, to clarify the dependence (or independence) of the A_{∞} category described in Chapter 1, we need to define a notion of homotopy equivalence of A_{∞} categories and hence we use the notion of A_{∞} functors. So we introduce the notion of A_{∞} functors in section 7. There we construct a representable A_{∞} functor and an A_{∞} category whose objects are A_{∞} functors.

In §8 we define homotopy equivalence of A_{∞} category. We prove in §8 that an A_{∞} functor which induces isomorphismes on cohomologies is a homotopy equivalence. This result is an algebraic analogue of J.H.C. Whitehead theorem in topology and is proved in [FOOO] §A5 in the case of A_{∞} algebra. (A similar results should had been proved in various cases in the reference. The author was unable to locate the first place where this kinds of results appeared.)

In §9, we prove another main result of the homological algebra of A_{∞} category, an A_{∞} analogue of Yoneda’s lemma. This result implies that we can embed any A_{∞} category C to the A_{∞} category of A_{∞} functors from C to CH. (Namely we identify an object of C to a functor represented by it.) As mentioned above, this point will be important to the further study of mirror symmetry and of Floer homology of 3 manifolds with boundary. An A_{∞} analogue of Yoneda’s lemma was proved in [Fu4] except signs. We give a proof with sign here and also we simplify the proof in [Fu4].

Those results on homological algebra of A_{∞} category will be used in future to further study the filtered A_{∞} category constructed in Chapter 1 of this article. For example we will prove that the filtered A_{∞} category constructed in Chapter 1 of this article is independent of the various choices involved up to homotopy equivalence. This results together with other results are not included in this Part II and is postponed to Part III etc. So two chapters of this article are yet rather independent in this article but will be unified in future.

The author would like to thank Professors Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, the joint authors of the book [FOOO]. He is happy to share many of the ideas described in this article with them.

The author also would like to thank Professor Seiki Nishikawa for his patience. The author apologize to other authors of this proceeding, since the delay of the completion of this article causes the delay of the publication of this proceeding.

Last but not the least, this article is dedicated to Prof. Katsuhiro Shiohama for his 60th birth day. The author owes much to Professor Shiohama for his encouragement and support especially at the time when the author started his career as a mathematician.

Chapter 1 : Floer homology and A_{∞} category.

1. A_{∞} category and filtered A_{∞} category.

To study the part of homological mirror symmetry conjecture we concern with in this article, we need to use the notion of A_{∞} category appeared in [Fu1] in the study of topological field theory. (A_{∞} structure had been studied in homotopy theory for a long time, especially in [St].) Actually we need to include the obstruction theory we developed in [FOOO], and to modify the definition of A_{∞} category a bit, in order to correct the flaw pointed out in [Ko2], [Oh1]. We call this modified version filtered A_{∞} category. We start with the usual definition of A_{∞} category. Most of the constructions in this section are straight forward generalizations of the definition of filtered A_{∞} algebra in [FOOO] Chapter 4.

We fix R, a commutative ring with unit. In our main application, R will be ℂ or ℚ.

Definition 1.1. An A_{∞} category, C, is a collection of a set 𝔒𝔟(C), the set of objects, a graded free R module C(c_{1}, c_{2}) for each c_{1}, c_{2} ∈ 𝔒𝔟(C), the operations

𝔪_{k} : C[1](c_{0}, c_{1}) ⊗ ⋯ ⊗ C[1](c_{k−1}, c_{k}) → C[1](c_{0}, c_{k}),

of degree +1 for k = 1, 2, ⋯ and c_{i} ∈ 𝔒𝔟(C). Here C[1](c_{0}, c_{1}) is C(c_{0}, c_{1}) with degree shifted. (C[1]^{m}(c_{0}, c_{1}) = C^{m+1}(c_{0}, c_{1})). They are assumed to satisfy the A_{∞} formula (1.3) described below.

To describe the A_{∞} formula, we introduce notations. Let a, b ∈ 𝔒𝔟(C), we put

(1.2)  B_{k}C[1](a, b) = ⨁_{a=c_{0},c_{1},··· ,c_{k−1},c_{k}=b} C[1](c_{0}, c_{1}) ⊗ ⋯ ⊗ C[1](c_{k−1}, c_{k}).

We define, in case k = 0,

B_{0}C[1](a, a) = R, B_{0}C[1](a, b) = 0 if a ≠ b.

We put

BC[1](a, b) = ⨁_{k} B_{k}C[1](a, b), B_{k}C[1] = ⨁_{a,b} B_{k}C[1](a, b), BC[1] = ⨁_{a,b} BC[1](a, b).

We define a homomorphism

Δ : B_{k}C[1](a, b) → ⨁_{k_{1}+k_{2}=k} ⨁_{c} B_{k_{1}}C[1](a, c) ⊗ B_{k_{2}}C[1](c, b)

by

Δ(x_{1} ⊗ ⋯ ⊗ x_{k}) = (x_{1} ⊗ ⋯ ⊗ x_{k_{1}}) ⊗ (x_{k_{1}+1} ⊗ ⋯ ⊗ x_{k}).

It induces maps Δ : B_{k}C[1] → ⊕_{k_{1}+k_{2}=k}B_{k_{1}}C[1] ⊗ B_{k_{2}}C[1], Δ : BC[1] → BC[1] ⊗ BC[1]. (BC[1](a, a), Δ) and (BC[1], Δ) are graded coalgebras. (They are coassociative but not cocomutative.)

Operations 𝔪_{k} define homomorphisms : B_{k}C[1](a, b) → C[1](a, b). It can be extended uniquely to coderivations

d̂_{k} : BC[1] → BC[1], d̂_{k} : BC[1](a, b) → BC[1](a, b)

by

\begin{matrix}d̂_{k}(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{ℓ} & (−1)^{(deg x_{1}+1)+···+(deg x_{ℓ−1}+1)} \\ x_{1} ⊗ ⋯ ⊗ 𝔪_{k}(x_{ℓ}, ⋯, x_{ℓ+k−1}) ⊗ ⋯ ⊗ x_{n}. \end{matrix}

We put

d̂= ∑_{k} d̂_{k}.

Now the A_{∞} formula is

(1.3)  d̂∘ d̂= 0.

We can expand it and rewrite it using 𝔪_{k}. We thus obtain, for example,

\begin{aligned}0 & = 𝔪_{1}𝔪_{1}, \\ 0 & = 𝔪_{1}𝔪_{2}(x ⊗ y) + 𝔪_{2}(𝔪_{1}(x) ⊗ y) + (−1)^{deg x+1}𝔪_{2}(x ⊗ 𝔪_{1}(y)). \end{aligned}

(See [GJ], [FOOO] Chapter 4.) Namely (C(a, b), 𝔪_{1}) is a chain complex and 𝔪_{2} is a derivation up to sign.

Remark 1.4. Here we follow the sign convention of [FOOO] and not of [Fu7].

Definition 1.5. Let c ∈ 𝔒𝔟(C). We say an element e_{c} ∈ C^{0}(c, c) = C^{1}[1](c, c) is a unit if

(1.5.1)  𝔪_{2}(e_{c}, x_{1}) = x_{1}, 𝔪_{2}(x_{2}, e_{c}) = (−1)^{deg x_{2}}x_{2}

for x_{1} ∈ C(c, c′), x_{2} ∈ C(c′, c) and

(1.5.2)  𝔪_{k+ℓ+1}(x_{1}, ⋯, x_{ℓ}, e_{c}, y_{1}, ⋯, y_{k}) = 0

for and k + ℓ ≠ 1.

Definition 1.6. A_{∞} category with one object is called an A_{∞} algebra.

Example-Lemma 1.7. Let (A, d, ·) be a differential graded algebra. Namely d : A^{k} → A^{k+1}, ∧ : A^{k} ⊗ A^{ℓ} → A^{k+ℓ} with

(1.8.1) d ∘ d = 0, (1.8.2) (x · y) · z = x · (y · z), (1.8.3) d(x · y) = dx · y + (−1)^{deg x}x · dy.

We put

(1.9.1) 𝔪_{1}(x) = (−1)^{deg x}dx, (1.9.2) 𝔪_{2}(x, y) = (−1)^{deg x(deg y+1)}x · y, (1.9.2) 𝔪_{k} = 0 for k > 2.

Then (A, 𝔪) is an A_{∞} algebra.

Proof. 𝔪_{1}𝔪_{1} = 0 follows from (1.8.1). We calculate using (1.8.3) and (1.9.1),(1.9.2) that

𝔪_{1}(𝔪_{2}(x, y)) = (−1)^{deg x+deg y+deg x(deg y+1)}d(x · y)
= −𝔪_{2}(𝔪_{1}(x), y) − (−1)^{deg x+1}𝔪_{2}(x, 𝔪_{1}(y)).

This is A^{∞} formula on B_{2}A[1]. We can also check

𝔪_{2}(𝔪_{2}(x, y), z) + 𝔪_{2}(x, 𝔪_{2}(y, z))
= (−1)^{deg x deg y+deg x deg z+deg x deg z+deg y}((x · y) · z − x · (y · z)),

which is zero by (1.8.2). □

Definition-Example 1.10. ([BoK]). A differential graded category C is a collection of a set 𝔒𝔟(C), the set of objects, a graded differential graded R module C(c_{1}, c_{2}) for each c_{1}, c_{2} ∈ 𝔒𝔟(C), the operations

∘ : C(c_{1}, c_{2}) ⊗ C(c_{2}, c_{3}) → C(c_{1}, c_{3}),

which is a chain map and is associative in the sense of (1.8). We then define 𝔪_{k} by (1.9). We obtain an A_{∞} category.

For later use, we introduce some other notations. Let C be an A_{∞} category and c be its object. We put

BC[1](c) = ⨁_{k} B_{k}C[1](c) = ⨁_{k} C[1](c, c)^{⊗k}.

(We remark that B_{k}C[1](c) ≠ B_{k}C[1](c, c).) BC[1](c) is a coalgebra and 𝔪_{k} defines a structure of A_{∞} algebra on it.

Actually our main example is a filtered A_{∞} category rather than A_{∞} category. We are going to define it. We first define a universal Novikov ring [No], [FOOO].

Definition 1.11. Let T be a formal parameter. We consider a formal power series

(1.12)  ∑_{i=1}^{∞} a_{i}T^{λ_{i}}

where a_{i} ∈ R, λ_{i} ∈ ℝ. We assume λ_{i} < λ_{i+1} and lim_{i→∞} λ_{i} = ∞. We denote by Λ_{R,nov} the set of all such series. It has an obvious ring structure.

We consider its subring consisting of (1.12) such that λ_{i} ≥ 0 in addition and denote it by Λ_{R,0,nov}. The ring Λ_{R,0,nov} has a maximal ideal consisting of all series (1.12) such that λ_{i} > 0 in addition. We denote it by Λ_{+,nov}.

We write Λ_{nov}, Λ_{0,nov}, Λ_{+,nov} in place of Λ_{R,nov}, Λ_{R,0,nov}, Λ_{R,+,nov} in case no confusion can occur.

For each λ, we define F^{λ}Λ_{nov} so that it consists of the elements (1.12) satisfying λ_{i} ≥ λ in addition. It induces a filtration on Λ_{nov}. Namely each F^{λ}Λ_{nov} is an additive subgroup and F^{λ_{1}}Λ_{nov} · F^{λ_{2}}Λ_{nov} ⊆ F^{λ_{1}+λ_{2}}Λ_{nov}. Filtration on Λ_{nov} induces ones on Λ_{0,nov} and Λ_{+,nov}.

Our filtration induces a uniform structure on Λ_{nov}, Λ_{0,nov} and Λ_{+,nov} in a usual way. Then these rings are complete with respect to it.

Remark 1.13. In [FOOO], we considered the set of series ∑_{i=1}^{∞} a_{i}T^{λ_{i}}e^{n_{i}} where n_{i} are integers, and denote by Λ_{nov} the set of all such series. However for the present purpose (that is to discuss Mirror symmetry where the case c^{1}(M) = 0 is important) it is more convenient to use Definition 1.11.

Now we define filtered A_{∞} category.

Definition 1.14. A filtered A_{∞} category C is a correction of a set 𝔒𝔟(C), the set of objects, a graded torsion free filtered Λ_{0,nov} module C(c_{1}, c_{2}) for each c_{1}, c_{2} ∈ 𝔒𝔟(C), the operations

𝔪_{k} : C[1](c_{0}, c_{1}) ⊗ ⋯ ⊗ C[1](c_{k−1}, c_{k}) → C[1](c_{0}, c_{k}),

of degree +1 for k = 0, 1, 2, ⋯, and c_{i} ∈ 𝔒𝔟(C). Note that k = 0 is included in the case of filtered A_{∞} category. Here 𝔪_{0} is a map

𝔪_{0} : Λ_{0,nov} → ⨁_{c} C[1](c, c).

(𝔪_{0} is not included in the case of A_{∞} category.) We assume that 𝔪_{k} respects the filtration in the sense that

(1.15)  𝔪_{k}(F^{λ_{1}}C[1](c_{0}, c_{1}) ⊗ ⋯ ⊗ F^{λ_{k}}C[1](c_{k−1}, c_{k})) ⊆ F^{∑ λ_{i}}C[1](c_{0}, c_{k}).

They induce coderivations

d̂_{k} : BC[1] → BC[1], d̂_{k} : BC[1](a, b) → BC[1](a, b)

in the same way as before. Our filtrations on C induces one on BC[1] and we let B̂C[1] be the completion. d̂_{k} induces a map : B̂C[1] → B̂C[1], which we denote by the same symbol. The sum

d̂= ∑_{k=0}^{∞} d̂_{k}

converges by virtue of (1.15). Now we assume

(1.16)  d̂∘ d̂= 0.

We assume also

(1.17)  𝔪_{0} ≡ 0 mod Λ_{+,nov}.

We define a unit of filtered A_{∞} category in the same way as Definition 1.5. A filtered A_{∞} category with one object is called a filtered A_{∞} algebra.

For each filtered A_{∞} category C and c ∈ 𝔒𝔟(C), the operations 𝔪_{k} : B_{k}C(c) → C(c, c) define a structure of filtered A_{∞} algebra on C(c, c).

We can construct an A_{∞} category (of Λ_{0,nov} module) from a filtered A_{∞} category C in the following way. Let c ∈ 𝔒𝔟(C). We define Definition 1.18. An element b of F^{+}C^{1}(c, c) is said to be a bounding chain if

d̂(e^{b}) = 0.

Here F^{+}C^{1}(c, c) = ∪_{λ>0}F^{λ}C^{1}(c, c) and

e^{b} = 1 + b + b ⊗ b + b ⊗ b ⊗ b + ⋯ ∈ B̂C(c, c).

We define M̃(c) be the set of all bounding chains of c.

Definition 1.19. Let b_{i} ∈ M̃(c_{i}), i = 0, ⋯, k, k > 0 we define

𝔪_{k}^{(b_{0},··· ,b_{k})} : C[1](c_{0}, c_{1}) ⊗ ⋯ ⊗ C[1](c_{k−1}, c_{k}) → C[1](c_{0}, c_{k})

by

𝔪_{k}^{(b_{0},··· ,b_{k})}(x_{1}, ⋯, x_{k}) = ∑_{ℓ_{0},··· ,ℓ_{k}} 𝔪_{k+ℓ_{0}+···+ℓ_{k}}(b_{0}^{ℓ_{0}}, x_{1}, b_{1}^{ℓ_{1}}, ⋯, b_{k−1}^{ℓ_{k−1}}, x_{k}, b_{k}^{ℓ_{k}}).

Here

b^{ℓ} = \overbrace{b ⊗ ⋯ ⊗ b}^{ℓ times} .

Proposition 1.20. We put

𝔒𝔟(C′) = ⋃_{c∈𝔒𝔟(C)} M̃(c) × {c},

C′((c, b), (c′, b′)) = C(c, c′) and let 𝔪_{k}^{(b_{0},··· ,b_{k})} be the operations. Then C′ is an A_{∞} category.

The proof (which is easy) goes in exactly the same was as the proof of [FOOO] Lemma 13.37. Hence we omit it.

Actually M̃(c) is too big and it is more reasonable to define an equivalence relation ∼ on it and divide M̃(c) by ∼. See [FOOO],[Fu5],[Ot] on it.

We can continue and study A_{∞} functor, natural transformations etc. We will do it later. We give our main example before continuing the discussion on algebraic formalism. If the reader is mainly interested in algebraic formalism then he can skip §2 ∼ §5 and proceed to Chapter 2.

2. Floer homology and A_{∞} category I - the set of objects -

The idea of this section is rather old. Inspired by S. Donaldson’s lecture [Do], the author [Fu1] found that Floer homology and counting of holomorphic polygons will define an A_{∞} category. However there was a trouble in defining Lagrangian intersection Floer homology as was mentioned in [Ko2], [Oh1]. We could overcome this trouble in [FOOO] and the construction is now presented here in a modified way. The notion of filtered A_{∞} category introduced in the last section is defined for this purpose.

We explain, in this section, the definition of the objects of our filtered A_{∞} category LAG and define a graded Λ_{0,nov} module LAG(c, c′) for two objects when c ≠ c′. Basically an object of LAG is a Lagrangian submanifold. However we need to add some topological data to it. One of the topological data to be added is related to the orientation problem of the moduli space of pseudoholomorhic discs. Another data to be added is related to the way to fix the degree of elements of LAG(c, c′). To motivate those data we mention some comments on how they will be used. During those comments we assume that the reader is familiar to the Floer homology and pseudoholomorphic curves. (Please skip them otherwise.)

Let (M, ω) be a symplectic manifolds. Let B be a closed 2 form which we call the B-field. We put Ω = ω + 2π\sqrt{−1}B.

Definition 2.1. Let 𝔒𝔟_{1}(LAG(M, Ω)) be the set of all pairs (L, L) such that :

(2.1.1) L is a Lagrangian submanifold. Namely dim L = dim M/2 and ω|_{L} = 0. (2.1.2) L is a complex line bundle equipped with a connection ∇ such that its curvature F_{∇} coincides with the restriction of 2π\sqrt{−1}B to L. (Here we identify the Lie algebra of U(1) with \sqrt{−1}ℝ.)

Remark 2.2. One may consider more general objects than 𝔒𝔟_{1}(LAG)(M, Ω). There are at least two generalizations.

(2.2.1) One may relax the condition on L so that L → M is a Lagrangian immersion. (2.2.2) One may consider the vector bundle L together with its unitary connection ∇ whose curvature is 2π\sqrt{−1}B times the unit matrix.

The modification of the construction to include these cases will be discussed elsewhere. (See [Ak] on (2.2.1).)

It seems impossible to define an A_{∞} category whose objects are all elements of 𝔒𝔟_{1}(LAG(M, Ω)) because of the transversality problem. (We use Bair’s category theorem to achieve transversality quite frequently.) So instead we take and fix a countable set of Lagrangian submanifolds and let 𝔒𝔟_{2}(LAG(M, Ω)) be the set of all elements (L, 𝔏) ∈ 𝔒𝔟_{1}(LAG(M, Ω)) such that L is in this countable set.

The module of morphisms LAG((L_{1}, 𝔏_{1}), (L_{2}, 𝔏_{2})) of our filtered A_{∞} category is Floer’s chain complex, which is the free Λ_{0,nov} module generated by the intersection points ∈ L_{1} ∩ L_{2}. More precisely

(2.3)  LAG((L_{1}, 𝔏_{1}), (L_{2}, 𝔏_{2})) = ⨁_{p∈L_{1}∩L_{2}} Hom((𝔏_{1})_{p}, (𝔏_{2})_{p}) ⊗_{ℂ} Λ_{0,nov}.

However there are two delicate points which will soon come to the story. The first of them is sign or orientation of the moduli space of pseudoholomorphic discs which we will use to define operations 𝔪_{k} (see §3,4), and the other is the degree in the Floer homology.

We start with the first point. We refer [FOOO] Chapter 6 for the thorough argument on the orientation and present only a sketch of it in this article. We first fix an element st ∈ H^{2}(M; ℤ_{2}). We take a 3-skeleton M^{(3)} of M. Then there exists a unique real rank 2 vector bundles V (st) on M^{(3)} such that w^{1}(V (st)) = 0, w^{2}(V (st)) = st, here w is the Stiefel-Whitney class.

Definition 2.4. ([FOOO]). L is said to be relatively spin in (M, st) if it is oriented and if the second Stiefel-Whitney class of (the tangent bundle of) L coincides with the restriction of st.

Let L be relatively spin in (M, st), and let L^{(2)} be the two skeleton of L. Then V ⊕ T L is trivial on L^{(2)}.

A (M, st)-relative spin structure of L is by definition a spin structure of the restriction of V ⊕ T L to L^{(2)}.

We remark that the two spin structures are equivalent if it is equivalent on the first skeleton. Moreover oriented vector bundle is trivial on two skeleton if it is spin. Therefore the set of (M, st)- relative spin structures of L corresponds one to one to the set of trivializations of the restriction of V ⊕ T L to L^{(1)} which can be extended to L^{(2)}. We use this remark to show the following :

Lemma 2.5. The group H^{1}(L; ℤ_{2}) acts simple transitively on the set of all (M, st)- relative spin structures of L if it is nonempty.

Proof. Let ψ ∈ C^{1}(L; ℤ_{2}) be a cocycle defining an element of H^{1}(L; ℤ_{2}). Let Ψ : V ⊕ T L|_{L^{(2)}} → L^{(2)} × ℝ^{n+2} be isomorphism of bundles whose restrictions to V ⊕ T L|_{L^{(1)}} define a relative spin structures of L. For each one cell Δ^{1} of L we define a map

g_{ψ,Δ^{1}} : (Δ^{1}, ∂Δ^{1}) → (SO(n + 2), I)

representing ψ(Δ^{1}) ∈ π_{1}(SO(n + 2)) = ℤ_{2}. (I ∈ SO(n + 1) is the unit matrix.) We put

Ψ′(x, v) = g_{ψ,Δ^{1}}(x)(x, v)

for (x, v) ∈ V ⊕ T L|_{L^{(1)}}, x ∈ Δ^{1}. Since ψ is a cocycle it follows that Ψ′ : V ⊕ T L|_{L^{(1)}} → L^{(1)} × ℝ^{n+2} can be extended to L^{(2)}. It is easy to see that the relative spin structure determined by Ψ′ depends only on the cohomology class of ψ and the relative spin structure Ψ. We put [ψ] · [Ψ] = [Ψ′].

Conversely, let Ψ_{i} : V ⊕T L|_{L^{(2)}} → L^{(2)} ×ℝ^{n+2} be isomorphism of bundles whose restrictions to V ⊕ T L|_{L^{(1)}} define two relative spin structures of L. There exists a map g : L^{(2)} → SO(n + 2) such that

Ψ_{2}(x, v) = g(x)(Ψ_{1}(x, v)).

Since π_{0}(SO(n + 2)) is trivial we may modify Ψ_{i} so that g(x) = 1 for x ∈ L^{(1)}. Then for each 1 cell Δ^{1} of L we have

[g|_{Δ^{1}}] ∈ π_{1}(SO(n + 2)) = ℤ_{2}.

We regard it as a cochain. It is a cocycle since g can be extended to L^{(2)}. We thus obtain ψ = g such that [ψ] · [Ψ_{1}] = [Ψ_{2}]. □

We denote by 𝔒𝔟_{3}(LAG(M, Ω, st)) the set of all pairs of (L, L) ∈ 𝔒𝔟_{2}(LAG(M, Ω)) and an (M, st)-relative spin structure on L. The reason we add relative spin structure is that it induces orientations of various moduli spaces we use in a canonical way. (See the next section and [FOOO] Chapter 6.)

Next we consider the degree problem. We use the notion of graded Lagrangian submanifold due to M. Kontsevich and P. Seidel [Se1] for this purpose. Let (ℝ^{2n}, ω) be a symplectic vector space of dimension 2n. We denote by Lag_{n} = Lag(ℝ^{2n}, ω) the set of all oriented linear subspaces V of ℝ^{2n} of dimension n such that ω|_{V} ≡ 0. We call it the oriented Lagrangian Grassmanian. It is well-known that π_{1}(Lag_{n}) ≅ ℤ and it has a generator called the (universal) Maslov class. (See [AG]). Let L̃ag_{n} be the universal covering space of Lag_{n}.

Let (M, ω) be a symplectic manifold. Then we have a fiber bundle Lag(M) → M whose fiber at p ∈ M is identified with Lag(T_{p}M). We remark that we can find an almost complex structure compatible with its symplectic structure, and it is unique up to homotopy. Hence the Chern classes of the tangent bundle of a symplectic manifolds are well defined.

Lemma 2.6. The following two conditions are equivalent.

(2.7.1) c^{1}(M) = 0 where c^{1}(M) is the first Chern class of T M. (2.7.2) There exists a covering space L̃ag(M) of Lag(M) such that its restriction to each fiber is identified with L̃ag_{n} → Lag_{n}.

Proof. Since L̃ag → Lag is a covering space, the obstruction to construct L̃ag(M) lies in the second cohomology. Hence it is easy to see that (2.7.2) is equivalent to the triviality of the bundle Lag(M) → M at the two skeleton M^{(2)} of M.

Now let us assume (2.7.1). Then the complex vector bundle T M is trivial on two skeleton. Since Symplectic group Symp(n) is homotopy equivalent to U(n), it follows that Lag(M) → M is trivial on two skeleton. The proof of the converse is similar. □

From now on, we assume c^{1}(M) = 0. (In the case when the compatible almost complex structure is integrable, this condition implies that M has a Ricci flat Kähler metric, due to Yau’s proof of Calabi conjecture.) We also fix a covering space L̃ag(M) of Lag(M) as in (2.7.2).

Remark 2.8. In a way similar to the proof of Lemma 2.5, we can show that the set of such lifts L̃ag(M) is an affine space over H^{1}(M; ℤ) if it is nonempty.

Let L be an oriented Lagrangian submanifold. We have a canonical section s of the restriction of Lag(M) to L. Namely

s(p) = T_{p}L ⊆ T_{p}M.

Definition 2.9. A graded Lagrangian submanifold of (M, L̃ag(M)) is a pair of oriented Lagrangian submanifold L and a lift of s to s̃ : L → L̃ag(M). We call s̃, the grading of L.

Definition 2.10. We denote by 𝔒𝔟_{4}(LAG(M, Ω, st, L̃ag(M))) the set of all triples (L, L, s̃) such that (L, L) ∈ 𝔒𝔟_{3}(LAG(M, Ω, st)) and that (L, s̃) is a graded Lagrangian submanifold.

Example 2.11. Let T^{∗}N → N be a cotangent bundle of an oriented manifold N with a canonical symplectic structure. The tangent bundle T T^{∗}N is isomorphic to the complexification of the pull back π^{∗}T N.

Hence its structure group is reduced to U(n) ∩ GL(n; ℝ) = O(n). Moreover, since N is oriented, it follows that the structure group is reduced to SO(n). Since Lag_{n} = U(n)/SO(n) it follows that the bundle Lag_{n}(T^{∗}N) is trivial. We thus obtain L̃ag(T^{∗}N) as in (2.5.2)

We remark that the zero section is a Lagrangian submanifold. Hence we have a section s_{0} of Lag(T^{∗}N) on the zero section. Since zero section ≅ N is homotopy equivalent to T^{∗}N it induces a section s_{0} of Lag(T^{∗}N). It then induces a trivialization of Lag(T^{∗}N) and hence a section s̃_{0} : T^{∗}N → L̃ag(T^{∗}N). We may choose s̃_{0} such that s̃_{0}(p) is transversal to the tangent space of the fibers of T^{∗}N → N.

Now let L be a Lagrangian submanifold of T^{∗}N transversal to the fiber. Then, for each p ∈ L, there exists a path ℓ_{p} which joints T_{p}L to s_{0}(p) in Lag(T_{p}T^{∗}N) such that ℓ_{p}(t) is transversal to the tangent space of the fiber for each t ≠ 0. The homotopy class of such ℓ_{p} is unique. We lift it so that s_{0}(p) will be lifted to s̃_{0}(p). In this way we obtain a lift s̃(p) ∈ L̃ag(T_{p}T^{∗}N). It is easy to see that this lift is independent of various choices.

We thus obtain a graded Lagrangian submanifold (L, s̃).

In a similar way, we can consider the case when (M, ω) has a singular Lagrangian fibration as follows. Let π : M → N be a smooth map with the following properties.

(2.12.1) There exists a subcomplex X ⊂ M of codimension > 2 such that π is a submersion on M − X. (2.12.2) The kernel of the differential of d_{p}π at p ∈ M −X is a Lagrangian vector subspace of T_{p}M. (2.12.3) π is proper.

Now we define a section s′ of Lag(M) on M − X by putting s′(p) = Ker d_{p}π. It defines a real vector bundle on M − X whose fiber at p is s′(p). This vector bundle tensored with ℂ is isomorphic to the tangent bundle of M − X. Thus, in the same way as above, we have a trivialization of Lag(M) on M − X. It induces a bundle L̃ag(M − X) on M − X. We can extend it to L̃ag(M) → M uniquely since the codimension of X in M is > 2. (Note the trivialization of the restriction of Lag(M) to M − X may not be extended.)

Actually the condition for a Lagrangian submanifold to be graded is related to the (absolute) Maslov index η : π_{2}(M, L) → ℤ as follows.

Let us first review Maslov index. Let φ : (D^{2}, ∂D^{2}) → (M, L) be a map representing an element of π_{2}(M, L). The pullback bundle φ^{∗}(T M) has a trivialization since D^{2} is contractible. We restrict this trivialization to ∂D^{2}. On the other hand, for each t ∈ ∂D^{2} we have a Lagrangian subspace T_{φ(p)}L ⊂ T_{φ(p)}M. Hence we have S^{1} → Lag_{n}. It determines an element of π_{1}(Lag_{n}) ≅ ℤ. We call it Maslov index and write η([φ]).

We consider the composition π_{2}(M) → π_{2}(M, L) → ℤ. One can verify easily that the composition coincides with the twice of the first Chern class c^{1} : π_{2}(M) → ℤ.

Therefore, in the case when c^{1}(M) = 0, the homomorphism η induces a homomorphism : Im(π_{2}(M, L) → π_{1}(L))) → ℤ. We can extend it to π_{1}(L) as follows. Using c^{1}(M) = 0 there exists a lift L̃ag(M) → M. Let ℓ : S^{1} → L be a loop representing an element of π_{1}(L). We define a map

ℓ^{+} : S^{1} → Lag(M)

by

ℓ^{+}(t) = T_{ℓ(t)}L ∈ Lag(T_{p}M).

Since L̃ag(M) → Lag(M) is a covering space we have a lift

ℓ̃^{+}(t) : [0, 1] → L̃ag(M)

of ℓ^{+}. Since L̃ag/ℤ = Lag there exists \overline{η}(ℓ) ∈ ℤ such that

\overline{η}(ℓ) · ℓ̃^{+}(0) = ℓ̃^{+}(1).

It is easy to see that \overline{η} defines a homomorphism

\overline{η} : π_{1}(L) → ℤ.

Now it is easy to show the following two lemmata.

Lemma 2.13. The composition of π_{2}(M, L) → π_{1}(L) with \overline{η} is η.

Lemma 2.14. There exists a lift s̃ of s : L → Lag(M) if and only if \overline{η} : π_{1}(L) → ℤ is 0.

We remark that \overline{η} depends on the choice of L̃ag(M), while η does not depend on it. Note the set of all choices of L̃ag(M) is an affine space over H^{1}(M; ℤ) = Hom(π_{1}(M), ℤ). In view of the exact sequence

π_{2}(M; L) → π_{1}(L) → π_{1}(M),

the group Hom(π_{1}(M), ℤ) controls the way to extend η (which is defined on the image of π_{2}(M; L) → π_{1}(L)) to \overline{η} which is defined on π_{1}(L).

Now let (L_{1}, s̃_{1}) and (L_{2}, s̃_{2}) be graded Lagrangian submanifolds which intersect transversally each other. Let p ∈ L_{1}∩L_{2}. We are going to define an index η_{L_{1},L_{2}}(p).

We first consider a pair of family of Lagrangian submanifolds ℓ_{0}(τ), ℓ_{1}(τ) ∈ Lag_{n}, τ ∈ ℝ so that it is constant for |τ| large. We assume that

(2.15)  ℓ_{0}(−∞) ∩ ℓ_{1}(−∞) = ℓ_{0}(∞) ∩ ℓ_{1}(∞) = {0}.

We want to associate an integer, Maslov-Viterbo index η(ℓ_{0}, ℓ_{1}), for such a pair. We need its relation to the index of Cauchy-Riemann operator also. Let us now describe it. We consider the product ℝ × [0, 1] ⊂ ℂ and identify τ to the first coordinate. We consider the operator

\overline{∂} : W^{1,p}(ℝ × [0, 1]; ℂ^{n}) → W^{0,p}(ℝ × [0, 1]; ℂ^{n} ⊗ Λ^{0,1}),

where W^{k,p} denotes the set of sections of Sobolev (k, p) class, that is the linear space of sections whose derivatives up to order k are of L^{p} class. (We take p > 2. So W^{1,p} ⊆ C^{0}.) For an element u ∈ W^{k+1,2}(ℝ × [0, 1]; ℂ^{n}), we put the boundary condition

(2.16)  u(τ, 0) ∈ ℓ_{0}(τ), u(τ, 1) ∈ ℓ_{1}(τ).

We denote by W^{1,p}(ℝ×[0, 1]; ℂ^{n}; ℓ_{0}, ℓ_{1}) the subset of W^{1,p}(ℝ×[0, 1]; ℂ^{n}) satisfying the boundary condition (2.16). Then the \overline{∂} operator induces an operator :

(2.17)  \overline{∂} : W^{1,p}(ℝ × [0, 1]; ℂ^{n}; ℓ_{0}, ℓ_{1}) → W^{0,p}(ℝ × [0, 1]; ℂ^{n} ⊗ Λ^{0,1}).

Lemma 2.18. (2.17) is a Fredholm operator.

The lemma is a consequence of (2.15). The proof is omitted.

We can calculate the index of (2.17) in the following way. We first take g(τ) such that g(τ)ℓ_{0}(τ) = ℓ_{0}(−∞). Then we consider the pair ℓ′_{0}(τ) ≡ ℓ_{0}(−∞), ℓ′_{1}(τ) ≡ g(τ)ℓ_{1}(τ). It is easy to see that the index of (2.17) does not change if we replace ℓ_{0}(τ), ℓ_{1}(τ) by ℓ′_{0}(τ), ℓ′_{1}(τ). Hence we may assume that ℓ_{0}(τ) is independent of τ. We put V = ℓ_{0}(τ).

For V ∈ Lag_{n}, let X(V) be the set of all V′ ∈ Lag_{n} such that V′ is not transversal to V. X(V) is a (real) codimension one subcomplex such that it is a smooth submanifold outside a set of (real) codimension > 1 in X(V).

Proposition 2.19. There exists an orientation of the regular part of X(V) such that the index of (2.17) is equal to the intersection number ℓ_{1}·X(V) when V ≡ ℓ_{0}(τ).

Sketch of a proof. We explain an idea of the proof of Proposition 2.19 by using the notion of spectral flow ([APS]). Let us consider the operator (2.17). It is an operator

(2.20)  \overline{∂} = \frac{∂}{∂τ} + J_{t,τ} ∘ \frac{∂}{∂t}.

J_{t,τ} ∘ \frac{∂}{∂t} is a family of elliptic operators on [0, 1] and is parametrized by τ. Also the boundary condition (2.16) is τ dependent. We choose an appropriate bundle automorphism of φ^{∗}T M on ℝ × [0, 1] and use it to regard J_{t,τ} ∘ \frac{∂}{∂t} as a family of elliptic operators whose boundary condition is τ independent. We thus obtain an operator

(2.21)  \frac{∂}{∂τ} + P_{τ}

where

P_{τ} : W^{1,p}([0, 1]; ℂ^{n}; Ξ) → W^{0,p}([0, 1]; ℂ^{n}).

Here Ξ is a boundary condition independent of τ. Then, as in [APS], the index of (2.21) is calculated by a spectral flow of the family of elliptic operators P_{τ}. We remark that the operator P_{τ} will have a kernel in a neighborhood of the points τ where ℓ_{1}(τ) are not transversal to ℓ_{0}.

Thus the index of the spectral flow coincides to the intersection number ℓ_{1} · X(V). □

For our purpose, we need to relax the condition (2.15) and consider the case

(2.22)  ℓ_{0}(−∞) = ℓ_{1}(−∞),  ℓ_{0}(∞) ∩ ℓ_{1}(∞) = {0}.

We perturb ℓ so that it is transversal to X(ℓ(0)) on (0, 1]. Then we will put

(2.23)  η(ℓ_{0}, ℓ_{1}) “ = ” ℓ_{1}|_{(0,1]} · X(ℓ_{0}(0))

where · is the intersection number. However, since ℓ_{0}(0) ∈ X(ℓ_{0}(0)), we need to be very careful to define the intersection pairing in (2.23). Actually ℓ_{0}(0) ∈ X(ℓ_{0}(0)) is a singular point of X(ℓ_{0}(0)) where n components of codimension one meet each other. To define right hand side of (2.23) rigorously, we modify ℓ_{1} in a neighborhood of 0 as follows. Let U(ℓ_{0}(0)) be a small neighborhood of ℓ_{0}(0) in Lag_{n}. We will define components U_{i}(ℓ_{0}(0)) such that

(2.24)  U(ℓ_{0}(0)) − X(ℓ_{0}(0)) = ⋃_{i=0}^{n} U_{i}(ℓ_{0}(0)),

as follows.

We take a symplectic isomorphism ℂ^{n} ≃ T^{∗}ℓ_{0}(0). For each

V ∈ U(ℓ_{0}(0)) − X(ℓ_{0}(0)),

there exists a non degenerate quadratic function f on ℓ_{0}(0) such that V is a graph of df. We define U_{i}(ℓ_{0}(0)) so that V ∈ U_{i}(ℓ_{0}(0)) if f is a quadratic function of index i.

We may choose U(ℓ_{0}(0)) so that U_{i}(ℓ_{0}(0)) is contractible. Note that \overline{U}_{i}(ℓ_{0}(0)) ∩ \overline{U}_{i+1}(ℓ_{0}(0)) is of codimension one in X(ℓ(0)) and they are all of the codimension 1 components of X(ℓ(0)) ∩ U(ℓ_{0}(0)).

Now we modify ℓ_{1} so that ℓ_{1}(ϵ) is in U_{n}(ℓ_{0}(0)) for small ϵ. Now we define η(ℓ_{0}, ℓ_{1}) by

(2.25)  η(ℓ_{0}, ℓ_{1}) = ℓ_{1}|_{[ϵ,1]} · X(ℓ_{0}(0)).

If we take another perturbation ℓ such that ℓ′_{1}(ϵ) is in U_{k}(ℓ_{0}(0)) for small k, then

(2.26)  ℓ_{1}|_{[ϵ,1]} · X(ℓ_{0}(0)) + n = ℓ′_{1}|_{[ϵ,1]} · X(ℓ_{0}(0)) + k.

Let −ℓ be the path such that −ℓ(t) = ℓ(1 − t) then we have

Lemma 2.27. η(ℓ_{0}, ℓ_{1}) + η(ℓ_{1}, ℓ_{0}) = −n, if (2.22) holds.

Proof. Let us perturb ℓ_{1} so that ℓ_{1}(ϵ) is in U_{n}(ℓ_{0}(0)) for small ϵ. Then we have

ℓ_{1}|_{(0,1)} · X(ℓ_{0}(0)) + ℓ_{0}|_{(0,1)} · X(ℓ_{1}(0)) = 0.

Let us change the pair ℓ_{1}, ℓ_{0} by the one parameter family of authomorphisms of ℂ^{n} as follows : Let us denote modified pair by ℓ′_{1}, ℓ′_{0} : we take it so that ℓ′_{1} is constant.

We then find that ℓ′_{0}(ϵ) ∈ U_{0}(ℓ′_{1}(0)) for ϵ small. Lemma 2.27 now follows from (2.26). □

Now we are going to define an index of the intersection point of two graded Lagrangian submanifolds (L_{1}, s̃_{1}), (L_{2}, s̃_{2}). Let p ∈ L_{1} ∩ L_{2}. We take a pair (ℓ̃_{0}, ℓ̃_{1}) of path in L̃ag(T_{p}M) such that ℓ̃_{0}(0) = ℓ̃_{1}(0), and ℓ̃_{0}(1) = s̃_{1}(p), ℓ̃_{1}(1) = s̃_{2}(p). We put ℓ_{i} = π ∘ ℓ̃_{i}. We then put

(2.28)  η_{(L_{1},s̃_{1}),(L_{2},s̃_{2})}(p) = η(ℓ_{0}, ℓ_{1}).

It is easy to see that the right hand side is independent of the path (ℓ̃_{0}, ℓ̃_{1}).

Definition 2.29. Let (L_{1}, s̃_{1}, 𝔏_{1}), (L_{2}, s̃_{2}, 𝔏_{2}) ∈ 𝔒𝔟_{4}(LAG(M, Ω, st, L̃ag(M))). We assume L_{1} ≠ L_{2}. We define :

(2.30)  \begin{aligned}LAG^{k} & ((L_{1}, s̃_{1}, 𝔏_{1}), (L_{2}, s̃_{2}, 𝔏_{2})) \\ & = ⨁_{{p∈L_{1}∩L_{2} \atop η_{(L_{1},s̃_{1}),(L_{2},s̃_{2})}(p)=k}} Hom((𝔏_{1})_{p}, (𝔏_{2})_{p}) ⊗_{ℂ} Λ_{0,nov}. \end{aligned}

We discuss the case L_{1} = L_{2} later. (We actually need one more small modification to (2.30) for the orientation problem. We will discuss it later. See (3.16).)

Lemma 2.27 implies :

(2.31)  η_{(L_{1},s̃_{1}),(L_{2},s̃_{2})}(p) = −n − η_{(L_{2},s̃_{2}),(L_{1},s̃_{1})}(p)

Example 2.32. Let u = df be an exact 1 form on N and L_{2} ⊂ T^{∗}N be its graph. Let L_{1} be a zero section. We assume that f is a Morse function. If p ∈ L_{1} ∩ L_{2} then p = (x, 0) and df(x) = 0. We define s̃_{1}, s̃_{2} as in Example 2.11. We find

η_{(L_{1},s̃_{1}),(L_{2},s̃_{2})}(p) = Index(∇_{x}^{2}f) − n.

Here ∇_{x}^{2}f is the Hessian of f at x. Namely in this case the Maslov index reduces to the Morse index.

Remark 2.33. Let (L_{1}, s̃_{1}) be a graded Lagrangian submanifold. We assume that L_{1} is connected. When we change the lift s̃_{1}, then the degree η changes only by an overall integer. Let us state it more precisely. We recall that L̃ag(M) → Lag(M) is a normal covering whose deck transformation group is ℤ. Let s̃′_{1} be another lift of s_{1}(p) = T_{p}(L). Then there exists an integer k such that s̃_{1} = k · s̃′_{1}. We have :

(2.34.1)  η_{(L_{1},s̃_{1}),(L_{2},s̃_{2})}(p) = η_{(L_{1},s̃′_{1}),(L_{2},s̃_{2})}(p) + k

and

(2.34.2)  η_{(L_{1},s̃_{2}),(L_{2},s̃_{1})}(p) = η_{(L_{1},s̃_{2}),(L_{2},s̃′_{1})}(p) − k

for every (L_{1}, s̃_{2}) and p ∈ L_{1} ∩ L_{2}. More precisely the identification of the deck transformation group of L̃ag(M) → Lag(M) and ℤ has an ambiguity Aut(ℤ) = {±1}. We can fix it by requiring (2.34).

So far, we restricted ourselves to a symplectic manifold with c^{1} = 0 and Lagrangian submanifold with Maslov index 0. This allows us to define Floer homology with ℤ grading. However it is useful for some other purposes to relax the condition c^{1} = Maslov index = 0. Let us discuss the general case here, following [Se1]. Let us define N ∈ ℤ_{≥0} by

Im(c^{1} : π_{1}(M) → ℤ) = Nℤ.

N is called the minimal Chern number of (M, ω). In a way similar to the proof of Lemma 2.6, we can construct a covering space

(2.35)  π : L̃ag^{N}(M) → Lag(M)

such that π induces 2N hold covering on each fiber. (We remark that two hold fiberwise covering of Lag^{N}(M) always exists because M is orientable.)

For any N′ dividing N, we put

L̃ag^{N′}(M) = L̃ag^{N}(M)/ℤ_{N/N′}

which is a 2N′ hold fiberwise covering of Lag(M).

Let L ⊂ M be a Lagrangian submanifold. We can define

\overline{η} : π_{1}(L) → ℤ_{2N}

using (2.35). Moreover the composition of π_{2}(M; L) → π_{1}(L) and \overline{η} coincides with the mod 2N reduction of η : π_{2}(M; L) → ℤ. (Note that the Maslov index η : π_{2}(M; L) → ℤ is defined for any pair of symplectic manifold M and its Lagrangian submanifold L.) Now we have :

Lemma 2.36. A lift

s̃: L → L̃ag^{N′}(M)|_{L}

of the map s : L → Lag(M)|_{L} exists if and only if the image of \overline{η} : π_{1}(L) → ℤ_{2N} is contained in 2N′ℤ_{2N}.

Such a lift s̃ is called an N′ grading of L′. Let L_{i} (i = 1, 2) be a pair of Lagrangian submanifolds with N′ grading. We assume they are transversal to each other. Let p ∈ L_{1} ∩L_{2}. Then we can define its index η(p) ∈ ℤ_{2N′} in a similar way to Definition 2.25. Thus we have a ℤ_{2N′} graded Floer homology.

3. Floer homology and A_{∞} category II - the operator 𝔪_{k} - the transversal case-

We next define

𝔪_{k} : LAG[1](c_{0}, c_{1}) ⊗ ⋯ ⊗ LAG[1](c_{k−1}, c_{k}) → LAG[1](c_{0}, c_{k})

where c_{i} = (L_{i}, s̃_{i}, 𝔏_{i}) ∈ 𝔒𝔟_{4}(LAG(M, Ω, st, L̃ag(M))), under the additional asumption that L_{i} ≠ L_{i+1}. (Here L_{k+1} = L_{0} by convention.) It follows from our assumption that L_{i} is transversal to L_{i+1}. Again the idea presented here is not new and is discussed already in [Fu1], [Fu4]. However we here give a more precise argument especially on degree and orientation. In fact there were several errors on those points in [Fu1].

We use moduli spaces of pseudoholomorphic discs. Let us consider a pair (D^{2}, z⃗) of 2 disc D^{2} with the canonical complex structure and z⃗ = (z_{0}, ⋯, z_{k}), the ordered set of k + 1 points on its boundary. We assume that (z_{0}, ⋯, z_{k}) respects the cyclic order. We denote by M̃_{k+1} the space of all such pairs. The group P SL(2; ℝ) = Aut(D^{2}, J) acts on it, and let M_{k+1} be the quotient space. It is well known that M_{k+1} is diffeomorphic to ℝ^{k−2} and carries a natural orientation. We can compactify it to CM_{k+1} in a way similar to the Deligne-Mumford compactification of the moduli space of marked closed Riemann surfaces. (See [FOh], [FOOO] §3.)

Let (D^{2}, z⃗) ∈ M̃_{k+1}. Let ∂_{i}D^{2} be the part of ∂D^{2} between z_{i−1} and z_{i}. (Here z_{−1} = z_{k+1} by notation.)

Let p_{i} ∈ L_{i} ∩ L_{i+1}. (L_{k+1} = L_{0} by notation.) We fix a compatible almost complex structure J on M. Namely we assume

ω(JX, JY) = ω(X, Y), ω(X, JX) ≥ 0.

We consider the set of all ((D^{2}, z⃗), φ) such that

(3.1.1) (D^{2}, z⃗) ∈ M_{k+1}, φ : D^{2} → M, (3.1.2) φ is pseudoholomorphic, (3.1.3) φ(∂_{i}D^{2}) ⊂ L_{i}, (3.1.4) φ(z_{i}) = p_{i}.

Figure 3.1

We denote by M̃_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) the set of all such ((D^{2}, z⃗), φ).

The group P SL(2; ℝ) acts on M̃_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) by

u · ((D^{2}, z⃗), φ) = ((D^{2}, u(z⃗)), φ ∘ u^{−1}).

Let M_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) be the quotient space. We can compactify our moduli space M_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) by using a notion of stable map from open Riemann surface. (See [FOOO] §3.) Let CM_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) be the compactification.

Theorem 3.2. There exists a Kuranishi structure (with corners) of dimension n + (k + 1) − ∑ η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p_{i}) − 3 on CM_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}).

The notion of Kuranishi structure is defined in [FOn2], to handle transversality problem appeared in the construction of fundamental chains of various moduli spaces in a uniform way. We do not try to define it here. Roughly speaking it is a way to restate the following imprecise statement in a rigorous way :

“Theorem”3.2′^{2}. By a “generic perturbation”, CM_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) will become a manifold with corner. Its dimension is n+(k+1)−∑ η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p_{i})− 3.

The word “generic perturbation” in “Theorem”3.2′ should be made precise. Kuranishi structure is a way to include as the most general perturbation as possible. Usually the reader do not have to be bothered with the detail of the study of a space with Kuranishi structure. The frame work of Kuranishi structure is designed so that desired fundamental chain (usually over ℚ) which has the properties expected from naive guess can be constructed in the situation we are interested in (that is the moduli space of pseudoholomorphic discs). However if the reader is interested in the most delicate parts of the proofs, he needs to investigate Kuranishi structures. (For example the discussion of Remark 3.3 requires a detail of the frame work of Kuranishi structure.) We pretend as if a statement like “Theorem”3.2′ is correct usually and give remarks on Kuranishi structure when necessary.

Remark 3.3. In our case, the fundamental chain of CM_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) can be defined over integer. In fact, by Lemma 2.10, the Maslov index is 0 in our case. Moreover by Lemma 2.6, c^{1}(M) = 0. It follows that our Lagrangian submanifold is semi-positive in the sense of [FOOO] Chapter 3. Therefore, we can apply the method of [FOOO] §9,§A3 to obtain a perturbation of our moduli space so that it is a simplicial complex with fundamental chain over ℤ. (It is actually a space with Whitney stratification.) Using this remark, our filtered A_{∞} category is defined over Λ_{ℤ,0,nov}. Also we can work over Λ_{ℤ_{2},0,nov} and forget all the orientation problems (as we did in [Fu4]). However to work over ℂ (or Λ_{ℂ,nov}) seems to be natural when our main interest is in mirror symmetry and not in applications of Floer homology to symplectic topology.

2We write Theorem in the quote since the statement as is stated is not correct.

The proof of Theorem 3.2 consists of two parts. One is to use implicit function theorem and Taubes’ type gluing argument to construct Kuranishi structure. This part is in fact a straightforward analogue of the argument presented in [FOn2] Chapter 3, [FOh] and [FOOO] Chapter 5. So we do not repeat it. The second part is a calculation of the (virtual) dimension, which is related to the study of Maslov index in the last section. We discuss this second point now. The argument here is also an analogy of one in [FOOO] Chapter 6.

First we remark that the number k + 1 − 3 is the dimension of the moduli space CM_{k+1}. There is a natural projection CM_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) → CM_{k+1}. Hence we are only to show that the virtual dimension of the fiber of the projection is n − ∑ η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p_{i}). (In the case of the space with Kuranishi structure, its virtual dimension is its dimension by definition. So it suffices to calculate the virtual dimension.)

Namely we fix (D^{2}, z⃗) ∈ M_{k+1} and study the moduli space of pseudoholomorphic maps φ. (Actually we need to study the case when the domain of φ is singular also. However since the study of it is similar to and is written in detail in [FOOO] Chapter 5, we omit it.)

The study of virtual dimension is a problem calculating the index. We first choose a metric on the domain and fix a function space to work with. For our purpose, it is convenient to use an alternative representative of (D^{2}, z⃗). Namely we take a one dimensional Kähler manifold Σ with the following properties :

(3.4.1) There exists a compact subset Σ_{0} such that the complement Σ − Σ_{0} is isometric to the disjoint union of k + 1 copies of (−∞, 0] × [0, 1]. (3.4.2) Σ is conformaly equivalent to D^{2} − {z_{1}, ⋯, z_{k+1}}

By using [FOh] Theorem 10.4, such Σ with a singular Riemannian metric is given in a canonical way. Also Σ and metric in [FOh] Theorem 10.4 depends smoothly on (D^{2}, z⃗). This fact is essential to work out the Taubes’ type gluing construction and the analytic detail of the proof of Theorem 3.2, as in [FOOO] Chapter 5. For our purpose, that is to calculate the index, we do not need it.

Note that k + 1 copies of (−∞, 0] × [0, 1] in (3.4.1) corresponds to the marked points z_{0}, ⋯, z_{k+1} by (3.4.2). We let End_{i} Σ ≡ (−∞, 0] × [0, 1] be the copy corresponding to z_{i}.

Let φ : D^{2} → M be a smooth map satisfying (3.1.1), (3.1.3), (3.1.4). It induces a map Σ → M, which we denote by the same symbol. We consider the operator

(3.5)  D_{φ}\overline{∂} : W^{1,p}(Σ, φ^{∗}T M; φ^{∗}T L) → W^{0,p}(Σ, φ^{∗}T M ⊗ Λ^{0,1}Σ).

Here (3.5) is a linearization operator of the pseudoholomorphic curve equation :

\overline{∂}φ = 0,

W^{k,p} in (3.5) denotes the space of all sections whose derivative up to order k is of L^{p} class, and

W^{1,p}(Σ, φ^{∗}T M; φ^{∗}T L) = {u ∈ W^{1,p}(Σ, φ^{∗}T M) | u(x) ∈ T_{φ(x)}L_{i} if x ∈ ∂_{i}Σ}

where ∂_{i}Σ is a part of ∂Σ which corresponds to ∂_{i}D^{2}. (We remark that elements of W^{1,p} is continuous since we assumed p > 2.)

Note that ends of Σ are of product type. We also assumed that L_{i} is transversal to L_{i+1} at p_{i}. Hence the operator (3.5) is nondegenerate at the end. Therefore, by a standard argument, we can prove that (3.5) is Fredholm. We are going to calculate its index.

We first trivialize φ^{∗}T M on Σ. The trivialization is unique up to homotoy since Σ is contractible. We next take a path s_{i} : ∂_{i}Σ → Lag_{n}. s_{i} is defied by s_{i}(x) = T_{φ(x)}T L ⊂ T_{φ(x)}T M = (φ^{∗}T M)_{x} ≅ ℂ^{n}.

We thus reduced the problem calculating the virtual dimension of the moduli space CM_{k+1}(L_{0}, ⋯, L_{k}; p_{1}, ⋯, p_{k}) to the problem calculating the index of the operator

(3.6)  \overline{∂} : W^{1,p}(Σ, ℂ^{n}; s_{0}, ⋯, s_{k}) → W^{0,p}(Σ, ℂ^{n} ⊗ Λ^{0,1}Σ),

where

W^{1,p}(Σ, ℂ^{n}; s_{0}, ⋯, s_{k}) = {u ∈ W^{1,p}(Σ, ℂ^{n}) | u(x) ∈ s_{i}(x) if x ∈ ∂_{i}Σ}.

The index of (3.6) does not change if we change s_{i} in a homotopy class. Hence we may assume :

(3.7)  \{ \begin{matrix}s_{i}(τ, 1) = T_{p_{i}}L & if τ < −T, (τ, 1) ∈ End_{i+1} Σ, \\ s_{i}(τ, 0) = T_{p_{i}}L & if τ < −T, (τ, 0) ∈ End_{i} Σ. \end{matrix}

We next construct an elliptic complex whose index is η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p). We put :

Y = D^{2} ∪ \{x + \sqrt{−1}y | x ≥ 0, y ∈ [−1, 1] \} ⊂ ℂ.

Figure 3.2

We remark ∂Y ≅ ℝ where −∞ corresponds to ∞ − \sqrt{−1} and ∞ corresponds ∞ + \sqrt{−1}. We also define a path ℓ̃_{i} : ℝ → L̃ag(T_{p_{i}}M) such that ℓ̃_{i}(−∞) = s̃_{i}(p) and ℓ̃_{i}(∞) = s̃_{i+1}(p), where s̃_{i} is the grading of the Lagrangian submanifold of L_{i} we have chosen. We may also assume that ℓ_{i}(t) is locally constant if |t| > T. We thus obtain a Fredholm operator

(3.8.i)  \overline{∂} : W^{1,p}(Y, T_{p_{i}}M; ℓ_{i}) → W^{0,p}(Y, T_{p_{i}}M ⊗ Λ^{0,1}Σ).

Here ℓ_{i} = π ∘ ℓ̃_{i} and W^{1,p}(Y, T_{p_{i}}M; ℓ_{i}) is defined by

W^{1,p}(Y, T_{p_{i}}M; ℓ_{i}) = {u ∈ W^{1,p}(Y, T_{p_{i}}M) | u(x) ∈ ℓ_{i}(x) if x ∈ ∂Y ≅ ℝ}.

Now we have the following two Lemmata 3.9 and 3.13.

Lemma 3.9. The index of (3.8.i) is η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p).

Proof. Take a biholomorphic map ψ : ℝ × [0, 1] → Y − {−1} such that (∞, t) corresponds ∞ + t\sqrt{−1} and (−∞, t) corresponds −1 ∈ ∂\overline{Y}. We pull back the boundary condition ℓ_{i} to ℝ × {0, 1} and obtain (ℓ_{i,0}, ℓ_{i,1}). We then consider the operator

(3.10)  \overline{∂} : W^{1,p}(ℝ × [0, 1]; ℂ^{n}; ℓ_{i,0}, ℓ_{i,1}) → W^{0,p}(ℝ × [0, 1]; ℂ^{n} ⊗ Λ^{0,1}).

The operator (3.10) is similar to (2.17). However (3.10) is not a Fredholm operator since the operator is degenerate when τ ∈ ℝ goes to −∞. In fact ℓ_{i,0}(−∞) = ℓ_{i,1}(−∞). So to obtain a Fredholm operator, we need to use a weighted Sobolev norm. We consider the weighted Sobolev norm

∥u∥_{1,p;δ}^{p} = ∥e^{δ|τ|}u∥_{p}^{p} + ∥e^{δ|τ|}∇u∥_{p}^{p}

where ∥∥_{p} is the usual L^{p} norm and ∇ is a covariant derivative. We define W^{1,p;δ}(ℝ× [0, 1]; ℂ^{n}; ℓ_{i,0}, ℓ_{i,1}) using this weighted Sobolev norm and the same boundary condition as W^{1,p}(ℝ × [0, 1]; ℂ^{n}; ℓ_{i,0}, ℓ_{i,1}). We then consider

(3.11.δ)  \overline{∂} : W^{1,p;δ}(ℝ × [0, 1]; ℂ^{n}; ℓ_{i,0}, ℓ_{i,1}) → W^{0,p;δ}(ℝ × [0, 1]; ℂ^{n} ⊗ Λ^{0,1}).

(3.11.δ) is a Fredholm operator for nonzero small δ. We can prove the following easily.

Sublemma 3.12. The index of (3.11.δ) is η(ℓ_{i,0}, ℓ_{i,1}) − n if δ > 0 and if |δ| is small, and is η(ℓ_{i,0}, ℓ_{i,1}) if δ < 0 and if |δ| is small.

In fact, if we study (3.10) using spectral flow as in the sketch of the proof of Proposition 2.19, then we find that n is the number of eigenvalues converging to 0 as τ → −∞. Hence the index changes by n when we move δ from negative to positive. This difference corresponds to the way to perturb ℓ_{i,1}(τ) for τ close to −∞. Namely if we perturb so that ℓ_{i,0}(−∞) is transversal to ℓ_{i,1}(−∞). (Then the operator (3.11.δ) will become elliptic.) We can prove the sublemma using this observation.

Now we consider the case when δ > 0. This means that we consider the solution of \overline{∂}u = 0 with u converges to 0 in the exponential order as τ → −∞. Then when we transform u by ψ, it will corresponds to an element of the kernel of (3.8.i) such that its value at −1 is zero. Therefore we find that the index of (3.11.δ) for δ > 0 is the index of (3.8.i) minus n. Lemma 3.9 follows from Sublemma 3.12. □

Lemma 3.13. The index of (3.6) plus the sum of the indices of (3.8.i) for i = 0, ⋯, k is n.

Proof. We glue the elliptic operator (3.6) with the elliptic operators (3.8.i) on their boundaries. Then we obtain an elliptic operator

(3.14)  \overline{∂} : W^{1,p}(D^{2}; ℂ^{n}; ℓ) → W^{0,p}(D^{2}; ℂ^{n} ⊗ Λ^{0,1}).

Here notations in (3.14) are defined as follows.

ℓ : S^{1} → ℂ^{n} is a path obtained by joining s_{0}, ℓ_{0}, s_{1}, ℓ_{1}, s_{2}, ℓ_{2}, ⋯, ℓ_{k}, s_{k} in this order. W^{1,p}(D^{2}; ℂ^{n}; ℓ) is the set of W^{1,p} sections u of ℂ^{n} on D^{2} such that u(t) ∈ ℓ(t) for t ∈ S^{1} = ∂D^{2}.

Now the index of (3.14) is equal to the index of (3.6) plus the sum of the indices of (3.8.i) for i = 0, ⋯, k by excision property of indices.

On the other hand, the homotopy invariance of index implies that the index of (3.14) depends only on the homotopy type of ℓ. Moreover, in fact, ℓ is lifted to L̃ag_{n} since s_{0}, ℓ_{0}, s_{1}, ℓ_{1}, s_{2}, ℓ_{2}, ⋯, ℓ_{k}, s_{k} are all lifted. Hence we may assume that ℓ is constant. Then it is easy to see that the index of (3.14) is n. The proof of Lemma 3.13 is complete. □

From Lemma 3.9 and Lemma 3.13 we find that the virtual dimension of the moduli space CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}) is as asserted in Theorem 3.2. □

We next discuss the orientation. To define an orientation of the moduli space CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}), we need one extra data, which we discuss now. We consider the complex (3.8.i). Its index as a virtual vector space depends only on the homotopy class of the path ℓ_{i}. Since ℓ_{i} = π ∘ ℓ̃_{i}, and ℓ̃_{i} is a path joining s̃_{i}(p) and s̃_{i+1}(p) which is unique up to homotopy, it follows that the homotopy class of ℓ_{i} is defined in a canonical way for graded Lagrangian submanifold. Therefore the virtual vector space, which is an index of (3.8.i), is well-defined. Now we choose the orientation of this virtual vector space, the index of (3.8.i).

Remark 3.15. In a similar context of finite dimensional Morse theory, we need to fix an orientation of the stable (or unstable) manifold of each critical point in order to fix an orientation of the moduli space of connecting orbit. (See for example [FOn2] §21.)

The orientation of the index virtual vector space of (3.8.i) corresponds to the choice of orientation of stable manifold in the finite dimensional Morse theory.

To describe the choice of the orientation of the index of (3.8.i) in more canonical way, we proceed as follows. We modify the definition in (2.30) and put

(3.16)  \begin{aligned}LAG^{k} & ((L_{1}, s̃_{1}, 𝔏_{1}), (L_{2}, s̃_{2}, 𝔏_{2})) \\ & = ⨁_{{p∈L_{1}∩L_{2} \atop η_{(L_{1},s̃_{1}),(L_{2},s̃_{2})}(p)=k}} Hom((𝔏_{1})_{p}, (𝔏_{2})_{p}) \\ & ⊗_{ℂ} Λ_{0,nov} ⊗_{ℝ} Λ^{top}(Index (3.8.i)). \end{aligned}

(3.16) is isomorphic to (2.30). The choice of isomorphism corresponds one to one to the choice of orientations of the index of (3.8.i). In case we fix a choice of the index virtual vector space of (3.8.i) for each p, we use (2.30) in place of (3.16).

Remark 3.17. In (3.16) we include the determinant of (3.8.i) in Floer’s chain complex. The necessity of it becomes more apparent if we consider more general situation. Namely let us consider the case when L_{1} may not transversal to L_{2} but is of clean intersection. (It means that the L_{1} ∩ L_{2} is a submanifold and the T_{p}L_{1} ∩ T_{p}L_{2} ≅ T_{p}(L_{1} ∩ L_{2}) for p ∈ L_{1} ∩ L_{2} is constant.) In that case the right hand side of (3.16) will become

LAG((L_{1}, s̃_{1}, 𝔏_{1}), (L_{2}, s̃_{2}, 𝔏_{2}))
= Γ(L_{1} ∩ L_{2}; Hom(𝔏_{1}|_{L_{1}∩L_{2}}, 𝔏_{2}|_{L_{1}∩L_{2}})
⊗_{ℂ} Λ_{0,nov} ⊗_{ℝ} Λ^{top}(Index (3.8.i)).

Here Λ^{top}(Index (3.8.i)) is one dimensional real vector bundle on L_{1} ∩ L_{2}, which corresponds to a local system π_{1}(L_{1} ∩ L_{2}) → {±1}. See [FOOO] Chapter 6 §25.6 for detail.

Theorem 3.18. A choice of (M, st)-relative spin structures on L_{i} and a choice of an orientation of the index virtual vector space of (3.8.i) induce an orientation of CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}), in a canonical way.

Here orientation of CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}) means orientation in the sense of Kuranishi structure [FOn2].

Proof. The idea of the proof is a combination of ones in [FOOO] Chapter 6 and [FOn] §21, (the later is suggested already in Floer’s paper [Fl3]). Let E_{i} be the index virtual vector space of (3.8.i). For each φ : (D^{2}, ∂D^{2}) → (M, L), let E(φ) be the index virtual vector space of (3.6). φ induces paths s_{i} : ∂_{i}Σ → Lag_{n} as explained during the proof of Theorem 3.2. Joining them with ℓ_{i} we obtain a loop ℓ : Σ → Lag_{n} as in the proof of Theorem 3.2. ℓ depends continuously on φ, so we write ℓ(φ). We thus find a family of elliptic operators

(3.19)  \overline{∂} : W^{1,p}(D^{2}; ℂ^{n}; ℓ(φ)) → W^{0,p}(D^{2}; ℂ^{n} ⊗ Λ^{0,1}).

parametrized by φ. The following result is proved in [FOOO] Chapter 6 Theorem 21.1 (see also D.Silva [Sil].)

Theorem 3.20. The choice of (M, st)-relative spin structures on L_{i} induces an orientation of the index bundle of (3.19).

Let E′(φ) be the virtual vector space which is the index of (3.19). Let E(φ) be the virtual vector space which is the index of (3.5). Let E_{i} be the index of (3.8.i). Then by family of index gluing theorem (see for example [Fu8] §4) we have an isomorphism

(3.21)  E′(φ) ⊕ ⨁_{i=0}^{k} E_{i} ≅ E(φ).

Here E′(φ), E(φ) are virtual vector bundles and E_{i} are virtual vector spaces. Theorem 3.18 follows from Theorem 3.20 and (3.21). □

Remark 3.22. We may take an orientation of CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}) such that (3.21) holds as an identify of oriented vector space. However in order to define an operator 𝔪_{k} satisfying A_{∞} formula, we need to change this orientation in a way depending on the dim E_{i} and n in a similar way to [FOOO] Chapter 6. Namely we put

\begin{matrix}E′(φ) ⊕ (−1)^{ϵ} ⨁_{i=0}^{k} E_{i} ≅ E(φ), \\ ϵ = ∑_{j=0}^{k} ∑_{ℓ=1}^{j} dim E_{i}. \end{matrix}

(Compare the above formula to [FOOO] Remark 25.2 (1).)

The problem of degree and orientation being understood, we are ready to define 𝔪_{k} in the case L_{i} ≠ L_{i+1}.

We consider the case when the dimension

\begin{matrix}dim & CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}) \\ = n + (k + 1) − ∑_{i=0}^{k} η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p_{i}) − 3 = 0. \end{matrix}

This (together with (2.37)) implies

(3.23)  1 + ∑_{i=0}^{k−1}(η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p_{i}) − 1) = η_{(L_{0},s̃_{0}),(L_{k},s̃_{k})}(p_{k}) − 1.

Let us put deg p_{i} = η_{(L_{i},s̃_{i}),(L_{i+1},s̃_{i+1})}(p_{i}) for i = 0, ⋯, k − 1, and deg p_{k} = η_{(L_{0},s̃_{0}),(L_{k},s̃_{k})}(p_{k}). Then (3.23) implies that

1 + ∑(deg p_{i} − 1) = deg p_{k} − 1.

We are going to define the matrix elements

(3.24) \begin{matrix}⟨ & 𝔪_{k}(p_{0}, ⋯, p_{k−1}), p_{k}⟩ \\ ∈ Hom (⨂_{i=0}^{k−1} Hom((𝔏_{i})_{p_{i}}, (𝔏_{i+1})_{p_{i}}), Hom((𝔏_{0})_{p_{k}}, (𝔏_{k})_{p_{k}})) ⊗_{ℂ} Λ_{nov} \end{matrix}

when (3.23) is satisfied.

Before defining (3.24), we explain how (3.24) defines 𝔪_{k}. We recall that that an element of LAG((L_{i}, s̃_{i}, 𝔏_{i}), (L_{i+1}, s̃_{i+1}, 𝔏_{i+1})) is a formal sum

∑_{j=0}^{∞} T^{λ_{i}^{(j)}}[v_{i}^{(j)}]

where

v_{i}^{(j)} ∈ Hom((𝔏_{i})_{p_{i}^{(j)}}, (𝔏_{i+1})_{p_{i}^{(j)}}), p_{i}^{(j)} ∈ L_{i} ∩ L_{i+1}.

Then we define

𝔪_{k}(v_{1}^{(j_{1})}, ⋯, v_{k−1}^{(j_{k−1})}),

by

(3.25)  \begin{matrix}𝔪_{k}(v_{1}^{(j_{1})}, ⋯, v_{k−1}^{(j_{k−1})}) \\ = ∑_{j_{1},··· ,j_{k}} ∑_{p_{k}∈L_{0}∩L_{k}} T^{λ_{1}^{(j_{1})}+···+λ_{k−1}^{(j_{k−1})}} \\ ⟨𝔪_{k}(p_{0}^{(j_{1})}, ⋯, p_{k−1}^{(j_{k−1})}), p_{k}⟩(v_{1}^{(j_{1})} ⊗ ⋯ ⊗ v_{k−1}^{(j_{k−1})}), \end{matrix}

using (3.24). Now we define (3.24).

Definition 3.26. We define (3.24) by

(3.27)  ∑_{φ∈CM_{k+1}(L_{0},··· ,L_{k};p_{0},··· ,p_{k})} T^{ℜE(φ)}e^{\sqrt{−1}ℑE(φ)}h_{∇}(φ(∂D^{2}))ϵ_{φ}

where E, h, ϵ are defined below. First we define E by

E(φ) = ∫_{D^{2}} φ^{∗}Ω.

ℜE and ℑE are its real and imaginary parts, respectively.

Next we define h. Let v_{i} ∈ Hom((𝔏_{i})_{p_{i}}, (𝔏_{i+1})_{p_{i}}). Then

(3.28)  \begin{aligned}h_{∇} & (φ(∂D^{2}))(v_{0} ⊗ ⋯ ⊗ v_{k−1}) \\ & = P_{∇_{k}}(φ(∂_{k}D^{2})) ∘ v_{k−1} ∘ ⋯ ∘ P_{∇_{1}}(φ(∂_{1}D^{2})) ∘ v_{0}. \end{aligned}

Here P_{∇_{i}}(φ(∂_{i}D^{2})) : (𝔏_{i})_{p_{i}} → (𝔏_{i})_{p_{i+1}} is the parallel transport along the path φ(∂_{i}D^{2}) of the bundle 𝔏_{i} with respect to the connection ∇_{i}.

Finally ϵ_{φ} ∈ {±1} is determined by the orientation of CM_{k+1}(L_{0}, ⋯, L_{k} ; p_{0}, ⋯, p_{k}) defined by Theorem 3.20. (See also Remark 3.22.)

Figure 3.3

Lemma 3.29. (3.27) is an element of Hom((𝔏_{0})_{p_{k}}, (𝔏_{k})_{p_{k}}) ⊗ Λ_{+,nov}.

Lemma 3.27 is a consequence of Gromov compactness theorem. (See [FOOO] Proposition 5.8 for the proof of a similar statement.) The following lemma is used in the proof of A_{∞} formula.

Lemma 3.30. T^{ℜE(φ)}e^{\sqrt{−1}ℑE(φ)}h_{∇}(φ(∂D^{2})) depends only on the homotopy class of φ.

Proof. We can prove that ℜE(φ) depends only on the homotopy class of φ by using Stokes’ theorem and the fact that L_{i} is a Lagrangian submanifold. Homotopy independence of e^{\sqrt{−1}ℑE(φ)}h_{∇}(φ(∂D^{2})) follows from the condition that the curvature of ∇_{i} is 2π\sqrt{−1}B (Condition (2.1.2)). □

Remark 3.31. We remark that we are working under the hypothesis L_{i} ≠ L_{i+1} in this section. In this case we put 𝔪_{0} = 0. We also remark that in this case the coefficient of the operator belongs to the maximal ideal Λ_{+,nov}, since the energy of nonconstant pseudoholomorphic map is always positive.

Remark 3.32. Theorem 3.2 implies that CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}) has a Kuranishi structure but it is in general not true that we can find a generic perturbation using J so that it is a manifold. So in order to make sense of (3.27) we need to take multivalued perturbation (multisection) as in [FOn2] and use it to define a fundamental chain over ℚ of our moduli space CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}). In the situation of (3.27) where the moduli space is of dimension 0, it means that we need to put multiplicity = ±1/integer in place of ϵ_{φ} ∈ {±1}.

However, in our situation, all of our Lagrangian submanifolds are semipositive in the sense defined in [FOOO] Chapter 3. Hence by using normally polynomial sections described in [FOOO] §A3, we can prove that CM_{k+1}(L_{0}, ⋯, L_{k}; p_{0}, ⋯, p_{k}) has a fundamental chain over ℤ. In other words our operation 𝔪_{k} is defined over Λ_{ℤ,0,nov}.

The next task to be carried out might be the proof of A_{∞} formula d̂∘ d̂ = 0. However the formula

(3.33)  ∑ ±𝔪_{∗}(x_{0}, ⋯, x_{ℓ−1}, 𝔪_{∗}(x_{ℓ}, ⋯, x_{m}), x_{m+1}, ⋯, x_{k}) = 0

does not holds if we consider only 𝔪_{∗} with ∗ > 0. In other words we can not prove A_{∞} formula when we consider only L_{i}’s with L_{i} ≠ L_{i+1}. We need to include the case L_{i} = L_{i+1} and 𝔪_{0}, which is nonzero in the general case. Namely, for example, we are not able to prove

(3.34)

0 =𝔪_{1}(𝔪_{3}(x_{0}, x_{1}, x_{2}))
+ 𝔪_{2}(𝔪_{2}(x_{0}, x_{1}), x_{2}) + (−1)^{deg x_{0}+1}𝔪_{2}(x_{0}, 𝔪_{2}(x_{1}, x_{2}))
+ 𝔪_{3}(𝔪_{1}(x_{0}), x_{1}, x_{2}) + (−1)^{deg x_{0}+1}𝔪_{3}(x_{0}, 𝔪_{1}(x_{1}), x_{2})
+ (−1)^{deg x_{0}+1+deg x_{1}+1}𝔪_{3}(x_{0}, x_{1}, 𝔪_{1}(x_{2})),

which is (3.33) in case k = 2, if we consider only 𝔪_{∗} with ∗ > 0. In place of (3.34), we will prove

(3.35)

0 =𝔪_{1}(𝔪_{3}(x_{0}, x_{1}, x_{2}))
+ 𝔪_{2}(𝔪_{2}(x_{0}, x_{1}), x_{2}) + (−1)^{deg x_{0}+1}𝔪_{2}(x_{0}, 𝔪_{2}(x_{1}, x_{2}))
+ 𝔪_{3}(𝔪_{1}(x_{0}), x_{1}, x_{2}) + (−1)^{deg x_{0}+1}𝔪_{3}(x_{0}, 𝔪_{1}(x_{1}), x_{2})
+ (−1)^{deg x_{0}+1+deg x_{1}+1}𝔪_{3}(x_{0}, x_{1}, 𝔪_{1}(x_{2}))
+ 𝔪_{4}(𝔪_{0}(1), x_{0}, x_{1}, x_{2}) + (−1)^{deg x_{0}+1}𝔪_{4}(x_{0}, 𝔪_{0}(1), x_{1}, x_{2})
+ (−1)^{deg x_{0}+1+deg x_{1}+1}𝔪_{4}(x_{0}, x_{1}, 𝔪_{0}(1), x_{2})
+ (−1)^{deg x_{0}+1+deg x_{1}+1+deg x_{2}+1}𝔪_{4}(x_{0}, x_{1}, x_{2}, 𝔪_{0}(1)).

Here, for example, the element 𝔪_{0}(1) which appeared in 𝔪_{4}(𝔪_{0}(1), x_{0}, x_{1}, x_{2}) is an element of the Λ_{0,nov} module LAG((L_{0}, s̃_{0}, 𝔏_{0}), (L_{0}, s̃_{0}, 𝔏_{0})). Thus 𝔪_{0} and 𝔪_{4} in (3.35) are not defined in this section. So we generalize the definition of this section and discuss the case when L_{i} = L_{i+1} for some i, in the next section.

4. Floer homology and A_{∞} category III

- the operator 𝔪_{k} - the general case-

The discussion of this section is a combination of the argument of [FOOO] (especially its Chapter 4) and one of the last section. Namely, in the case when c_{0} = ⋯ = c_{k+1} = c = (L, s̃, 𝔏), the operator

𝔪_{k} : LAG[1](c, c) ⊗ ⋯ ⊗ LAG[1](c, c) → LAG[1](c, c)

(k = 0, 1, ⋯) is the operator of the filtered A_{∞} algebra structure constructed in [FOOO] Theorem 13.22.

We consider the following situation. Let L_{(j)}, j = 1, ⋯, m be Lagrangian submanifolds such that L_{(j)} ≠ L_{(j+1)}. Let ℓ_{j} ∈ ℤ_{>0}. We put L_{0} = ⋯ = L_{ℓ_{1}−1} = L_{(0)}, L_{ℓ_{1}} = ⋯ = L_{ℓ_{1}+ℓ_{2}−1} = L_{(1)}, …, L_{∑_{j=0}^{m−1} ℓ_{j}} = ⋯ = L_{∑_{j=0}^{m} ℓ_{j}−1} = L_{(m)}. We take s̃_{i}, 𝔏_{i} so that

c_{i} = (L_{i}, s̃_{i}, 𝔏_{i}) ∈ 𝔒𝔟_{4}(LAG(M, Ω, st, L̃ag(M))).

We write also

s_{i}^{(j)} = s_{ℓ_{0}+···+ℓ_{i−1}+j}, 𝔏_{i}^{(j)} = 𝔏_{ℓ_{0}+···+ℓ_{i−1}+j},

sometimes. We are going to define :

𝔪_{k} : LAG[1](c_{0}, c_{1}) ⊗ ⋯ ⊗ LAG[1](c_{k−1}, c_{k}) → LAG[1](c_{0}, c_{k}).

(Here k = 0, 1 · · ·.) Note that we defined the graded Λ_{0,nov} module LAG[1](c_{i}, c_{i+1}) if L_{i} ≠ L_{i+1} in Definition 2.29. But in case L_{i} = L_{i+1} we need to start with the definition of LAG[1](c_{i}, c_{i+1}). Let us put L = L_{i} = L_{i+1}. We have (s̃_{i}, 𝔏_{i}), (s̃_{i+1}, 𝔏_{i+1}) on L_{i}, L_{i+1}.

Roughly speaking the graded Λ_{0,nov} module LAG((L, 𝔏_{i}), (L, 𝔏_{i+1})) we are going to define is a singular chain complex with local coefficient Hom(𝔏_{i}, 𝔏_{i+1}). (We remark that the curvature of 𝔏_{i} coincides with the curvature of 𝔏_{i+1} by (2.2.2). Hence Hom(𝔏_{i}, 𝔏_{i+1}) is a flat U(1) bundle.) But we need to be a bit careful in defining it, so that the definition of 𝔪_{k}, which is based on the smooth correspondence, works. (See also [FOOO] §A1.)

Let σ : Δ^{q} → L be a smooth (singular) simplex. Let (1, 0, ⋯, 0) ∈ Δ^{q} be the base point. We take a trivialization σ^{∗}𝔏_{i} ≅ Δ^{q} × (𝔏_{i})_{σ(1,0,··· ,0)} by using a parallel transport along the σ image of the straight line joining (1, 0, ⋯, 0) with a given point on Δ^{q}. A smooth singular simplex with local coefficient Hom(𝔏_{i}, 𝔏_{i+1}) is a pair (σ, u) such that σ : Δ^{q} → L and u ∈ Hom((𝔏_{i})_{σ(1,0,··· ,0)}, (𝔏_{i+1})_{σ(1,0,··· ,0)}). We now put :

(4.1)  S_{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) = {∑ a_{i}(σ_{i}, u_{i})| a_{i} ∈ ℂ}/ ∼_{1} .

Here (σ_{i}, u_{i}) are smooth singular chain complexes with local coefficient, ∑ is a finite sum, and ∼_{1} is defined by a_{i}(σ_{i}, u_{i}) + a′_{i}(σ_{i}, u′_{i}) ∼_{1} (σ_{i}, a_{i}u_{i} + a′_{i}u′_{i}). We can define a boundary operator ∂ on it using a trivialization σ^{∗}𝔏_{i} ≅ Δ^{q} ×(𝔏_{i})_{σ(1,0,··· ,0)}.

We need to divide it further by an appropriate equivalence relation and then take a countably generated subcomplex, in order to obtain the chain complex we use. Let us now describe this process.

Let Hom(𝔏_{i}, 𝔏_{i+1})^{∗} be the dual bundle of Hom(𝔏_{i}, 𝔏_{i+1}). We consider the vector bundle Hom(𝔏_{i}, 𝔏_{i+1}) ⊗ Λ^{q}(L) on L and let

W^{−∞}(Hom(𝔏_{i}, 𝔏_{i+1}) ⊗ Λ^{q}(L))

be the set of all distribution valued sections of it. Note an element of it can be identified with a linear map

C^{∞}(Hom(𝔏_{i}, 𝔏_{i+1})^{∗} ⊗ Λ^{n−q}(L)) → ℂ

by

T ↦ (w ↦ ∫_{L} T ∧ w) .

(We assume that L is oriented.) We use this identification to define a map

T : S_{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) → W^{−∞}(Hom(𝔏_{i}, 𝔏_{i+1}) ⊗ Λ^{n−q}(L))

by putting

(4.2)  ∫_{L} T (σ_{i}, u_{i}) ∧ w = ∫_{Δ^{q}}⟨u_{i}, σ^{∗}w⟩,

and extending it to a complex linear map in an obvious way. Note that we used the trivialization σ^{∗}𝔏_{i} ≅ Δ^{q} × (𝔏_{i})_{σ(1,0,··· ,0)} mentioned before to define the right hand side of (4.2).

We define an equivalence relation S_{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) such that x ∼_{2} y if and only if T (x) = T (y). We now put

\overline{S}_{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) = S_{q}(L; Hom(𝔏_{i}, 𝔏_{i+1}))/ ∼_{2} .

Let P be a simplicial complex of dimension q and f : P → L be a piecewise smooth map. We assume that P has a base point. We assume that the pull back f^{∗}Hom(𝔏_{i}, 𝔏_{i+1}) is trivial and we fix a trivialization. Let u be an element of the fiber of f^{∗}Hom(𝔏_{i}, 𝔏_{i+1}) at the base point. Then, from f and u, we obtain an element of \overline{S}_{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) in an obvious way. This element is independent of the subdivision of the simplicial complex P (as a chain). We denote it by [P, f, u]. Every element of \overline{S}_{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) is realized in this way. We remark that, if we do not divide by the equivalence relation ∼_{2}, then the element we obtain will depend on the subdivision of the simplicial complex P.

We use cohomology rather than homology (since the product structure is the main issue here). We write

\overline{S}^{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) = \overline{S}_{n−q}(L; Hom(𝔏_{i}, 𝔏_{i+1})).

Next we need to take a countably generated (over ℂ) subcomplex of \overline{S}^{q}(L; Hom (𝔏_{i}, 𝔏_{i+1})). We recall that we have fixed a countable set of Lagrangian submanifolds and we assumed that the Lagrangian submanifold part of the objects of our category is always in this set. We next choose a countably generated subcomplex of \overline{S}^{∗}(L) for each member L of the countable set of Lagrangian submanifolds we have chosen.

The condition that this subcomplex is assumed to satisfy, is rather delicate and is not mentioned here. (See [FOOO] §A1, §A5.) The reason we need to choose a countably generated subcomplex is that we need to use frequently Bair’s category theorem to achieve transversality in various situations and in Bair’s category theorem countability is an essential issue. The transversality here is not only a technical problem but also is related to many essential points of the story. It is related to the fact that, for example, the square of delta function is ill-defined, and hence is also related to the problem of infinity in quantum field theory.

We denote by C^{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) the countably generated subcomplex we have chosen. We put

LAG^{q}((L, 𝔏_{i}), (L, 𝔏_{i+1})) = C^{q}(L; Hom(𝔏_{i}, 𝔏_{i+1})) ˆ⊗_{ℂ}Λ_{0,nov}.

Here ˆ⊗_{ℂ} means that we take a completion by using topology induced by the filtration of Λ_{0,nov}. (Note we assumed L_{i} = L_{i+1} = L.) In other words an element of LAG^{q}((L, 𝔏_{i}), (L, 𝔏_{i+1})) is realized by countable sum

∑ T^{λ_{i}}[P_{i}, f_{i}, u_{i}]

where λ_{i} → ∞. Using the grading of our Lagrangian submanifold, we shift the degree as follows. Let s̃_{i} and s̃_{i+1} be two gradings of our Lagrangian submanifold L. Then, as mentioned in section 2, there exists a unique integer k such that s̃_{i+1} = k · s̃_{i}. Here ℤ acts on L̃ag_{n} as a deck transformation group. We put

(4.3)  k = s̃_{i+1} − s̃_{i} ∈ ℤ.

Now we define :

Definition 4.4.

LAG^{q}((L, s̃_{i}, 𝔏_{i}), (L, s̃_{i+1}, 𝔏_{i+1})) = C^{q+(s̃_{i+1}−s̃_{i})}(L; Hom(𝔏_{i}, 𝔏_{i+1})) ˆ⊗_{ℂ}Λ_{0,nov}.

We thus defined LAG(c, c′) in our general situation. We turn to the definition of our operators 𝔪_{k}.

Let L_{(j)}, ℓ_{j}, (j = 1, ⋯, m), c_{i} = (L_{i}, s̃_{i}, 𝔏_{i}) (i = 0, ⋯, ∑_{j=0}^{m} ℓ_{j}) be as in the beginning of this section. For i with L_{(j)} = L_{i} ≠ L_{i+1} = L_{(j+1)}, we take p_{i} = p_{(j)} ∈ L_{(j)} ∩ L_{(j+1)} and

v_{i} ∈ Hom((𝔏_{i})_{p_{(j)}}, (𝔏_{i+1})_{p_{(j)}}).

For i with L_{(j)} = L_{i} = L_{i+1}, we take

x_{i} = [P_{i}, f_{i}, u_{i}] ∈ C^{g_{i}+(s̃_{i+1}−s̃_{i})}(L_{(j)}; Hom(𝔏_{i}, 𝔏_{i+1})).

We are going to define

(4.5)  \begin{matrix}𝔪_{∑_{j=0}^{m−1} ℓ_{j}}(x_{0}, ⋯, & x_{ℓ_{0}−1}, v_{ℓ_{0}}, x_{ℓ_{0}+1}, ⋯, x_{ℓ_{0}+ℓ_{1}−1}, v_{ℓ_{0}+ℓ_{1}}, \\ x_{ℓ_{0}+ℓ_{1}−1}, ⋯, x_{ℓ_{0}+···+ℓ_{m−1}−1}, \\ v_{ℓ_{0}+···+ℓ_{m−1}}, x_{ℓ_{0}+···+ℓ_{m−1}+1}, ⋯, x_{ℓ_{0}+···+ℓ_{m}−1}). \end{matrix}

Hereafter we write (4.5) as 𝔪(x_{0}, ⋯, x_{ℓ_{0}+···+ℓ_{m}−1}), for simplicity. There are two cases :

(4.6.1) L_{(m)} ≠ L_{(0)}. (4.6.2) L_{(m)} = L_{(0)}.

Case (4.6.1) : In this case we take p_{(m)} ∈ L_{(m)} ∩ L_{(0)} and are going to define the matrix element

(4.7)  ⟨𝔪(x_{0}, ⋯, x_{ℓ_{0}+···+ℓ_{m}−1}), p_{(m)}⟩ ∈ Hom((𝔏_{ℓ_{0}+···+ℓ_{m}−1})_{p_{(m)}}, (𝔏_{0})_{p_{(m)}}).

We need to define (4.7) only in case the degree is correct. Namely in case

(4.8)  ∑(g_{i} + 1) + ∑(η(p_{(j)}) + 1) = η(p_{(m)}) + 1.

To define (4.7) in case (4.8) is satisfied, we use a moduli space which is similar to but a bit more complicated than one we used in the last section. Let us define it now. Let us consider the system ((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ) such that

(4.9.1) ((D^{2}, z⃗), φ) ∈ M̃_{m+1}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m)}), where the right hand side is as in (3.1). (4.9.2) w^{(j)} = (w_{1}^{(j)}, ⋯, w_{ℓ_{j}−1}^{(j)}). w_{i}^{(j)} ∈ ∂_{j}D^{2}. Here ∂_{j}D^{2} is as in the last section. (4.9.3) If i ≠ i′ then w_{i}^{(j)} ≠ w_{i′}^{(j)}. w_{1}^{(j)}, ⋯, w_{ℓ_{j}−1}^{(j)} respects the order of ∂_{j}D^{2}.

Figure 4.1 The totality of such ((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ) is denoted by

M̃_{m+1}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m)}; ℓ_{0}, ⋯, ℓ_{m}).

We divide it by an obvious action of P SL(2; ℝ) and denote the quotient space by

M_{m+1}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m)}; ℓ_{0}, ⋯, ℓ_{m}).

We can compactify it by using the notion of stable maps. (See [FOOO] §3 for its definition in the case Riemann surface has a boundary. We denote the compactification by

CM_{m+1}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m)}; ℓ_{0}, ⋯, ℓ_{m}).

We define evaluation maps

e⃗v = (e⃗v^{(0)}, ⋯, e⃗v^{(m)}), e⃗v^{(j)} = (ev_{1}^{(j)}, ⋯, ev_{ℓ_{j}−1}^{(j)}),

such that

e⃗v^{(j)} : CM_{m+1}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m)}; ℓ_{0}, ⋯, ℓ_{m}) → L_{(j)}^{ℓ_{j}−1}

is

e⃗v_{i}^{(j)}[(D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ] = φ(w_{i}^{(j)}).

We consider the fiber product

(4.10)  \begin{matrix}CM_{m+1}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, & ⋯, p_{(m)}; ℓ_{0}, ⋯, ℓ_{m}) \\ e⃗v & ×_{f_{∗}} ∏_{j} ∏_{i=1}^{ℓ_{j}−1} P_{ℓ_{0}+···+ℓ_{j−1}+i}. \end{matrix}

Now we have :

Lemma 4.11. If (4.8) is satisfied then (4.10) has a Kuranishi structure of dimension 0. The choice of relative spin structure on L_{(j)} and the choices of the orientations of the index virtual bundle of (3.8.i), determine an orientation of (4.10) in a canonical way.

Proof. The proof of Lemma 4.11 is a straight forward combination of the argument of [FOOO] Chapters 5,6 and one in the last section. We remark that to prove Lemma 4.11 we need to restrict the chains P_{i} so that only countably many of them are studied, since we need to use Bair’s category theorem to prove Lemma 4.11.

We need to choose the orientation by modifying the fiber product orientation in a way combining [FOOO] Chapter 6 and (3.21) as follows :

We remark that each P_{i} is regarded as a cochain rather than a chain. This means that we coorient it. Namely for each x ∈ P_{i} such that f_{i} : P_{i} → L ( f_{i} is the map defining P_{i} as a differential form valued distribution) is an immersion at x, we have an orientation of the normal bundle N_{f_{i}(x)}f_{i}(P_{i}).

For each marked point z_{i} we define E_{i} as follows. If L_{i} ≠ L_{i+1} then we define E_{i} as in (3.21) namely it is the index of (3.8.i). In case L_{i} ≠ L_{i+1}, then z_{i} corresponds to some P_{i′}. Then we put E_{i} = N_{f_{j}(x)}f_{j}(P_{j}) here f_{j}(x) = φ(z_{i}). Then (3.21) holds. So we define an orientation by (3.22). □

Now to define the matrix element (4.7), we need to define a weight. We put

E((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ) = ∫_{D^{2}} φ^{∗}ω.

Then the “absolute value” of the weight we put is T^{E(((D^{2},z⃗;w^{(0)},··· ,w^{(m)}),φ))}. We next are going to define the “phase factor”

H(((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ); u⃗, v⃗) ∈ Hom((𝔏_{ℓ_{0}+···+ℓ_{m}−1})_{p_{(m)}}, (𝔏_{0})_{p_{(m)}}).

Here u⃗ denotes the totality of u_{i} ∈ Hom((𝔏_{i})_{q_{i}}, (𝔏_{i+1})_{q_{i}}) here q_{i} is the image by f_{i} of the base point of P_{i}, and v⃗ denotes the totality of v_{i} ∈ Hom((𝔏_{i})_{p_{(j)}}, (𝔏_{i+1})_{p_{(j)}}). We put

α_{i} = \{ \begin{matrix}u_{i} & if L_{i} = L_{i+1}, \\ v_{i} & if L_{i} ≠ L_{i+1}. \end{matrix}

Now we put

P_{∇}(φ(∂D^{2})) = P_{∇_{∑{m \atop j=0} ℓ_{j}}} (φ(∂_{∑_{j=0}^{m} ℓ_{j}}D^{2})) ∘ α_{∑_{j=0}^{m} ℓ_{j}−1} ∘ ⋯ ∘ α_{1} ∘ P_{∇_{1}}(φ(∂_{1}D^{2})).

Figure 4.2.

We remark here that we use the trivialization of f_{i}^{∗}Hom(𝔏_{i}, 𝔏_{i+1}) to regard u_{i} as an element of Hom((𝔏_{i})_{w_{i′}^{(j)}}, (𝔏_{i+1})_{w_{i′}^{(j)}}) where ℓ_{0} + ⋯ + ℓ_{j−1} + i′ = i. (A priori,

u_{i} is a homomorphism between the fibers at q_{i} = f_{i}(1, 0, ⋯, 0).)

Now we define

H(((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ); u⃗, v⃗) = exp (2π\sqrt{−1} ∫_{D^{2}} φ^{∗}B) P_{∇}(φ(∂D^{2})).

Definition 4.12. We assume that (4.6.1) and (4.8) are satisfied. Then we define the matrix element (4.7) by

(4.13)  \begin{matrix}∑_{[(D^{2},z⃗;w^{(0)},··· ,w^{(m)}),φ]∈(4.10)} & ϵ_{φ}T^{E(((D^{2},z⃗;w^{(0)},··· ,w^{(m)}),φ))} \\ H(((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ); u⃗, v⃗). \end{matrix}

Here ϵ_{φ} = ±1 is determined by the orientation of the moduli space (4.10).

It is a consequence of Gromov compactness that (4.13) is an element of

Hom((𝔏_{ℓ_{0}+···+ℓ_{m}−1})_{p_{(m)}}, (𝔏_{0})_{p_{(m)}}) ˆ⊗Λ_{nov}.

We thus defined the operator 𝔪_{k} in Case (4.6.1).

Case (4.6.2) : This case is similar to the case of (4.6.1) and we proceed as follows. Let us define a moduli space

M_{m+1}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m−1)}; ℓ_{0}, ⋯, ℓ_{m})

as follows. We have p_{(j)} ∈ L_{(j)} ∩ L_{(j+1)} for j = 0, ⋯, m − 1. We consider the system ((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}, u), φ) such that

(4.14.1) Put z⃗ = (z_{0}, ⋯, z_{m}), z⃗_{−} = (z_{0}, ⋯, z_{m−1}). Then ((D^{2}, (z_{−})), φ) ∈ M̃_{m}(L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m−1)}). Where the right hand side is as in (3.1). (4.14.2) w^{(j)} = (w_{1}^{(j)}, ⋯, w_{ℓ_{j}−1}^{(j)}). w_{i}^{(j)} ∈ ∂_{j}D^{2}. Here ∂_{j}D^{2} is as in the last section. (4.14.3) If i ≠ i′ then w_{i}^{(j)} ≠ w_{i′}^{(j)}. w_{1}^{(j)}, ⋯, w_{ℓ_{j}−1}^{(j)} respects the order of ∂_{j}D^{2}.

Figure 4.3.

We split this moduli space according to the homotopy type of the map φ. We denote a homotopy class by β and let

M_{m+1}(β; L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m−1)}; ℓ_{0}, ⋯, ℓ_{m})

be the corresponding component of the moduli space. We define an evaluation map

e⃗v^{(j)} : CM_{m+1}(β; L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m−1)}; ℓ_{0}, ⋯, ℓ_{m}) → L_{(j)}^{ℓ_{j}−1},

in a way similar to the case (4.6.1). Using it we define the fiber product.

(4.15)  \begin{matrix}𝔐_{β}((p_{(1)}, ⋯, p_{(m−1))}; P_{1}, ⋯, P_{∑ ℓ_{j}}) \\ = CM_{m+1}(β; L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m−1)}; ℓ_{0}, ⋯, ℓ_{m}) \\ _{e⃗v} ×_{f_{∗}} ∏_{j} ∏_{i=1}^{ℓ_{j}−1} P_{ℓ_{0}+···+ℓ_{j−1}+i}. \end{matrix}

We also define another evaluation map at the remaining mark point z_{m}. Namely we define

ev : CM_{m+1}(β; L_{(0)}, ⋯, L_{(m)}; p_{(1)}, ⋯, p_{(m−1)}; ℓ_{0}, ⋯, ℓ_{m}) → L_{(m)} = L_{(0)}

by

ev((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}, u), φ) = φ(z_{m}).

It induces

ev : 𝔐_{β}((p_{(1)}, ⋯, p_{(m−1))}; P_{1}, ⋯, P_{∑ ℓ_{j}}) → L_{(0)}.

Now we put

(4.16) \begin{aligned}∑_{β} T^{E(((D^{2},z⃗;w^{(0)},··· ,w^{(m)}),φ_{β}))} & H(((D^{2}, z⃗; w^{(0)}, ⋯, w^{(m)}), φ_{β}); u⃗, v⃗) \\ & ev_{∗}[𝔐_{β}((p_{(1)}, ⋯, p_{(m−1))}; P_{1}, ⋯, P_{∑ ℓ_{j}})]. \end{aligned}

where φ_{β} is a map with homotopy class β. (4.16) is an element of

C^{g+(s̃_{k}−s̃_{0})}(L; Hom(𝔏_{(0)}, 𝔏_{(m)}))

(g = ℓ_{0} + ⋯ + ℓ_{m} − 1.) We define

𝔪(x_{0}, ⋯, x_{ℓ_{0}+···+ℓ_{m}−1}) = (4.16).

Now we have :

Theorem 4.17. The operation 𝔪_{k} defined above satisfies A_{∞} relations.

Since the detail of the definition of module of morphisms and operations are already discussed, the proof of Theorem 4.17 is in fact a straight forward generalization of the argument of [FOOO] and one of [Fu1], [Fu4]. So we do not repeat it. The main idea is that the degeneration of elements of M_{k+1} can be described as in the Figure below and they correspond to the terms in the A_{∞} formula.

Figure 4.4.

5. Unit and Homotopy unit.

We have thus constructed operations 𝔪_{k} which satisfy the A_{∞} relations. To complete the construction of our filtered A_{∞} category LAG(M, ω) we need to construct a unit. There is a delicate problem related to transversality to construct a unit of our A_{∞} category LAG(M, ω). This problem is solved in [FOOO] §20. We discuss an outline of it here (together with its slight generalization. Namely we generalize from the case A_{∞} algebra discussed in [FOOO] to A_{∞} category).

Let (L, s̃, 𝔏) be an object of LAG(M, ω). Then, by definition, the module of morphisms, LAG((L, s̃, 𝔏), (L, s̃, 𝔏)) is a subcomplex of the de-Rham complex W^{−∞}(L; ℂ⊗Λ^{∗}(L)) of distribution valued forms, since Hom(𝔏, 𝔏) together with its connection is trivial. Then 1 ∈ W^{−∞}(L; ℂ⊗Λ^{0}(L)) is an element of LAG^{0}((L, s̃, 𝔏), (L, s̃, 𝔏)). Actually 1 is the element corresponding to the fundamental chain [L] by our identification (4.2), the Poincaré duality. So we write [L] rather than 1.

“Theorem” 5.1^{3}. [L] is a unit of our filtered A_{∞} category LAG(M, ω).

“Proof”^{4}. We need to show

(5.2)  𝔪_{k}(x_{1}, ⋯, x_{i−1}, [L], x_{i+1}, ⋯, x_{k}) = 0

for k > 2. We also need to show

(5.3)  𝔪_{1}([L]) = 0, (−1)^{deg x}𝔪_{2}([L], x) = 𝔪_{2}(x, [L]) = x.

(5.3) can be “proved” in a way similar to (5.2). So we explain the idea of the “proof” of (5.2). There are two cases to study. Namely the transversal case (which corresponds §3) and the nontransversal case (which corresponds to §4). Actually the general case is a mixture. In the case when L_{0} = ⋯ = L_{k} the argument is explained in detail in [FOOO] Chapter 5 (and is outlined in [FOOO] Chapter 2 section 7). So we restrict ourselves to the following special case. Suppose we have mutually transversal three Lagrangian submanifolds L_{0}, L_{1}, L_{2}. We put L_{3} = L_{2}. We also assume 𝔏_{3} = 𝔏_{2}. Let p_{i} ∈ L_{i} ∩ L_{i+1}, i = 0, 1 and let p_{2} ∈ L_{1} ∩ L_{2}. Let u_{i} ∈ Hom(𝔏_{i}, 𝔏_{i+1})_{p_{i}}, i = 0, 1 and u_{2} ∈ Hom(𝔏_{0}, 𝔏_{2})_{p_{2}}. We want to show that the matrix element

(5.4)  ⟨𝔪_{3}((p_{0}, u_{0}), (p_{1}, u_{1}), [L_{2}]), (p_{2}, u_{2})⟩

3We put this statement in the quote since it is not correct as it is stated. We will state a precise theorem later (Theorem 5.13).

4We put proof in the quote since it contains a gap. The correct proof of the theorem will be explained later. We need to go into the detail of the framework of Kuranishi structure to make it precise.

is zero. (This is a part of Formula (5.2) to be shown.) We recall that the matrix element (5.4) is defined as follows. We consider the set of (φ, (z_{0}, z_{1}, z_{2}), w) such that the following holds :

(5.5.1) φ : D^{2} → M is a pseudoholomorphic map. (5.5.2) z_{0}, z_{1}, w, z_{2} ∈ ∂D^{2}. And they are in this order, with respect to the usual cyclic orientation on ∂D^{2}. (5.5.3) φ(\overline{z_{0}, z_{1}}) ∈ L_{1}, φ(\overline{z_{1}, z_{2}}) ∈ L_{2}, φ(\overline{z_{2}, z_{0}}) ∈ L_{0}.

We divide the set of all such (φ, (z_{0}, z_{1}, z_{2}), w) by an obvious P SL(2; ℝ) action and let

M_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1)

be the quotient space. (This definition is a special case of the definition in the last section.) We compactify it to CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1). We define an evaluation map

ev : CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1) → L_{2}

by

ev(φ, (z_{0}, z_{1}, z_{2}), w) = φ(w).

Now, by definition, the matrix element (5.4) is given by the order counted with sign of the (finite) set

(5.6)  CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1) ×_{L_{2}} L_{2},

in the case when the moduli space and the fiber product is transversal and the virtual dimension of (5.6) is zero. We need to show that this number is zero.

To “prove” it we consider the moduli space

CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}) = CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 0)

defined in section 3. This moduli space consists of isomorphism classes of (φ, (z_{0}, z_{1}, z_{2})) satisfying the same conditions as (5.5). So we can define a map

(5.7)  π : CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1) → CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2})

by

π(φ, (z_{0}, z_{1}, z_{2}), w) = (φ, (z_{0}, z_{1}, z_{2})).

It is easy to see that the fiber of the map (5.7) is one dimensional and is parametrized by the position of w ∈ \overline{p_{2}p_{0}}.

It follows that

Virdim CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1) = Virdim CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}) + 1.

One the other hand, by the definition of fiber product and virtual dimension, we have, in general

Virdim A ×_{B} C = Virdim A + Virdim C − Virdim B.

Hence

Virdim CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1)×_{L_{2}}L_{2}
= Virdim CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0,1).

We are studying the case when the virtual dimension of (5.6) is zero. Therefore we have

(5.8)  Virdim CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}) = −1.

(5.8) implies that (if everything is transversal then) CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}) is empty. Hence CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}; 0, 0, 1) is also empty.

Thus we are done ? □

Actually there is a gap in the above argument. (5.8) implies that the space CM_{3}(L_{0}, L_{1}, L_{2}; p_{0}, p_{1}, p_{2}) is empty, only in case it is transversal. Using the theory of Kuranishi structure and multivalued perturbation, we can always make it transversal. However the trouble is whether we can make this perturbation to be compatible with the map π in (5.7). Namely if we can take a perturbation compatible with (5.7), then it is not consistent with other maps we mention below.

The boundary of moduli spaces CM_{m}(L_{0}, ⋯, L_{m}; p_{(0)}, ⋯, p_{(m)}; ℓ_{1}, ⋯, ℓ_{m}) can be described by products of similar moduli spaces (but with smaller m or ℓ_{i}). This consistency is essential to show A_{∞} formula, Theorem 4.17. It is possible to make the perturbations consistent with this identification of the boundary to other moduli spaces. However, we cannot find, in general, a perturbation which is compatible to both. Namely in general there is no perturbation which is compatible with π and the identification of the boundary of the moduli space with the product of the other moduli spaces.

This problem looks rather technical. But it is quite delicate and essential point. It is explained in detail in [FOOO] Chapter 5 and is solved there. In [FOOO], the case we have only one Lagrangian submanifold is discussed. But the general case is completely parallel. So we only give a statement here.

To state the precise version of “Theorem” 5.1, we need to introduce a notion of homotopy unit. To explain it, let us start with a geometric model of this notion. We first review the following well know notion in topology.

Definition 5.9. An H-space is a space X with a map p : X × X → X and a base point e ∈ X such that p(x, e) = p(e, x) = x.

Homotopy H space is (X, p, e) such that the restrictions of p to {e} × X ≅ X and to X × {e} ≅ X are homotopic to identity.

Lemma 5.10. Homotopy H space is homotopy equivalent to an H space.

Proof. Let us consider X ∪ [0, 1] and identify 0 with e. We get a space X_{+} which is obviously homotopy equivalent to X. By assumption there exists a homotopies

H_{l} : X × [0, 1] → X, H_{r} : X × [0, 1] → X

such that

H_{l}(x, 0) = p(e, x), H_{r}(x, 0) = p(x, e), H_{l}(x, 1) = H_{r}(x, 1) = x.

We extend p to X_{+} × X_{+} by putting

p(t, x) = H_{l}(x, t), p(x, t) = H_{r}(x, t).

Then (X_{+}, p, 1) is an H space. □

Now we translate the definition of X_{+} in the proof above into the algebraic language and define homotopy unit as follows. Let C be an A_{∞} category without unit. Let

e_{c} ∈ C^{0}(c, c)

be elements. We define

C(c, c)_{+} = C(c, c) ⊕ R · e_{c+} ⊕ R · f_{c+}

such that deg e_{c+} = 0, deg f_{c} = −1. We extend 𝔪_{1} to C(c, c)_{+} by

\begin{aligned}𝔪_{1}(f_{c}) & = e_{c+} − e_{c}, \\ 𝔪_{1}(e_{c+}) & = 0. \end{aligned}

We put C(c, c′)_{+} = C(c, c′). We then define BC_{+} as in §1.

Definition 5.11. We say that e_{c} is a homotopy unit if we can extend 𝔪_{k} to BC_{+} so that it will become an A_{∞} category with unit e_{c+}.

We remark that, by the definition of unit, the extension of 𝔪_{k} to BC_{+} is automatically determined in the case e_{c+} is included in the formula. So to extend 𝔪_{k} we only need to define

(5.12)  \begin{aligned}𝔪_{k}(x_{1,1}, ⋯, x_{1,ℓ_{1}} & , f_{c}, x_{2,1}, ⋯, x_{2,ℓ_{2}}, \\ & , f_{c}, x_{3,1}, ⋯, x_{m−1,ℓ_{m−1}}, f_{c}, x_{m,1}, ⋯, x_{m,ℓ_{m}}), \end{aligned}

where k = ∑_{i=1}^{m}(ℓ_{i} + 1) − 1. We can write down the equation they are supposed to satisfy by rewriting A_{∞} formula. See [FOOO] §20 for such a formula.

We can define a notion of homotopy unit of filtered A_{∞} category in the same way. Now we have :

Theorem 5.13. [L] is a homotopy unit of LAG(M, ω).

The proof is a straight forward generalization of the argument in [FOOO] §20. So we discuss it only very briefly. As we mentioned above, the reason that [L] fails to be a unit (or in other words the reason (5.2) can be nonzero) is that the perturbation is not compatible with maps (5.7) (and its analogues). Let us restrict ourselves to the case of (5.4).

On the other hand, we can find another perturbation so that it is compatible with π but is not compatible with other operations. We now choose these two perturbations and take a homotopy between them. We now have a moduli space using this one parameter family of perturbations. Taking its fundamental chain (or counting its order with sign) we obtain a matrix element

⟨𝔪_{3}((p_{0}, u_{0}), (p_{1}, u_{1}), f_{[L_{2}]}), (p_{2}, u_{2})⟩ .

The other operations in (5.12) can be defined in a similar way.

Using Theorem 5.13 we can modify our filtered A_{∞} category LAG(M, ω), (which has only a homotopy unit) so that it has an (exact) unit. From now one, we write LAG(M, ω) for this modified one.

Chapter 2 : Homological algebra of A_{∞} category.

6. Twisted complex and derived A_{∞} category.

We have constructed our main example in §§2,3,4,5. So we go back to the continuation of section 1 and further study algebraic formalism. In this section we follow Bondal-Kapranov [BoK] and Kontsevich [Ko1] (see also [Fu7] §16), to define twisted complex and derived A_{∞} category.

In the case of abelian category, its derived category C is constructed in the following way. First, we consider the category CC of chain complex of objects of C. We then consider the weak equivalence between objects of CC. (Weak equivalence is a chain map which induces an isomorphism in cohomology.) We divide our category CC by weak equivalence and obtain 𝔻C. The category 𝔻C is not an abelian category, but has an operation which is an algebraic version of the construction of the mapping cone (in topology). This construction of mapping cone gives a notion of distinguished triple on 𝔻C. Then 𝔻C will become a triangulated category. (See [GM],[KaS],[Ha] etc. for detail.)

To generalize this construction to A_{∞} category, we need to generalize the notion of chain complex (of elements of C) to twisted complex.

Remark 6.1. The twisted complex is defined by Bondal-Kapranov [BoK] in the case when 𝔪_{k} = 0 for k ≥ 3. Kontsevitch [Ko1] mentioned its generalization to A_{∞} category and suggested its application to mirror symmetry. The twisted complex was also applied in [Fu7] to mirror symmetry. (The author learned some part of the contents of this section from P. Seidel’s talks and papers [Se2],[Se3].)

Let C be an A_{∞} category. We first increase the objects of it a bit in the following way. Let c be an object of C and k be an integer. We add an object c[k] and put

(6.2)  C(c′[ℓ], c[k]) = C(c′, c)[ℓ − k].

We add all of c[k] and define 𝔪_{k} as follows. Let

x_{i} ∈ C^{deg x_{i}−k_{i−1}+k_{i}}(c_{i−1}, c_{i}). ≅ C^{deg x_{i}}(c_{i−1}[k_{i−1}], c_{i}[k_{i}])

We write s^{∗}x_{i} when we regard it as an element of C^{deg x_{i}}(c_{i−1}[k_{i−1}], c_{i}[k_{i}]), and write x_{i} when we regard it as an element of C^{deg x_{i}+k_{i}−k_{i−1}}(c_{i−1}, c_{i}). Now we put

(6.3)  𝔪_{k}(s^{∗}x_{1}, ⋯, s^{∗}x_{k}) = (−1)^{k_{0}}s^{∗}𝔪_{k}(x_{1}, ⋯, x_{k}).

Then we have

∑_{1≤ℓ<m≤k}(−1)^{deg s^{∗}x_{1}+···+deg s^{∗}x_{ℓ−1}+ℓ−1}

𝔪_{k−m+ℓ+1}(s^{∗}x_{1}, ⋯, 𝔪_{m−ℓ+1}(s^{∗}x_{ℓ}, ⋯, s^{∗}x_{m}), ⋯, s^{∗}x_{k})

= ∑_{1≤ℓ<m≤k}(−1)^{deg x_{1}+···+deg x_{ℓ−1}+ℓ−k_{0}+k_{ℓ−1}−1}

𝔪_{k−m+ℓ+1}(s^{∗}x_{1}, ⋯, 𝔪_{m−ℓ+1}(s^{∗}x_{ℓ}, ⋯, s^{∗}x_{m}), ⋯, s^{∗}x_{k})

= ∑_{1≤ℓ<m≤k}(−1)^{deg x_{1}+···+deg x_{ℓ−1}+ℓ−1}

s^{∗}(𝔪_{k−m+ℓ+1}(x_{1}, ⋯, 𝔪_{m−ℓ+1}(x_{ℓ}, ⋯, x_{m}), ⋯, x_{k})) = 0.

Thus A_{∞} formula holds.

We add more objects to C as follows. Let us consider formally the direct sum c_{1} ⊕ ⋯ ⊕ c_{k} and regard it as an object of our A_{∞} category. We then define

C(c_{1} ⊕ ⋯ ⊕ c_{k}, c′_{1} ⊕ ⋯ ⊕ c′_{m}) = ⨁_{i=1}^{k} ⨁_{j=1}^{m} C(c_{i}, c′_{j}).

We define operations 𝔪_{k} in an obvious way.

From now on, we always extend the set of objects of A_{∞} category in this way. Then, for any objects c, c′, we have c[k] and c ⊕ c′. The object c ⊕ c′ is the direct sum in the sense of additive category.

Actually in our main example, the object c[k] exists already since it corresponds to the same Lagrangian submanifold, U(1) bundle, etc. as c but with different grading s̃.

On the other hand, the direct sum c⊕c′ (already) exists if c and c′ corresponds to Lagrangian submanifolds L,L′ such that L ∩ L′ = ∅. In this case, c ⊕ c′ corresponds L∪L′ together with line bundle, framing, relative spin structure induced ones from c and c′.

However, in case L ∩ L′ ≠ ∅, the disjoint union L ∪ L′ is not an embedded Lagrangian submanifold and hence is not an objects of our A_{∞} category. (If we include immersed Lagrangian submanifold as an objects, then again c ⊕ c′ will be included (geometrically).) Here we simply add c ⊕ c′ formally.

Now we start the construction of twisted complex and derived A_{∞} category. Let k_{1} < k_{2} be integers. We consider a finitely many objects c_{i} ∈ 𝔒𝔟(C), (i = k_{1}, ⋯, k_{2}) and elements

x_{i,j} ∈ C[1]^{0}(c_{i}[i], c_{j}[j]) ≅ C^{1+i−j}(c_{i}, c_{j})

for each i < j.

Definition 6.4. We say (c_{k_{1}}, ⋯, c_{k_{2}}; (x_{i,j})) is a twisted complex if for each i < j, the equation :

(6.5)  ∑_{m≥1} ∑_{i=ℓ_{0}<···<ℓ_{m}=j} 𝔪_{m}(x_{ℓ_{0},ℓ_{1}}, ⋯, x_{ℓ_{m−1},ℓ_{m}}) = 0

is satisfied. The set of all twisted complex is denoted by 𝔒𝔟(DC).

Example 6.6. If x_{i,j} = 0 for j ≠ i + 1 then (6.5) is

(6.7.1)  𝔪_{2}(x_{i,i+1}, x_{i+1,i+2}) = 0,
(6.7.2)  𝔪_{1}(x_{i,i+1}) = 0.

(6.7.2) implies that x_{i,i+1} is a cocycle. Since 𝔪_{2} is the composition of morphisms (upto sign), (6.7.1) implies that

0 → c_{k_{1}} ⟶^{x_{k_{1},k_{1}+1}} ⋯ ⟶^{x_{k_{2}−1,k_{2}}} c_{k_{2}} → 0

is a chain complex in our additive category C. (Note deg x_{i,i+1} = 0 if we regard it as an element of C(c_{i}, c_{i+1}) (and not of C(c_{i}[i], c_{i+1}[i + 1])).

We are going to show that twisted complex consist a triangulated category. (See [GM], [Ha], [KaS] for the definition of triangulated category.)

We first define an additive category whose object is a twisted complex. Let c^{(1)} = (c_{k_{1}}^{(1)}, ⋯, c_{k_{2}}^{(1)}; (x_{i,j}^{(1)})), c^{(2)} = (c_{k_{1}}^{(2)}, ⋯, c_{k_{2}}^{(2)}; (x_{i,j}^{(2)})) be twisted complexes. We first define a morphism between them. We put

(6.8)  C^{k}(c^{(1)}, c^{(2)}) = ⨁_{i,j} C[1]^{k}(c_{i}^{(1)}[i], c_{j}^{(2)}[j]).

We define a boundary operator 𝔪̂_{1} on it by

𝔪̂_{1}(y_{i,j}) = (z_{i,j})

where

\begin{aligned}z_{i,j} = ∑_{i≤a<b≤j} & ∑_{m_{1}≥1,m_{2}≥1} ∑_{i=ℓ_{0}^{(1)}<···<ℓ_{m_{1}}^{(1)} =a<b=ℓ_{0}^{(2)}<···<ℓ_{m_{2}}^{(2)} =j} \\ & 𝔪_{m} (x_{ℓ_{0}^{(1)},ℓ_{1}^{(1)}}^{(1)}, ⋯, x_{ℓ_{m_{1}−1}^{(1)},ℓ_{m_{1}}^{(1)}}^{(1)} , y_{a,b}, x_{ℓ_{0}^{(2)},ℓ_{1}^{(2)}}^{(2)}, ⋯, x_{ℓ_{m_{2}−1}^{(2)},ℓ_{m_{2}}^{(2)}}^{(2)} ) . \end{aligned}

Lemma 6.9. 𝔪̂_{1} ∘ 𝔪̂_{1} = 0.

Since the formulas which we need to prove Lemma 6.9 are rather long, we develop some notations before starting calculation. We define

x_{i,j}^{(1)} ∈ BC[1]^{0}(c_{i}^{(1)}[i], c_{j}^{(1)}[j])

by

x_{i,j}^{(1)} = ∑_{m≥1} ∑_{i=ℓ_{0}<···<ℓ_{m}=j} x_{ℓ_{0},ℓ_{1}}^{(1)} ⊗ ⋯ ⊗ x_{ℓ_{m−1},ℓ_{m}}^{(1)}.

Then (6.5) is

(6.10)  𝔪(x_{i,j}^{(1)}) = 𝔪(x_{i,j}^{(2)}) = 0.

Here and hereafter 𝔪 : BC[1] → B_{1}C[1] is the operation which is 𝔪_{k} on B_{k}C[1]. The definition of 𝔪̂_{1} is rewritten as

(6.11)  z_{i,j} = ∑_{i≤a<b≤j} 𝔪(x_{i,a}^{(1)} ⊗ y_{a,b} ⊗ x_{b,j}^{(2)}).

We may further simplify the notation as follows. We consider BC[1] and define a product • on it as follows. Let x ∈ BC[1](a, b), y ∈ BC[1](c, d), Then :

x • y = \{ \begin{matrix}x ⊗ y & if b = c, \\ 0 & if not. \end{matrix}

We put

x_{i,∗} = ∑_{j} x_{i,j}, x_{∗,j} = ∑_{i} x_{i,j}, x_{∗,∗} = ∑_{i,j} x_{i,j}.

We define y_{i,∗}, y_{∗,j}, y_{∗,∗}, z_{i,∗}, z_{∗,j}, z_{∗,∗} in a similar way. Then (6.10) is equivalent to

(6.12)  d̂(x_{∗,∗}^{(0)}) = d̂(x_{∗,∗}^{(1)}) = 0.

(6.11) can be written as

(6.13)  z_{∗,∗} = 𝔪(x_{∗,∗}^{(1)} • y_{∗,∗} • x_{∗,∗}^{(2)}).

Proof of Lemma 6.9. We have

𝔪̂_{1}(𝔪̂_{1}(y_{i,j})) = (w_{i,j})

where

w_{∗,∗} = 𝔪(x_{∗,∗}^{(1)} • 𝔪(x_{∗,∗}^{(1)} • y_{∗,∗} • x_{∗,∗}^{(2)}) • x_{∗,∗}^{(2)}).

(Here we assumed c_{i} ≠ c_{j} for i ≠ j for simplicity.) Note the degree of x_{i,j} are all zero (after shifted). Hence A_{∞} formula implies

𝔪(x_{∗,∗}^{(1)} • 𝔪(x_{∗,∗}^{(1)} • y_{∗,∗} • x_{∗,∗}^{(2)}) • x_{∗,∗}^{(2)})

+ 𝔪(d̂(x_{∗,∗}^{(1)}) • y_{∗,∗} • x_{∗,∗}^{(2)}) + (−1)^{deg y_{∗,∗}+1}𝔪(x_{∗,∗}^{(1)} • y_{∗,∗} • d̂(x_{∗,∗}^{(2)})) = 0.

The second and the third terms vanish by (6.10). The lemma follows. □

Definition 6.14. We say an element of C(c^{(1)}, c^{(2)}) a morphism in DC. It is said to be a closed morphism if it is 𝔪̂_{1} closed. We also put :

𝔻C(c^{(1)}, c^{(2)}) = H^{∗}(C(c^{(1)}, c^{(2)}); 𝔪̂_{1}).

We next define (higher) compositions :

(6.15)  𝔪̂_{k} : C(c^{(0)}, c^{(1)}) ⊗ ⋯ ⊗ C(c^{(k−1)}, c^{(k)}) → C(c^{(0)}, c^{(k)})

by

(6.16)  𝔪̂_{k}(y_{∗,∗}^{(1)}, ⋯, y_{∗,∗}^{(k)}) = 𝔪(x_{∗,∗}^{(0)} • y_{∗,∗}^{(1)} • x_{∗,∗}^{(1)} • ⋯ y_{∗,∗}^{(k)} • x_{∗,∗}^{(k)})

Theorem 6.17. We have an A_{∞} category DC such that the set of objects is 𝔒𝔟(DC), the set of morphisms is given by (6.8), and the operations are given by (6.16).

Proof. We are only to check the A_{∞} formula. The proof of it is quite similar to the proof of Lemma 6.9 and is left to the reader. □

We next are going to define the mapping cone in our category DC. For this purpose, we introduce a systematic way to associate an object of DC to each object of DDC. (More precisely we can define an A_{∞} functor DDC → DC. See the next section for the definition of A_{∞} category.)

Let c^{(k)} = ((c_{i}^{(k)}; i ∈ I^{k}), (x_{i,j}^{(k)})) be a twisted complex, where k ∈ I. (I, I^{k} be subsets of ℤ of the form {a, a + 1, a + 2, ⋯, b}.) Let

y^{(k,n)} ∈ (DC)[1]^{0}(c^{(k)}[k], c^{(n)}[n]) = ⨁_{ℓ,m} C[1]^{0}(c_{ℓ}^{(k)}[k + ℓ], c_{m}^{(n)}[n + m]).

We write its C[1]^{0}(c_{ℓ}^{(k)}[k + ℓ], c_{m}^{(n)}[n + m]) component by y_{ℓ,m}^{(k,n)}.

We assume 𝔠 = ((c^{(k)}; k ∈ I), (y^{(k,n)})) is a twisted complex of the A_{∞} category DC. We are going to construct an object |𝔠| of DC. The construction is an analogy of the construction of double complex.

We put

(6.18)  |𝔠|_{i} = ⨁_{k+ℓ=i} c_{ℓ}^{(k)}.

Let i < j. We are going to define z_{i,j} ∈ C^{0}[1](|𝔠|_{i}[i], |𝔠|_{j}[j]). Let k, ℓ, m, n be integers such that k + ℓ = i, m + n = j. We define z_{i,j}^{(k,ℓ,m,n)}, the C(c_{ℓ}^{(k)}, c_{m}^{(n)}) component of z_{i,j} by :

(6.19)  z_{i,j}^{(k,ℓ,m,n)} = \{ \begin{matrix}y_{ℓ,m}^{(k,n)} & if k < n \\ x_{ℓ,m} & if k = n, ℓ < m \\ 0 & otherwise. \end{matrix}

Lemma 6.20. |𝔠| = (|𝔠|_{i}, z_{i,j}) is a twisted complex.

Proof. The condition (6.5) for |𝔠| reduces to the condition (6.5) for 𝔠 and ones for c_{i}. □

Suppose we have two twisted complexes c, c′ and y ∈ C^{0}(c, c′) = C[1]^{0}(c[−1], c′), be a closed morphism. We put c_{−1} = c and c_{0} = c′. Then (c_{−1}, c_{0}; y) is an object of DDC. Hence |c_{−1}, c_{0}; y| is an object of DC. We call it the mapping cone of y : c → c′ and write it as Cone(c, c′; y).

Lemma 6.21. There exists I : c′ → Cone(c, c′; y), J : Cone(c, c′; y) → c[1] such that, for any twisted complex b, we have a long exact sequence

H^{∗}(DC(b, c), 𝔪_{1}) →^{R_{y}} H^{∗}(DC(b, c′), 𝔪_{1})

→^{I} H^{∗}(DC(b, Cone(c, c′; y)), 𝔪_{1})

→^{J} H^{∗}(DC(b, c[1]), 𝔪_{1})

→^{R_{y}} H^{∗}(DC(b, c′[1]), 𝔪_{1}) → ⋯

Here R_{y} is induced by the right multplication by y.

Proof. We remark that

Cone(c, c′; y)_{i} = c_{i+1} ⊕ c′_{i}.

Hence we can define I as an inclusion c′_{i} → c_{i+1} ⊕ c′_{i} ≅ Cone(c, c′; y)_{i}. And we define J as a projection Cone(c, c′; y)_{i} ≅ c_{i+1} ⊕ c′_{i} → c_{i+1}. It is easy to see that I and J are morphisms in DC. By definition we have an exact sequence of chain complex

0 → (DC(b, c′), 𝔪_{1}) →^{I} (DC(b, Cone(c, c′; y)), 𝔪_{1}) →^{J} (DC(b, c[1]), 𝔪_{1}) → 0.

By definition, the operator H^{∗}(DC(b, c), 𝔪_{1}) → H^{∗}(DC(b, c′), 𝔪_{1}) of the associated long exact sequence is R_{y}. □

So far we defined a notion corresponding to chain complex and to mapping cone, in the case of A_{∞} category. Usually to construct the derived category from the category of chain complex, we need to divide it by weak equivalence. We define a similar notion, homotopy equivalence between two objects of A_{∞} category.

Definition 6.22. Let C be an A_{∞} category and c, c′ ∈ 𝔒𝔟(C). Let x ∈ C^{0}(c, c′). We say that x is a homotopy equivalence if there exists y ∈ C^{0}(c′, c) such that

(6.23.1) 𝔪_{1}(x) = 𝔪_{1}(y) = 0. (6.23.2) 𝔪_{2}(y, x) − e_{c} ∈ Im 𝔪_{1}, 𝔪_{2}(x, y) − e_{c′} ∈ Im 𝔪_{1}.

Two objects c, c′ ∈ 𝔒𝔟(C) are said to be homotopy equivalent to each other if there exists a homotopy equivalence between them.

Lemma 6.24. Let c, c′ ∈ 𝔒𝔟(C), x ∈ C^{0}(c, c′) with 𝔪_{1}(x) = 0. Then the following five conditions are equivalent to each other.

(6.25.1) x is a homotopy equivalence. (6.25.2) For each b ∈ 𝔒𝔟(C) the map R_{x} : C(b, c) → C(b, c′), z ↦ 𝔪_{2}(z, x) induces an isomorphism on homology. (6.25.3) The map R_{x} : C(c′, c) → C(c′, c′) induces a surjection on homology and R_{x} : C(c, c) → C(c, c′) induces an injection on homology. (6.25.4) For each b ∈ 𝔒𝔟(C) the map L_{x} : C(c′, b) → C(c, b), z ↦ 𝔪_{2}(x, z) induces an isomorphism on homology. (6.25.5) The map L_{x} : C(c′, c) → C(c, c) induces a surjection on homology and L_{x} : C(c′, c′) → C(c, c′) induces an injection on homology.

Proof. (6.25.1)⇒(6.25.2) : 𝔪_{1}(x) = 0 together with A_{∞} formulae implies that R_{x} is a chain map. Let y be as in (6.23). Then it is easy to see that R_{x} ∘ R_{y}, R_{y} ∘ R_{x} induces identity in homology. (6.25.2) follows. (6.25.2)⇒(6.25.3) : Obvious. (6.25.3)⇒(6.25.1) : By (6.25.3) there exists y such that 𝔪_{2}(y, x) − e_{c′} ∈ Im 𝔪_{1}. Hence

𝔪_{2}(𝔪_{2}(x, y), x) ≡ −(−1)^{deg′ x}𝔪_{2}(x, 𝔪_{2}(y, x)) mod Im 𝔪_{1}
≡ (−1)^{deg x}𝔪_{2}(x, e_{c′}) mod Im 𝔪_{1}
≡ x mod Im 𝔪_{1}
≡ 𝔪_{2}(e_{c}, x) mod Im 𝔪_{1}.

Hence 𝔪_{2}(x, y) ≡ e_{c} mod Im 𝔪_{1}.

The proof of equivalence between (6.25.1),(6.25.4),(6.25.5) is similar. □

Lemma 6.24 implies that the composition of homotopy equivalences is a homotopy equivalence.

We define a category 𝔻C as follows. (It is a category in the usual sense and is an additive category.) Its object is a homotopy equivalence class of the objects of A_{∞} category DC. Morphism between them is defined by Definition 6.14. By (6.25.2),(6.25.4), a homotopy equivalence induces an isomorphism on Hom(c^{(1)}, c^{(2)}) in Definition 6.14 hence the set of morphisms is well defined. The composition of the morphisms is induced by 𝔪_{2}. The A_{∞} formula implies that the composition in the homology level is (exactly) associative.

(Actually to perform this construction in a rigorous way we need to define and use a notion of quotient category. Since it is standard we do not discuss it. See for example [GM], [KaS].)

We next define distinguished triangles in 𝔻C and prove that 𝔻C will be a triangulated category. We use the notion of mapping cone for this purpose. To do so, we need to show that the homotopy equivalence classes of Cone(c^{(1)}, c^{(2)}; y) depends only of the homotopy class of (c^{(1)}, c^{(2)}; y). More precisely we prove the following.

44 KENJI FUKAYA

Lemma 6.26. Let u_{1} ∈ DC(c^{(1)}, c^{′(1)}), u_{2} ∈ DC(c^{(2)}, c^{′(2)}) be homotopy equivalences. We assume 𝔪_{2}(u_{1}, y′) − 𝔪_{2}(y, u_{2}) ∈ Im 𝔪_{1}. Then Cone(c^{(1)}, c^{(2)}; y) is homotopy equivalent to Cone(c^{′(1)}, c^{′(2)}; y′) in the A_{∞} category DC. Proof. We put

𝔪_{2}(u_{1}, y′) − 𝔪_{2}(y, u_{2}) = 𝔪_{1}(z).

Then (u_{1}, u_{2}, z) defines a closed morphism DC((c^{(1)}, c^{(2)}; y), (c^{′(1)}, c^{′(2)}; y′)). Hence we have a closed morphism

(6.27)  Cone(c^{(1)}, c^{(2)}; y) → Cone(c^{′(1)}, c^{′(2)}; y′).

Then we obtain a commutative diagram :

Diagam 6.1.

comparing exact sequences in Lemma 6.21. Then Lemma 6.21 and five lemma implies that (6.27) induces an isomorphism

DC(b, Cone(c^{(1)}, c^{(2)}; y)) ≅ DC(b, Cone(c^{′(1)}, c^{′(2)}; y′))

for any b. The lemma now follows from Lemma 6.24. □

Using Lemma 6.26 we can define a notion of distinguished triangle in 𝔻C as follows. Definition 6.28.

[c^{(1)}] → [c^{(2)}] → [Cone(c^{(1)}, c^{(2)}; y))] → [c^{(1)}[1]]

is said to be a distinguished triangle. Theorem 6.29. 𝔻C is a triangulated category.

We omit the proof, which is rather a straightforward check of the axiom. Definition 6.30. 𝔻C is called the derived A_{∞} category of our A_{∞} category C.

In case we start with filtered A_{∞} category C we first construct an A_{∞} category as in Definition 1.14 and then construct derived A_{∞} category. We call it also the derived A_{∞} category of C and write it also as 𝔻C.

Note that the objects of derived A_{∞} category of a filtered A_{∞} category C can be regarded as c = ((c_{i}; i ∈ {k_{1}, k_{1} + 1, ⋯, k_{2}}), (x_{i,j}^{(k)})). But, in this case, x_{i,i} may be nonzero, (x_{i,i} should be in C[1]^{0}(c_{i}, c_{i}) and moreover x_{i,i} ∈ ∪_{λ>0}F^{λ}C[1]^{0}(c_{i}, c_{i}).) x_{i,j} is supposed to satisfy an equation

(6.31)  ∑_{m} ∑_{ℓ_{0},··· ,ℓ_{m}} 𝔪_{m}(x_{ℓ_{0},ℓ_{1}}, ⋯, x_{ℓ_{m−1},ℓ_{m}}) = 0,

which is similar to (6.5). However, in (6.31), m = 0, 1, ⋯, ℓ_{0} ≤ ℓ_{1} ≤ ⋯ ≤ ℓ_{m}. (Namely the case ℓ_{i} = ℓ_{i+1} is included.) Actually, in the case ℓ_{0} = ⋯ = ℓ_{m} = ℓ, (6.31) is

𝔪(e^{x_{ℓ,ℓ}}) = 0,

that is the definition x_{ℓ,ℓ} to be a bounding chain.

An element c = ((c_{i}; i ∈ {k_{1}, k_{1} + 1, ⋯, k_{2}}), (x_{i,j}^{(k)}; i ≤ j)) can be described by the following Figure 6.1. On the other hand, an element ((c_{i}; i ∈ {k_{1}, k_{1} + 1, ⋯, k_{2}}), (x_{i,j}^{(k)}; i < j)) can be described by the following Figure 6.2.

Figure 6.1, Figure 2.

7. A_{∞} functor and natural transformation.

In sections 6, we consider more general objects than the original objects (that is the twisted complex) of our A_{∞} category C. In this section, we further generalize it and define a notion, A_{∞} functor C → CH, which was defined in [Fu4] over ℤ_{2} coefficient. We simplify the description of [Fu4] using Bar complex and also we put here the precise sign, which was not discussed in [Fu4].

Definition 7.1. Let C_{1}, C_{2} be A_{∞} categories. An A_{∞} functor F from C_{1} to C_{2} is a collection of F_{k}, k = 1, 2, ⋯ such that

(7.2.1) F_{0} : 𝔒𝔟(C_{1}) → 𝔒𝔟(C_{2}) is a map. (7.2.2) For c_{1}, c_{2} ∈ 𝔒𝔟(C_{1}), F_{k}(c_{1}, c_{2}) : B_{k}C_{1}[1](c_{1}, c_{2}) → C_{2}[1](F_{0}(c_{1}), F_{0}(c_{2})) is a homomorphism of degree 0. (7.2.3) We extend F_{k}(c_{1}, c_{2}) to a coalgebra homomorphism

(7.3.1)  F̂(c_{1}, c_{2}) : BC_{1}[1](c_{1}, c_{2}) → BC_{2}[1](F_{0}(c_{1}), F_{0}(c_{2})).

Then F̂(c_{1}, c_{2}) is a chain map with respect to the boundary operator d in Definition 1.1.

Note

\begin{aligned}F̂(x_{1} ⊗ ⋯ ⊗ x_{k}) = ∑_{m} ∑_{0=ℓ_{1}<ℓ_{2}<···<ℓ_{m}=k} & F_{ℓ_{2}−ℓ_{1}−1}(x_{ℓ_{1}+1} ⊗ ⋯ ⊗ x_{ℓ_{2}}) ⊗ ⋯ \\ & ⋯ ⊗ F_{ℓ_{m}−ℓ_{m−1}−1}(x_{ℓ_{m−1}+1} ⊗ ⋯ ⊗ x_{ℓ_{m}}). \end{aligned}

Our homomorphism F̂ on B_{0}C_{1}[1] is defined as follows. We remark

B_{0}C_{1}[1](c_{1}, c_{2}) ≅ \{ \begin{matrix}R & c_{1} ≠ c_{2}, \\ 0 & c_{1} = c_{2}. \end{matrix}

We put

(7.3.2)  F̂(x) = \{ \begin{matrix}x & if x ∈ B_{0}C_{1}[1](c, c), \\ 0 & if x ∈ B_{0}C_{1}[1](c_{1}, c_{2}), c_{1} ≠ c_{2}. \end{matrix}

We next give an example of A_{∞} functor, that is a representable functor. We first define an A_{∞} category CH for this purpose.

Definition 7.4. 𝔒𝔟(CH) is the set of (all) chain complexes of free R nodules^{5}. Let (C, d), (C′, d) ∈ 𝔒𝔟(CH). Then

CH^{k}((C, d), (C′, d)) = ⨁_{ℓ} Hom_{R}(C^{ℓ}, C^{′ℓ+k}).

We define

(7.5.1)  𝔪_{1}(x) = d ∘ x + (−1)^{deg x+1}x ∘ d,
(7.5.2)  𝔪_{2}(x, y) = (−1)^{deg x(deg y+1)}y ∘ x.

We put 𝔪_{k} = 0 for k ≥ 3.

Remark 7.6. The sign in (7.5.2) is the same as one we need to regard differential graded algebra as an A_{∞} algebra. (See Example-Lemma 1.7.)

5To avoid Russell paradox in set theory, we fix a sufficiently large set (a universe) and consider only free R modules contained in this set.

46 KENJI FUKAYA

Proposition 7.7. CH is an A_{∞} category. Proof. It is easy to check 𝔪_{1} ∘ 𝔪_{1} = 0. We calculate

(𝔪_{1} ∘ 𝔪_{2})(x, y) = (−1)^{deg x(deg y+1)}d ∘ y ∘ x
+ (−1)^{deg x+deg y+1}(−1)^{deg x(deg y+1)}y ∘ x ∘ d
= (−1)^{deg x deg y+deg x}d ∘ y ∘ x
+ (−1)^{deg x(deg y+1)+deg y+1}y ∘ d ∘ x
− (−1)^{(deg x+1)(deg y+1)}y ∘ d ∘ x
− (−1)^{deg x+deg y}(−1)^{deg x(deg y+1)}y ∘ x ∘ d
= −(−1)^{deg x+1}𝔪_{2}(x, 𝔪_{1}(y)) − 𝔪_{2}(𝔪_{1}(x), y).

This implies 𝔪 ∘ d̂= 0 on B_{2}. We next calculate

𝔪_{2}(𝔪_{2}(x, y), z) + (−1)^{deg x+1}𝔪_{2}(x, 𝔪_{2}(y, z))
= (−1)^{deg x(deg y+1)+(deg x+deg y)(deg z+1)}z ∘ (y ∘ x)
+ (−1)^{deg x+1+deg y(deg z+1)+deg x(deg y+deg z+1)}(z ∘ y) ∘ x
= 0.

which is the third and the last part of the A_{∞} formulae to be checked. □

We next define : Definition 7.8. Let C be an A_{∞} category. We define its opposite A_{∞} category C^{o} as follows. (7.9.1) 𝔒𝔟(C^{o}) = 𝔒𝔟(C). (7.9.2) Let c, c′ ∈ 𝔒𝔟(C^{o}) = 𝔒𝔟(C). We put C^{o}(c, c′) = C(c′, c). (7.9.3) We define (higher) composition operators 𝔪_{k}^{o} of C^{o} by :

𝔪_{k}^{o}(x_{1}, ⋯, x_{k}) = (−1)^{ϵ}𝔪_{k}(x_{k}, ⋯, x_{1}),

where

ϵ = ∑_{1≤i<j≤k}(deg x_{i} + 1)(deg x_{j} + 1) + 1.

Lemma 7.10. C^{o} is an A_{∞} category. Proof. First we introduce some notations to simplify the formula. We put x = x_{1} ⊗ ⋯ ⊗ x_{k} ∈ B_{k}C and

(7.11)  Δ^{m−1}x = ∑_{a} x_{a}^{(1)} ⊗ ⋯ ⊗ x_{a}^{(m)}.

Here

Δ^{m−1} = ⋯ ∘ (Δ ⊗ 1 ⊗ 1) ∘ (Δ ⊗ 1) ∘ Δ.

Let

(7.12.1)  deg x = deg x_{1} + ⋯ + deg x_{k}

is a degree of x and

(7.12.2)  deg′ x = deg x_{1} + ⋯ + deg x_{k} + k

is its degree after shifted. We use notations (7.11),(7.12.1),(7.12.2) frequently for the rest of this article. We put

(7.13.1)  x^{op} = x_{k} ⊗ ⋯ ⊗ x_{1},

and

(7.13.2)  ϵ(x) = ∑_{1≤i<j≤k}(deg x_{i} + 1)(deg x_{j} + 1).

The A_{∞} formula for 𝔪 can be written as

(7.14)  ∑_{a}(−1)^{deg′ x{(1) \atop a}}𝔪(x_{a}^{(1)}, 𝔪(x_{a}^{(2)}), x_{a}^{(3)}) = 0.

We have

(7.15) 𝔪(x_{a}^{(1)}, 𝔪(x_{a}^{(2)}), x_{a}^{(3)}) =(−1)^{ϵ(x_{a}^{(1)})+ϵ(x_{a}^{(2)})+ϵ(x_{a}^{(3)})+ϵ_{1}(a)}

𝔪^{o}(x_{a}^{(3)op}, 𝔪^{o}(x_{a}^{(2)op}), x_{a}^{(1)op}),

where ϵ(x_{a}^{(j)}) are as in (7.13.2) and

ϵ_{1}(a) = (deg′ x_{a}^{(1)} + deg′ x_{a}^{(3)})(deg′ x_{a}^{(2)} + 1) + deg′ x_{a}^{(1)} deg′ x_{a}^{(3)}.

We remark that

ϵ(x_{a}^{(1)}) + ϵ(x_{a}^{(2)}) + ϵ(x_{a}^{(3)}) + (deg′ x_{a}^{(1)} + deg′ x_{a}^{(3)}) deg′ x_{a}^{(2)}
+ deg′ x_{a}^{(1)} deg′ x_{a}^{(3)} = ϵ(x),

and is independent of a. Hence (7.14) and (7.15) imply

∑_{a}(−1)^{deg′ x{(3)op \atop a}}𝔪^{o}(x_{a}^{(3)op}, 𝔪^{o}(x_{a}^{(2)op}), x_{a}^{(1)op}) = 0.

This is the A_{∞} formula of 𝔪^{o} to be checked. □

Definition 7.16. Let C be an A_{∞} category and c ∈ 𝔒𝔟(C). We construct an A_{∞} functor F^{c} = Hom(·, c) : C → CH^{o} as follows.

(7.17.1) F_{0}^{c}(b) = C(b, c). (We take 𝔪_{1} as the boundary operator.) (7.17.2) F_{k}^{c}(x_{1}, ⋯, x_{k})(y) = 𝔪_{k+1}(x_{1}, ⋯, x_{k}, y). Here y ∈ C(b, c), b_{1}, ⋯, b_{k+1} = b ∈ 𝔒𝔟(C), x_{i} ∈ C(b_{i}, b_{i+1}).

48 KENJI FUKAYA

Proposition 7.18. F^{c} is an A_{∞} functor. Proof. We calculate

𝔪_{2}^{o}(F(x_{a}^{(1)}), F(x_{a}^{(2)}))

= −(−1)^{(deg F(x_{a}^{(1)})+1)(deg F(x_{a}^{(2)})+1)}𝔪_{2}(F(x_{a}^{(2)}), F(x_{a}^{(1)}))

= −(−1)^{(deg F(x_{a}^{(1)})+1)(deg F(x_{a}^{(2)})+1)+(deg F(x_{a}^{(1)})+1) deg F(x_{a}^{(2)})}F(x_{a}^{(1)}) ∘ F(x_{a}^{(2)}) = −(−1)^{deg′ x{(1) \atop a}}F(x_{a}^{(1)}) ∘ F(x_{a}^{(2)}).

We recall 𝔪_{k} = 0 for k ≥ 3 in CH. Hence the condition that F̂ is a chain map is

(7.19)

0 = ∑_{a}(−1)^{deg′ x{(1) \atop a}}F(x_{a}^{(1)}) ∘ F(x_{a}^{(2)})

+ (−1)^{deg′ x}F(x) ∘ 𝔪_{1} + 𝔪_{1} ∘ F(x)

+ ∑_{a}(−1)^{deg′ x{(1) \atop a}}F(x_{a}^{(1)}, 𝔪(x_{a}^{(2)}), x_{a}^{(3)}).

(We use also 𝔪_{1}^{o} = −𝔪_{1} to deduce (7.19).) We plug in y to the right hand side of (7.19). Then the first term of the right hand side will be

(7.20)  ∑_{a}(−1)^{deg′ x{(1) \atop a}}𝔪(x_{a}^{(1)}, 𝔪(x_{a}^{(2)}, y)).

The second and the third terms of the right hand side will be

(7.21)  (−1)^{deg′ x}𝔪(x, 𝔪_{1}(y)) + 𝔪_{1}(𝔪(x, y)).

The fourth term will be

(7.22)  ∑_{a}(−1)^{deg′ x{(1) \atop a}}𝔪(x_{a}^{(1)}, 𝔪(x_{a}^{(2)}), x_{a}^{(3)}, y).

(7.20) + (7.21) + (7.22) = 0 is the A_{∞} formula for 𝔪. We thus proved (7.19). □

We next define a similar but a bit different A_{∞} functor ℜ𝔢𝔭_{0}(c) : C^{o} → CH. (At this stage ℜ𝔢𝔭_{0}(c) is just a symbol. We will define ℜ𝔢𝔭_{k} in section 9.) For this purpose, we prove the following. Definition-Lemma 7.23. For each A_{∞} functor F : C_{1} → C_{2}, we can construct its opposite A_{∞} functor F^{o} : C_{1}^{o} → C_{2}^{o} as follows. (7.24.1) F_{0}^{o} = F_{0}. (7.24.2) F_{k}^{o}(x) = (−1)^{ϵ(x)}F_{k}(x^{op}). Here we use notation (7.13). Proof. We need to check

(7.25)

∑_{ℓ} ∑_{a} 𝔪_{ℓ}^{o}(F^{o}(x_{a}^{(1)}), ⋯, F^{o}(x_{a}^{(ℓ)}))

= ∑_{a}(−1)^{deg′ x{(1) \atop a}}F^{o}(x_{a}^{(1)}, 𝔪^{o}(x_{a}^{(2)}), x_{a}^{(3)}).

FLOER HOMOLOGY AND MIRROR SYMMETRY II 49

The left hand side of (7.25) is

(7.26)  ∑_{ℓ} ∑_{a}(−1)^{ϵ(x)+1}𝔪_{ℓ}(F(x_{a}^{(ℓ)op}) ⋯, F(x_{a}^{(1)op})).

In a way similar to the proof of Lemma 6.8, we can check that the right hand side of (7.25) is

(7.27)  ∑_{a}(−1)^{deg′ x{(3) \atop a}}F(x_{a}^{(3)op}, 𝔪(x_{a}^{(2)op}), x_{a}^{(1)op}).

(7.26) = (7.27) is the condition that F is an A_{∞} functor. □

In view of (7.24) and (7.17), we can define an A_{∞} functor ℜ𝔢𝔭_{0}(c) : C^{o} → CH as follows. Definition 7.28.

(7.29.1)  ℜ𝔢𝔭_{0}(c)_{0}(b_{0}) = C(b_{0}, c).

Let x ∈ B_{k}C^{o}(b_{0}, b_{k}) = B_{k}C(b_{k}, b_{0}), y ∈ ℜ𝔢𝔭_{0}(c)_{0}(b_{0}) = C(c, b_{0}). Then

(7.29.2)  ℜ𝔢𝔭_{0}(c)_{k}(x)(y) = (−1)^{ϵ(x)}F^{c}(x^{op})(y) = (−1)^{ϵ(x)}𝔪(x^{op}, y).

We next apply the construction of F^{c} and ℜ𝔢𝔭_{0}(c) to the opposite A_{∞} category C^{o} and define ^{c}F, 𝔒𝔭ℜ𝔢𝔭_{0}(c) as follows. Definition 7.30. ^{c}F : C → CH is defined by (7.31.1) ^{c}F_{0}(b_{0}) = C(c, b_{0}). (7.31.2) ^{c}F_{k}(x)(y) = 𝔪_{k+1}^{o}(x, y) = −(−1)^{ϵ(x)+deg′ y deg′ x}𝔪_{k+1}(y, x^{op}), where x ∈ B_{k}C^{o}(b_{0}, b_{k}) = B_{k}C(b_{k}, b_{0}), y ∈ ^{c}F_{0}(b_{0}) ∈ C(c, b_{0}).

𝔒𝔭ℜ𝔢𝔭_{0}(c) : C^{o} → CH^{o} is defined by (7.32.1) 𝔒𝔭ℜ𝔢𝔭_{0}(c)(b_{0}) = C(c, b_{0}). (7.32.2) 𝔒𝔭ℜ𝔢𝔭_{k}(x)(y) = (−1)^{ϵ(x)} ^{c}F_{k}(x)(y) = −(−1)^{deg′ y deg′ x}𝔪_{k+1}(y, x^{op}).

It follows from construction that ^{c}F and ℜ𝔢𝔭_{0}(c) are A_{∞} functors. Definition 7.33. We say an A_{∞} functor : C → CH^{o}, C^{o} → CH, C → CH, C^{o} → CH^{o} to be representable if it is homotopic to F^{c}, ℜ𝔢𝔭_{0}(c), ^{c}F and ℜ𝔢𝔭_{0}(c), respectively. (Homotpy between A_{∞} functors will be defined in the next section.)

We next generalize the constructions above to the case when c is a twisted complex. For this purpose we first define a composition of two A_{∞} functors. Let C_{1}, C_{2}, C_{3} be A_{∞} categories and F : C_{1} → C_{2}, G : C_{2} → C_{3} be A_{∞} functors. Definition 7.34. The composition G ∘ F : C_{1} → C_{3} is defined as follows.

(7.35.1)  (G ∘ F)_{0} = G_{0} ∘ F_{0}
{ \atop G ∘ F}(c_{1}, c_{2}) = Ĝ(F_{0}(c_{1}), F_{0}(c_{2})) ∘ F̂(c_{1}, c_{2})

: BC_{1}(c_{1}, c_{2}) → BC_{3}(G_{0}(F_{0}(c_{1})), G_{0}(F_{0}(c_{2}))). (7.35.2)

It is easy to see that the composition is an A_{∞} functor.

We remark that there is an obvious A_{∞} functor C → DC.

Definition 7.36. Let c be an twisted complex of A_{∞} category C. We then consider the composition

(7.37)  C → DC →^{F^{c}} CH^{o}

where F^{c} is defined by applying Definition 6.15 to A_{∞} category DC. We write also F^{c} the composition (7.37). Similarly we define ^{c}F as the composition

C^{o} → (DC)^{o} →^{^{c}F} CH^{o},

where c is a twisted complex of C^{o}. (Actually the twisted complex of C^{o} can be constructed from one on C by changing the sign of the maps x_{i,j} appropriately. We leave the reader the problem to find the correct sign.) We define ℜ𝔢𝔭_{0}(c), 𝔒𝔭ℜ𝔢𝔭_{0}(c) in a similar way.

Definition 7.38. An A_{∞} functor : C → CH^{o}, C^{o} → CH, C → CH, C^{o} → CH^{o} is said to be derived representable if it is homotopic to F^{c}, ℜ𝔢𝔭_{0}(c), ^{c}F, 𝔒𝔭ℜ𝔢𝔭_{0}(c), respectively.

We next explain that A_{∞} functors are natural generalization of the notion of differential graded modules. Let (A, d, ∧) be a differential graded algebra. It defines an A_{∞} algebra as in Example-Lemma 1.7. We write it (A, 𝔪_{1}, 𝔪_{2}). (𝔪_{3} = ⋯ = 0.) We may regard it as an A_{∞} category A with one object.

Lemma 7.39. Homotopy classes of A_{∞} functors F : A → CH^{0} such that F_{k} = 0 for k ≥ 2, correspond one to one to the homotopy equivalence classes of graded differential left module over (A, d, ∧).

Proof. Let (D, d, ·) be a graded differential module over (A, d, ∧). We define

F_{0}^{d}(c_{0}) = (D, d).

Here c_{0} is the unique object of A. we define

F_{1}^{d} : A → Hom((D, d), (D, d))

by

F_{1}^{d}(x)(v) = (−1)^{deg x(deg v+1)}v · x.

As in Example-Lemma 1.7, we can easily check that F_{1}^{d} is a chain map and

F_{1}^{d}(𝔪_{2}(x, x′)) = F_{1}^{d}(x) ∘ F_{1}^{d}(x′).

Hence by putting F_{k}^{d} = 0, k > 1 we find an A^{∞} functor F : A → CH^{0}. The converse can be proved in a similar way. □

We remark that a representable functor F : A → CH^{0} corresponds to A itself (regarded as an A module) by Lemma 7.39.

Definition 7.40. A left A_{∞} module of an A_{∞} algebra C is an A_{∞} functor C → CH^{0}. A right A_{∞} module of an A_{∞} algebra C is an A_{∞} functor C → CH.

In a way similar to the proof of Lemma 7.39, we can check that this definition coincides with one given in [FOOO] §14.

Let C be an A_{∞} category and c, c′ be objects of it. There is an obvious A_{∞} functor C(c) → C. (Here we regard C(c) an A_{∞} algebra, that is an A_{∞} category with single object c.) We compose it with F^{c′} : C → CH and we obtain a left A_{∞} C(c) module, which is C(c, c′) as an R module. In a similar way, C(c, c′) has a structure of right C(c′) module. In other words, C(c, c′) is a left C(c) and right C(c′) bimodule in the sense defined in [FOOO] §14.

In the case of LAG(M, ω), this implies that we have a LAG(L, b), LAG(L′, b′) bimodule LAG((L, b), (L′, b′)). (Here b and b′ are bounding chains.) This is what is constructed in [FOOO] §14. The homology of LAG((L, b), (L′, b′)) is the Floer homology between two Lagrangian submanifolds.

Remark 7.41. Let X be a scheme over R. We can associate the following category C(X), which is an A_{∞} category. (Note that it satisfies 𝔪_{k} = 0 for k ≠ 2, and that 𝔪_{2} is commutative.)

The object of C(X) is an affine open subsets U_{A} = Spec(A) ⊂ X. The set of morphisms from U_{A} to U_{B} is {0} unless U_{B} ⊆ U_{A}. In the case U_{B} ⊆ U_{A}, the set of the morphisms C(X)(U_{A}, U_{B}) is the ring A. If U_{C} ⊆ U_{B} ⊆ U_{A} then A ⊆ B ⊆ C. 𝔪_{2} is defined as A ⊗ B → B (the product of ring B).

A functor from C(X) to the category of R modules can be identified with a presheaf on X.

We further study relations of A_{∞} functor to twisted complex.

Lemma 7.42. Let F : C_{1} → C_{2} be an A^{∞} functor. Then there exists an A^{∞} functor DF : DC_{1} → DC_{2} such that the following diagram commutes.

Digram 7.1

Sketch of the proof. Let (c_{k_{1}}, ⋯, c_{k_{2}}; (x_{i,j})) to be a twisted complex of C_{1}. We put

y_{a,b} = ∑_{k} ∑_{a=i_{1}<···<i_{k}=b} F_{k}(x_{i_{1}}, ⋯, x_{i_{k}}).

It is easy to see that (F_{0}(c_{k_{1}}), ⋯, F_{0}(c_{k_{2}}); (y_{i,j})) is a twisted complex of C_{2}. We put

(DF)_{0}(c_{k_{1}}, ⋯, c_{k_{2}}; (x_{i,j})) = (F_{0}(c_{k_{1}}), ⋯, F_{0}(c_{k_{2}}); (y_{i,j})).

We omit the definition of (DF)_{k}, k ≥ 1. □

Lemma 7.43. There exists an A^{∞} functor P : DDF → DF such that P_{0}(c) = |c|. Here |c| is defined in Lemma 6.20.

The proof is straightforward and is left to the reader.

We now proceed to the definition of an A_{∞} category 𝔉𝔲𝔫𝔨(C_{1}, C_{2}) whose objects are A_{∞} functors. (Here C_{1}, C_{2} are A_{∞} categories.) This section is almost the same as [Fu4] §10. However we put signs to every formula and check the formula with signs. (In [Fu4] we worked over ℤ_{2}.) The presentation of the proof is improved also.

Let F^{1}, F^{2} be A_{∞} functors from C_{1} to C_{2}. Let a, b ∈ 𝔒𝔟(C_{1}). Let ℓ be an integer. We consider family of operators

(7.44)  T_{k}(a, b) : B_{k}C_{1}[1](a, b) → C_{2}[1](F_{0}^{1}(a), F_{0}^{2}(b))

of degree t. (Here k = 1, 2, ⋯ for a ≠ b and k = 0, 1, 2, ⋯ for a = b.) We write t−1 = 𝔡𝔢𝔤T, t = 𝔡𝔢𝔤′T. We use 𝔡𝔢𝔤 in place of deg here to avoid confusion. 𝔡𝔢𝔤T will be a degree of T as a pre natural transformation as we will define below (Definition 7.49). deg T_{k}(a, b) is a degree as a homomorphism between graded modules. We remark that 𝔡𝔢𝔤′T = deg T_{k}(a, b). We also remark that, for x ∈ B_{k}C[1](a, b), we have

deg′ T_{k}(a, b)(x) = 𝔡𝔢𝔤′T + deg′ x = 𝔡𝔢𝔤T + deg′ x + 1.

For a′, b′ ∈ 𝔒𝔟(C_{2}), let

π_{a′,b′} : BC_{2}[1](a′, b′) → C_{2}[1](a′, b′)

be the projection.

Lemma 7.45. For each family T_{k}(a, b) there exists uniquely a family

T̂(a, b) : BC_{1}[1](a, b) → BC_{2}[1](F_{0}^{1}(a), F_{0}^{2}(b)),

of homomorphisms with the following properties.

π_{F_{0}^{1}(a),F_{0}^{2}(b)} ∘ T̂(a, b) = T_{k}(a, b) on B_{k}C_{1}[1](a, b). (7.46.1)

Δ ∘ T̂(a, b) = ∑_{c}(F̂^{1} ˆ⊗T̂(c, b) + T̂(a, c) ˆ⊗F̂^{2}) ∘ Δ. (7.46.2)

Here ˆ⊗ is defined by (A ˆ⊗B)(x, y) = (−1)^{deg B deg′ x}A(x) ⊗ B(y). (Note deg B = 𝔡𝔢𝔤′B and deg′ x is the degrees after shifted.)

Proof. Let x ∈ BC_{1}[1](a, b). We use notation (7.11). We define

(7.47)  T̂(x) = ∑_{a}(−1)^{deg T deg′ x_{a}^{(1)}} F̂^{1}(x_{a}^{(1)}) ⊗ T (x_{a}^{(2)}) ⊗ F̂^{3}(x_{a}^{(3)}).

It is easy to check (7.46). Uniqueness is also easy to show. □

Hereafter we write T etc. in place of T_{k}(a, b) etc. in case no confusion can occur.

Lemma 7.48. For each family T_{k}(a, b) with t = 𝔡𝔢𝔤′T_{k}(a, b) = deg T, there exists a family (δT)_{k}(a, b) such that

δ̂T = d̂∘ T̂ + (−1)^{t+1}T̂ ∘ d̂.

Proof. We calculate

Δ∘(d̂∘ T̂ + (−1)^{t+1}T̂ ∘ d̂)
= (d̂ˆ⊗1 + 1 ˆ⊗d̂) ∘ Δ ∘ T̂ + (−1)^{t}Δ ∘ T̂ ∘ d̂
= (d̂ˆ⊗1 + 1 ˆ⊗d̂) ∘ (F̂^{1} ˆ⊗T̂ + T̂ ˆ⊗F̂^{2}) ∘ Δ
+ (−1)^{t+1}(F̂^{1} ˆ⊗T̂ + T̂ ˆ⊗F̂^{2}) ∘ (d̂ˆ⊗1 + 1 ˆ⊗d̂) ∘ Δ
= (F̂^{1} ˆ⊗(d̂∘ T̂ + (−1)^{t+1}T̂ ∘ d̂) + (d̂∘ T̂ + (−1)^{t+1}T̂ ∘ d̂) ˆ⊗F̂^{2}) ∘ Δ.

Lemma 7.48 then follows from Lemma 7.45. □

Definition 7.49. We say a family T as in (7.44), a pre natural transformation from F^{1} to F^{2}. We let 𝔉𝔲𝔫𝔨(F^{1}, F^{2}) be the set of all pre natural transformations. (It is a graded module over the commutative ring R we work on.)

We define a boundary operator 𝔐_{1} = −δ on it by Lemma 7.48.

We say that T is a natural transformation or A_{∞} transformation if it is δ closed.

We remark that 𝔡𝔢𝔤(T) = deg(T_{0}(1)), where T_{0}(1) ∈ C[1](c, c).

We put minus sign in 𝔐_{1} = −δ since we need it to show Theorem 9.1 later.

Corollary 7.50. δ∘δ = 0. In other words, (𝔉𝔲𝔫𝔨(F^{1}, F^{2}), 𝔐_{1}) is a chain complex.

Corollary 7.50 is immediate from Lemma 7.45. We use the symbol 𝔐 to denote the operations on 𝔉𝔲𝔫𝔨(F^{1}, F^{2}) in order to distinguish it from operations on C_{1}, C_{2}.

Remark 7.51. If T is a natural transformation, then T_{0}(a) = T_{0}(a)(1) is a closed morphism in C_{2}(F_{0}^{1}(a), F_{0}^{2}(a)). (Here 1 ∈ B_{0}C_{1}[1] = the coefficient ring R.) Moreover we have

𝔪_{2}(T_{1}(a, b)(x), T_{0}(b)) ≡ ±𝔪_{2}(T_{0}(a), T_{1}(a, b)(x)) mod Im 𝔪_{1}.

for each x ∈ C_{1}(a, b). Thus natural transformation in our sense defines a natural transformation in the usual sense in homology level.

We next define 𝔐_{k}, k ≥ 2. Let F^{i}, i = 0, ⋯, k be A_{∞} functors from C_{1} to C_{2} and T^{i} ∈ 𝔉𝔲𝔫𝔨(F^{i−1}, F^{i}) be pre natural transformations. We put t_{i} = 𝔡𝔢𝔤′T^{i}. Let x ∈ B(C_{1}[1]). We consider

Δ^{2k+1}x = ∑_{a} x_{a}^{(1)} ⊗ ⋯ ⊗ x_{a}^{(2k+1)}.

and put

(7.52) T̂(x) = − ∑(−1)^{ϵ_{a}}𝔪 (F̂^{0}(x_{a}^{(1)}), T^{1}(x_{a}^{(2)}), ⋯, T^{k}(x_{a}^{(2k)}), F̂^{k}(x_{a}^{(2k+1)})) ,

where

(7.53)  ϵ_{a} = ∑_{j=1}^{k} ∑_{i=1}^{2j−1} t_{j} deg′ x_{a}^{(i)}.

It is easy to see that T̂ satisfies (7.46.2). Hence we can use it to define a pre natural transformation T. We also remark that 𝔡𝔢𝔤T = t_{1} + ⋯ t_{k} and is 𝔡𝔢𝔤′T = t_{1} + ⋯ t_{k} + 1.

Definition 7.54. We put 𝔐_{k}(T^{1}, ⋯, T^{k}) = T.

We remark that 𝔐_{k} is of degree one in the sense of 𝔡𝔢𝔤′. Namely we have

𝔡𝔢𝔤′T = 𝔡𝔢𝔤′T^{1} + ⋯ + 𝔡𝔢𝔤′T^{k} + 1.

We remark that the overall minus sign in (7.52) will be necessary for Theorem 9.1 to hold.

Theorem-Definition 7.55. There exists an A_{∞} category 𝔉𝔲𝔫𝔨(C_{1}, C_{2}) such that its object is an A_{∞} functor : C_{1} → C_{2}, the set of morphisms are the set of pre natural transformations, and that (higher) compositions are defined by Definition 7.54.

Proof. The A_{∞} formula we are going to check is :

(7.56) \begin{aligned}0 = ∑_{1≤m≤ℓ≤k}(−1)^{t_{1}+···+t_{m−1}} & 𝔐_{k−ℓ+m}(T^{1}, ⋯, T^{m−1}, \\ & 𝔐_{ℓ−m+1}(T^{m}, ⋯, T^{ℓ}), T^{ℓ+1}, ⋯, T^{k})(x), \end{aligned}

where t_{i} = 𝔡𝔢𝔤′T^{i}. To prove (7.56), we compare it with

(7.57) 0 = (−1)^{ϵ_{1}(a)}(𝔪 ∘ d̂)(F̂^{0}(x_{a}^{(1)}), T^{1}(x_{a}^{(2)}), ⋯, T^{k}(x_{a}^{(2k)}), F̂^{k}(x_{a}^{(2k+1)})),

where

(7.58)  ϵ_{1}(a) = ∑_{j=1}^{k} ∑_{i=1}^{2j−1} t_{j} deg′ x_{a}^{(i)}.

The formula (7.57) follows from the A_{∞} formula of C_{2}. We study the terms appearing in (7.56) and (7.57).

We first study the terms appearing in (7.57). There are two types of them. One is :

Type 1: Let 1 ≤ m < ℓ ≤ k. Then we have

(7.59)

𝔪 (F̂^{0}(x_{m}^{(1)}), T^{1}(x_{a}^{(2)}), ⋯, F̂^{m−1}(x_{a,b}^{(2m−1),(1)}),
𝔪 (F̂^{m−1}(x_{a,b}^{(2m−1),(2)}), T^{m}(x_{a}^{(2m)})), ⋯
⋯, T^{ℓ−1}(x_{a}^{(2ℓ−2)}), F̂^{ℓ}(x_{a,b}^{(2ℓ−1),(1)}))
F̂^{ℓ}(x_{a,b}^{(2ℓ−1),(2)}), ⋯, T^{k}(x_{a}^{(2k)}), F̂^{k}(x_{a}^{(2k+1)})) .

Here we put

Δ(x_{a}^{(i)}) = ∑_{b} x_{a,b}^{(i),(1)} ⊗ x_{a,b}^{(i),(2)}.

For a moment we do not put sign in the formula. We will check the sign carefully later.

Type 2: This is an analogue of (7.59) in the case when 1 ≤ m = ℓ ≤ k. That is :

(7.60) \begin{matrix}𝔪(F̂^{0}(x_{a}^{(1)}), & T^{1}(x_{a}^{(2)}), ⋯, T^{m}(x_{a}^{(2m)}), \\ d̂(F̂^{m}(x_{a}^{(2m+1)})), T^{m+1}(x_{a}^{(2m+2)}), ⋯, T^{k}(x_{a}^{(2k)}), F̂^{k}(x_{a}^{(2k+1)})). \end{matrix}

Now we turn to the terms appearing in (7.56).

Type 3: This is exactly the same as (7.59).

The other types of terms in (7.56) are the cases when 𝔐_{1} appears. We remark that 𝔐_{1} appears in (7.56) in case ℓ = m + k − 1 or ℓ = m. The terms of Types 4, 5 below correspond to the case either ℓ = m + k − 1 and the terms of Type 6 correspond to the case ℓ = m.

Let us first consider the case ℓ = m + k − 1. We recall

(7.61) \begin{matrix}(𝔐_{1}(𝔐_{k}(T^{1}, ⋯, T^{k})))(x) = & (𝔪 ∘ 𝔐_{k}(T^{1}, ⋯, T^{k})^{∧})(x) \\ + (−1)^{t_{1}+···+t_{k}}(𝔐_{k}(T^{1}, ⋯, T^{k}) ∘ d̂)(x). \end{matrix}

We remark that 𝔡𝔢𝔤′𝔐_{k}(T^{1}, ⋯, T^{k}) = t_{1} + ⋯ + t_{k} + 1. Note that the first term of (7.61) is already included in Type 3. So we only need to consider the second term. They are one of the following two types.

Type 4:

(7.62) \begin{matrix}𝔪(F̂^{0}(x_{a}^{(1)}), & T^{1}(x_{a}^{(2)}), ⋯, T^{m}(x_{a}^{(2m)}), \\ F̂^{m}(d̂(x_{a}^{(2m+1)})), T^{m+1}(x_{a}^{(2m+2)}), ⋯, T^{k}(x_{a}^{(2k)}), F̂^{k}(x_{a}^{(2k+1)})). \end{matrix}

Type 5:

(7.63)  \begin{matrix}𝔪(F̂^{0}(x_{a}^{(1)}), & T^{1}(x_{a}^{(2)}), ⋯, F̂^{m−1}(x_{a}^{(2m−1)}), \\ T^{m}(d̂(x_{a}^{(2m)})), ⋯, T^{k}(x_{a}^{(2k)}), F̂^{k}(x_{a}^{(2k+1)})). \end{matrix}

We next consider the case ℓ = m. we recall

(7.64)  \begin{matrix}𝔐_{k}(T^{1}, ⋯, & 𝔐_{1}(T^{m}), ⋯, T^{k})(x) \\ = & 𝔐_{k}(T^{1}, ⋯, 𝔪 ∘ T̂^{m}, ⋯, T^{k})(x) \\ + (−1)^{t_{m}}𝔐_{k}(T^{1}, ⋯, T^{m} ∘ d̂, ⋯, T^{k})(x). \end{matrix}

The first term again is included in Type 3. The second term then gives :

Type 6: This type is actually the same as Type 5.

We have finished describing all the types of the terms in (7.56). We now can prove the case when the coefficient ring is ℤ_{2} immediately. Namely terms of Types 1 and 3 cancel each other and terms of Types 2 and 4 cancel each other, since F̂ is a chain map, and terms of Types 5 and 6 cancel each other.

Let us now check the sign.

The sign in Type 3 is given by (−1)^{ϵ_{2}(a)+t_{1}+···+t_{m−1}} where

(7.65)

ϵ_{2}(a) = ∑_{j=m}^{ℓ} t_{j} (deg′ x_{a,b}^{(2m−1),(2)} + ∑_{i=2m}^{2j−1} deg′ x_{a}^{i})

+ ∑_{j=1}^{m−1} ∑_{i=1}^{2j−1} t_{j} deg′ x_{a}^{i}

+ (t_{m} + ⋯ + t_{ℓ} + 1)×

× ( ∑_{i=0}^{2m−2} deg′ x_{a}^{i} + deg′ x_{a,b}^{(2m−1),(1)})

+ ∑_{j=ℓ+1}^{k} ∑_{i=1}^{2j−1} t_{j} deg′ x_{a}^{i}.

Note that the first line in (7.65) comes from 𝔐_{ℓ−m+1}(T^{m}, ⋯, T^{ℓ}) in (7.56) and the rest comes from 𝔐_{k−ℓ+m}.

We calculate (7.65) and obtain

ϵ_{2}(a) − ϵ_{1}(a) = ∑_{i=0}^{2m−2} deg′ x_{a}^{i} + deg′ x_{a,b}^{(2m−1),(1)}.

On the other hand, the sign in the case of Type 1 in (7.57) is (−1)^{ϵ_{1}(a)+ϵ_{3}(a)} where

ϵ_{3}(a) = deg′ x_{m}^{(1)} + deg′ T^{1}(x_{a}^{(2)}) + ⋯ + deg′ x_{a,b}^{(2m−1),(1)}

≡ ∑_{i=0}^{2m−2} deg′ x_{a}^{i} + deg′ x_{a,b}^{(2m−1),(1)} + ∑_{i=1}^{m−1} t_{i} mod 2.

Thus Types 1 and 3 coincides together with sign.

We next consider Types 2 and 4. We find using (7.61) that the sign of both of them is

ϵ_{1}(a) + ∑_{i=1}^{m} t_{i} + ∑_{j=1}^{2m} deg′ x_{a}^{j}

and coincides to each other.

Next the sign of the terms of Type 5 is (−1)^{ϵ_{4}(a)}, where

ϵ_{4}(a) = ϵ_{1}(a) + ∑_{i=1}^{m} t_{i} + ∑_{j=1}^{2m+1} deg′ x_{a}^{j}.

We study Type 6. Note that we have sign (−1)^{t_{1}+···+t_{m−1}} in (7.56). So using (7.64) we find that the sign of the terms of Type 6 is (−1)^{ϵ_{4}(a)+1}. The proof of A_{∞} formula (7.56) is now complete.

Finally we define unit, that is the identity transform Id^{F} for each F ∈ 𝔒𝔟(C_{1}, C_{2}). Let e_{c} ∈ C_{2}^{0}(c, c) be the unit in C_{2}. Namely we assume that (1.5) is satisfied for it. We put

(7.66.1)  Id_{0}^{F}(a) = −e_{F_{0}(a)} ∈ C_{2}^{0}(F_{0}(a), F_{0}(a)),
(7.66.2)  Id_{k}^{F} = 0,  for k ≥ 1.

(Note that Id_{1}^{F}(x) ≠ x.) It is easy to see from definition that Id^{F} satisfies (1.5) for 𝔐. (We remark that we need minus sign in (7.66) since we put overall minus sign in the definition of 𝔐_{k}.) □

8. Homotopy equivalence and and A_{∞} analogue of J.H.C. Whitehead Theorem.

We now define homotopy equivalence between A_{∞} categories.

Definition 8.1. Two A_{∞} functors F^{1}, F^{2} : C_{1} → C_{2} are said to be homotopic to each other if they are homotopy equivalent as objects of 𝔉𝔲𝔫𝔨(C_{1}, C_{2}) in the sense of Definition 6.22.

Definition 8.2. The identity functor Id^{C} : C → C is defined as follows.

(8.3.1) Id_{0}^{C}(c) = c. (8.3.2) Id_{1}^{C}(x) = x. (8.3.3) Id_{k}^{C} = 0 for k ≥ 2.

Remark 8.4. The identity functor is similar to but is different from the identity transformation defined at the end of the last section.

Definition 8.5. An A_{∞} functor F : C_{1} → C_{2} is said to be a homotopy equivalence if there exists an A_{∞} functor F′ : C_{2} → C_{1} such that the composition F ∘ F′ is homotopic to Id^{C_{2}} and that F′ ∘ F is homotopic to Id^{C_{1}}.

Two A_{∞} categories are said to be homotopy equivalent to each other if there exists a homotopy equivalence between them.

Now the main result of this section is :

Theorem 8.6. Let F : C_{1} → C_{2} be an A_{∞} functor such that :

(8.7.1) F_{1} : C_{1}(c_{1}, c′_{1}) → C_{2}(F_{0}(c_{1}), F_{0}(c′_{1})) induces an isomorphism on 𝔪_{1} homology : (8.7.2) For any c_{2} ∈ 𝔒𝔟(C_{2}) there exists c_{1} ∈ 𝔒𝔟(C_{1}) such that F_{0}(c_{1}) is homotpy equivalent to c_{2}.

Then F is a homotopy equivalence.

Remark 8.8. In the case of A_{∞} algebras, Theorem 8.9 was proved in [FOOO] §A5. (8.7) is used as a definition of homotopy equivalence in [KoS1]. It seems that Kontsevich-Soibelman announced in [KoS1] that they will prove in [KoS2] a result similar to Theorem 8.6.

To prove Theorem 8.6, we start with the following special case.

Proposition 8.9. We assume (8.7.1) and that F_{0} : 𝔒𝔟(C_{1}) → 𝔒𝔟(C_{2}) is a bijection. Then F is a homotopy equivalence.

Proof. The proof is similar to the argument of [FOOO] §A5. We need to construct an A_{∞} functor F′ : C_{2} → C_{1} and a natural transformation T : F ∘ F′ → Id^{C_{2}}. For this purpose, we construct F′_{k} : B_{k}C_{2}[1] → B_{1}C_{1}[1] and T_{k} : B_{k}C_{2}[1] → B_{1}C_{2}[1] inductively on k. To describe the induction hypothesis, we define the notions, A_{k} functor and A_{k} transformation.

Let C, C′ be A_{∞} categories,= G_{0} : B_{0}C → B_{0}C′ be a map and G_{ℓ} : B_{ℓ}C[1](c_{1}, c_{2}) → B_{1}C′[1](c_{1}, c_{2}), 1 ≤ i ≤ k be R module homomorphisms of degree 0.

We put :

B_{a,··· ,b}C[1] = \frac{⨁_{i=0}^{a} B_{i}C[1]}{⨁_{i=0}^{b−1} B_{i}C[1]}.

The boundary operator of the chain complex BC[1] induces a boundary operator on B_{a,··· ,b}C[1]. Hence B_{a,··· ,b}C[1] is a chain complex.

Lemma 8.10. There exists uniquely a coalgebra homomorphism

Ĝ: BC[1] → BC′[1]

such that (8.11.1) Ĝ = Ĝ_{0,··· ,k} on ⨁_{i=0}^{k} B_{i}C[1]. (8.11.2) B_{1}C′[1] component of Ĝ(B_{i}C[1]) is 0 for i > k.

Proof. We define G : BC[1] → B_{1}C′[1] by G = G_{ℓ} on B_{ℓ}C[1] 1 ≤ ℓ ≤ k, and G = 0 on B_{ℓ}C[1] ℓ > k. We now define Ĝ in the same way as (7.3). Namely

Ĝ(x) = ∑_{i,a} G(x_{i,a}^{(1)}) ⊗ ⋯ ⊗ G(x_{i,a}^{(i)}).

Here

Δ^{i}(x) = ∑_{a} x_{i,a}^{(1)} ⊗ ⋯ ⊗ x_{i,a}^{(i)}.

(The definition of Ĝ on B_{0}C[1] is the same as (7.3.2).) It is easy to check (8.11). □

Ĝ induces a homomorphism : B_{a,··· ,b}C[1] → B_{a,··· ,b}C[1] for each a < b. We denote it by G_{a,··· ,b}.

Definition 8.12. We say G_{i} i ≤ k is an A_{k} functor if G_{0,··· ,k} is a chain map.

Lemma 8.13. If G_{i}, i ≤ k is an A_{k} functor then G_{a,··· ,b} is a chain map for b − a ≤ k − 1.

The proof is easy and is left to the reader.

Let us next denfine an A_{k} transformation. Let G and G′ be A_{k} functors from C to C′. Let S_{ℓ} : B_{ℓ}C[1](c_{1}, c_{2}) → B_{1}C′[1](G_{0}(c_{1}), G′_{0}(c_{2})), 1 ≤ ℓ ≤ k be R module homomorphisms of degree s + 1. It induces

(8.14)  S_{0,··· ,k} : ⨁_{i=0}^{k} B_{i}C[1] → ⨁_{i} B_{i}C′[1]

as follows. We define S : BC[1] → B_{1}C′[1] by S = S_{ℓ} on B_{ℓ}C[1], 0 ≤ ℓ ≤ k and S = 0 on B_{ℓ}C, ℓ > k. We then put :

(8.15)  S_{0,··· ,k}(x) = ∑_{a}(−1)^{s deg′ x_{a}^{(1)}}G_{0,··· ,k}(x_{a}^{(1)}) ⊗ S(x_{a}^{(2)}) ⊗ G′_{0,··· ,k}(x_{a}^{(3)}).

We remark here in (8.15) the case x_{a}^{(2)} = 1 ∈ B_{0}C[1] is included. As a consequence, the image of S_{0,··· ,k} is not contained in ⨁_{i=0}^{k} B_{i}C[1].

Definition 8.16. We say S_{ℓ}, 0 ≤ ℓ ≤ k, is an A_{k} transformation if

d̂∘ S_{0,··· ,k} + (−1)^{s+1}S_{0,··· ,k} ∘ d̂= 0,

on ⨁_{i=0}^{k} B_{i}C[1]. We put 𝔡𝔢𝔤′S = s.

Now we go back to the proof of Proposition 8.9. We are going to construct an A_{k} functor F′_{ℓ} ℓ ≤ k, and an A_{k} transformation T_{ℓ} ℓ ≤ k, from F ∘ F′ to Id^{C_{2}} by induction on k. (We remark that the composition of two A_{k} functors is well defined and is an A_{k} functor.)

We start with the case k = 0. We put

(8.17)  F′_{0} = F_{0}^{−1} : 𝔒𝔟(C_{2}) → 𝔒𝔟(C_{1}).

We remark that

B_{0}C[1] = ⨁_{c∈𝔒𝔟(C)} R

hence a map 𝔒𝔟(C_{2}) → 𝔒𝔟(C_{1}) induces a homomorphism B_{0}C_{2}[1] → B_{0}C_{1}[1]. We next put

(8.18)  T_{0}(c) = e_{c} ∈ C_{1}(F_{0}(F′_{0}(c)), c) = C_{1}(c, c).

Now we assume that we have an A_{k−1} transformation F′_{ℓ}, 0 ≤ ℓ ≤ k − 1 and an A_{k−1} transformation T_{ℓ}, 0 ≤ ℓ ≤ k − 1 and will consider the case of k. It follows from Lemma 8.13 that F′_{ℓ}, 0 ≤ ℓ ≤ k − 1 determine a chain map

(8.19)  F′_{2,··· ,k} : B_{2,··· ,k}C_{2}[1] → B_{2,··· ,k}C_{1}[1].

Using the obvious R module isomorphism

B_{2,··· ,k}C[1] = ⨁_{i=2}^{k} B_{i}C[1]

(8.19) induces an R module homomorphism

(8.20)  F̃′_{2,··· ,k} : ⨁_{i=2}^{k} B_{i}C[1] → ⨁_{i=1}^{k} B_{i}C[1].

Note however that (8.20) is not a chain map in general.

It is easy to see that (8.20) coincides with F̃′_{1,··· ,k−1} on ⨁_{i=2}^{k−1} B_{i}C. Hence we extend (8.20) to

(8.21)  F̃′_{1,··· ,k} : ⨁_{i=1}^{k} B_{i}C → ⨁_{i=1}^{k} B_{i}C.

We remark however that (8.21) is not a chain map in general. Nevertheless, using the fact that F′_{2,··· ,k} and F′_{1,··· ,k−1} are chain maps (Lemma 8.13), we can easily prove the following :

Lemma 8.22. The image of F̃′_{1,··· ,k} ∘ d̂− d̂∘ F̃′_{1,··· ,k} is contained in B_{1}C_{1}[1]. Moreover F̃′_{1,··· ,k} ∘ d̂− d̂∘ F̃′_{1,··· ,k} vanishes on ⨁_{i=1}^{k−1} B_{i}C_{2}[1].

We put

Ξ = F̃′_{1,··· ,k} ∘ d̂− d̂∘ F̃′_{1,··· ,k} : B_{k}C_{2}[1] → B_{1}C_{1}[1].

Note that 𝔪_{1} defines an R module homomorphism

d_{1} : B_{k}C_{2}[1] → B_{k}C_{2}[1],

by

d_{1}(x_{1} ⊗ ⋯ ⊗ x_{k}) = ∑_{i}(−1)^{deg′ x_{1}+··· deg′ x_{i+1}}x_{1} ⊗ ⋯ ⊗ 𝔪_{1}(x_{i}) ⊗ ⋯ ⊗ x_{k}.

We can prove d_{1}d_{1} = 0 easily.

We define a boundary operator on Hom(B_{k}C_{2}[1], B_{1}C_{1}[1]) by

(8.23)  dϕ = 𝔪_{1} ∘ ϕ + (−1)^{deg′ ϕ}ϕ ∘ d_{1}.

Lemma 8.22 then implies

dΞ = 0.

We moreover have the following :

Lemma 8.24. Ξ is contained in the image of d.

Proof. We define a boundary operator d on Hom(B_{k}C_{2}[1], B_{1}C_{2}[1]) in a way similar to (8.23). Then F_{1} induces a chain homomorphism F_{1,∗} : Hom(B_{k}C_{2}, B_{1}C_{1}) → Hom(B_{k}C_{2}, B_{1}C_{2}). By assumption F_{1,∗} induces an isomorphism on homology. So it suffices to show that F_{1} ∘ Ξ is a d boundary.

We next rewrite the Definition 8.16 (Sublemma 8.26). We need some notations for it.

We define T : BC_{1}[1] → B_{1}C_{1}[1] by T = T_{ℓ} on B_{ℓ}C_{1}[1], 0 ≤ ℓ ≤ k − 1 and T = 0 on B_{ℓ}C_{1}[1], ℓ > k. We define \overline{T} : BC_{1}[1] → B_{1}C_{1}[1] by \overline{T} = T_{ℓ} on B_{ℓ}C_{1}[1], 1 ≤ ℓ ≤ k − 1 and T = 0 on B_{ℓ}C_{1}[1], ℓ > k or ℓ = 0.

We then define T̂ and ˆ \overline{T} in the same way as (8.17). Namely we put

(8.25)

T̂(x) = ∑_{a}(−1)^{deg′ x{(1) \atop a}} F̂′(F̂(x_{a}^{(1)})) ⊗ T (x_{a}^{(2)}) ⊗ x_{a}^{(3)},
ˆ\overline{T}(x) = ∑_{a}(−1)^{deg′ x{(1) \atop a}} F̂′(F̂(x_{a}^{(1)})) ⊗ \overline{T} (x_{a}^{(2)}) ⊗ x_{a}^{(3)}.

We consider a filtration 𝔉 (the number filtration) of BC_{1}[1] such that 𝔉^{k}BC_{1}[1] = ⨁_{i=0}^{k} B_{i}C_{1}[1]. Note that T̂ does not preserve this filtration since T_{0}(B_{0}C_{1}[1]) ⊂ B_{1}C_{1}[1] is nonzero. But ˆ \overline{T} does preserve this filtration. Therefore ˆ \overline{T} induces an R module homomorphism B_{a,··· ,b}C_{1}[1] → B_{a,··· ,b}C_{1}[1]. We write it as \overline{T}_{a,··· ,b}.

We next recall that F′ : BC_{2}[1] → B_{1}C_{2}[1] is equal to F′_{ℓ} on B_{ℓ}C_{2}[1] 1 ≤ ℓ ≤ k−1, and is equal to 0 on B_{ℓ}C_{2}[1] ℓ ≥ k.

Sublemma 8.26. T_{ℓ} 0 ≤ ℓ ≤ k − 1, is an A_{k−1} transformation, if and only if

(8.27)  F ∘ F̃′_{1,··· ,k−1} − id − 𝔪 ∘ \overline{T}_{1,··· ,k−1} − \overline{T} ∘ d̂= 0.

Here (8.27) is a formula for an element in Hom(B_{1,··· ,k−1}C_{2}[1], B_{1}C_{2}[1]), and id ∈ Hom(B_{1}C_{2}[1], B_{1}C_{2}[1]) ⊂ Hom(B_{1,··· ,k−1}C_{2}[1], B_{1}C_{2}[1]).

Proof. We consider

(8.28)  Ψ̃ = −d̂∘ T_{0,··· ,k−1} + T_{0,··· ,k−1} ∘ d̂∈ Hom(B_{0,··· ,k−1}C_{2}[1], BC_{2}[1]).

Let Ψ be its Hom(B_{0,··· ,k−1}C_{2}[1], B_{1}C_{2}[1]) component. It is easy to see

Ψ̃(x) = ∑_{a} x_{a}^{(1)} ⊗ Ψ(x_{a}^{(2)}) ⊗ x_{a}^{(3)}.

Hence Ψ̃ is zero if and only if Ψ is zero. On the other hand Ψ̃ is zero if and only if T_{ℓ}, 0 ≤ ℓ ≤ k − 1 is an A_{k−1} transformation, by definition. Hence, to prove Sublemma 8.26, it suffices to show that Ψ is the right hand side of (8.27). This can be easily seen by using (8.18) and the fact that e_{c} is the unit as follows.

Let us consider the Hom(B_{0,··· ,k−1}C_{2}[1], B_{1}C_{2}[1]) component of (8.28). We find that the sum of the terms of it which contains T_{0} is

(8.29)  x ↦ ∑_{a} 𝔪 ((−1)^{deg′ x{(1) \atop a}} F̂(F̂′(x_{a}^{(1)})) ⊗ e ⊗ x_{a}^{(2)}) .

The terms of the right hand side of (8.29) are zero unless x_{a}^{(2)} = 1 or x_{a}^{(1)} = 1. In the first case it is

(−1)^{deg′ x}𝔪_{2}(F̂(F̂′(x), e) = −F̂(F̂′(x)).

In the second case, it is

𝔪(e ⊗ x) = \{ \begin{matrix}x & if x ∈ C_{1}[1], \\ 0 & otherwise. \end{matrix}

Thus they correspond to the first and second terms of (8.27). The terms of the Hom(B_{0,··· ,k−1}C_{2}[1], B_{1}C_{1}[1]) component of (8.28) which do not contain T_{0} correspond to the third and fourth terms of (8.27). (Note 𝔡𝔢𝔤T = 0.)

The proof of Sublemma 8.26 is complete. □

We go back to the proof of Lemma 8.24. (8.27) implies

F_{a,··· ,b} ∘ F′_{a,··· ,b} − id − d̂∘ \overline{T}_{a,··· ,b} − \overline{T}_{a,··· ,b} ∘ d̂= 0,

if b − a < k − 1. (Here id is the indentity on B_{a,··· ,b}C_{2}[1].) We put

Φ = F_{1,··· ,k} ∘ F̃′_{1,··· ,k} − id − d̂∘ \overline{T}_{,··· ,k} − \overline{T}_{1,··· ,k} ∘ d̂.

Since Φ induces zero on B_{2,··· ,k}C_{2}[1] and on B_{1,··· ,k−1}C_{2}[1] it follows that

Φ ∈ Hom(B_{k}C_{2}[1], B_{1}C_{2}[1]).

In other words, Φ = 0 on B_{1,··· ,k−1}C_{2}[1] and the image of Φ is contained in B_{1}C_{2}[1]. Now we calculate :

F_{1} ∘ Ξ = F_{1,··· ,k} ∘ Ξ
= F_{1,··· ,k} ∘ F̃′_{1,··· ,k} ∘ d̂− F_{1,··· ,k} ∘ d̂∘ F̃′_{1,··· ,k}
= F_{1,··· ,k} ∘ F̃′_{1,··· ,k} ∘ d̂− 𝔪_{1} ∘ F_{1,··· ,k} ∘ F̃′_{1,··· ,k}
= (id + d̂∘ \overline{T}_{1,··· ,k−1} + \overline{T}_{1,··· ,k−1} ∘ d̂+ Φ) ∘ d̂
− d̂∘ (id + d̂∘ \overline{T}_{1,··· ,k−1} + \overline{T}_{1,··· ,k−1} ∘ d̂+ Φ)
= −dΦ.

The proof of Lemma 8.24 is complete. □

We define F′_{k} ∈ Hom(B_{k}C_{2}[1], B_{1}C_{1}[1]) such that

(8.30)  dF′_{k} = −Ξ.

It is now easy to see that F′_{ℓ}, 0 ≤ ℓ ≤ k is an A_{k} functor.

We remark that F′_{k} satisfying (8.30) is not unique. Namely we can take any element Ψ in the kernel of d : Hom(B_{k}C_{2}[1], B_{1}C_{1}[1]) → Hom(B_{k}C_{2}[1], B_{1}C_{1}[1]) and can replace F′_{k} by F′_{k} + Ψ. We will use this freedom in the next step.

Now we are going to construct T_{k}. Using T_{ℓ}, ℓ = 1, ⋯, k − 1 (and F′_{ℓ}) we define

\overline{T}_{a,··· ,b} : B_{a,··· ,b}C_{2}[1] → B_{a,··· ,b}C_{2}[1].

Using T_{ℓ}, ℓ = 0, 1, ⋯, k − 1 (and F′_{ℓ}) we define

T̂_{1,··· ,k−1} : ⨁_{ℓ=0}^{k−1} B_{ℓ}C_{2}[1] → ⨁_{ℓ=0}^{∞} B_{ℓ}C_{2}[1].

T̂_{1,··· ,k−1} is a chain map by induction hypothesis. We consider

\overline{T}_{1,··· ,k} : ⨁_{i=1}^{k} B_{i}C_{2}[1] → ⨁_{i=1}^{k} B_{i}C_{2}[1].

We put

(8.31)  𝔗 = F_{1,··· ,k} ∘ F′_{1,··· ,k} − id − d̂∘ \overline{T}_{1,··· ,k} − \overline{T}_{1,··· ,k} ∘ d̂.

In the same way as the proof of Sublemma 8.26, we can use the fact that T̂_{1,··· ,k−1} is a chain map, to show the following :

Lemma 8.32. The image of 𝔗 is contained in B_{1}C_{2}[1]. Moreover 𝔗 vanishes on ⨁_{i=1}^{k−1} B_{i}C_{2}[1].

We may regard 𝔗 ∈ Hom(B_{k}C_{2}[1], B_{1}C_{2}[1]) by Lemma 8.32. Then, by definition and Lemma 8.32 we have d𝔗 = 0.

Lemma 8.33. We can choose F′_{k} so that 𝔗 ∈ Im d.

Proof. If we replace F′_{k} by F′_{k} + Ψ, then, by (8.31), 𝔗 is replaced by 𝔗 + F_{1} ∘ Ψ. The lemma now follows from the fact that F_{1} induces an isomorphism

H(Hom(B_{k}C_{2}[1], B_{1}C_{1}[1]), d) ≅ H(Hom(B_{k}C_{2}[1], B_{1}C_{2}[1]), d).

□

Lemma 8.33 and Sublemma 8.26 immediately imply that we can choose T_{k} such that T_{ℓ}, ℓ = 0, ⋯, k is an A_{k} transformation.

Thus we have constructed F′ and T : F ∘ F′ → Id^{C_{2}}. We next show that T is a homotopy equivalence. We prove it by using the following general result :

Proposition 8.34. Let G, G′ be A_{∞} functors C → C′. Let T : G → G′ be a natural transformation. We assume that T_{0}(c) : G_{0}(c) → G′_{0}(c) is an homotopy equivalence for any c.

Then there exits a natural transformation T′ : G′ → G, and prenatural transformations S : G → G, S′ : G′ → G′ such that

Id^{G′} −𝔐_{2}(T′, T) = 𝔐_{1}(S), Id^{G} −𝔐_{2}(T, T′) = 𝔐_{1}(S′).

Proof. The proof is similar to the construction of F′ and T above. Namely we construct

T′_{k} : B_{k}C[1] → B_{1}C′[1], S′_{k} : B_{k}C[1] → B_{1}C′[1]

inductively. More precisely we prove the following lemma by induction on k Lemma 8.35.k. Suppose T′_{ℓ}, for ℓ = 0, ⋯, k is an A_{k} transformation and S′_{ℓ} : B_{ℓ}C[1] → B_{1}C[1] for ℓ = 0, ⋯, k be R module homomorphisms. We define T′_{(k)}, S′_{(k)} by putting T′_{(k),i} = S′_{(k),i} = 0 for i > k, and T′_{(k),i} = T′_{i}, S′_{(k),i} = S′_{i} for i ≤ k. Then we have :

(8.36)  Id^{G′} −𝔐_{2}(T′_{(k)}, T) = 𝔐_{1}(S′_{(k)}), on B_{k}C_{2}[1].

Proof. For k = 0, we let T′_{0}(c) ∈ C_{2}(G′(c), G(c)) be a homotoy inverse to T_{0}(c). Then there exists S′_{0}(c) ∈ C_{2}(G′(c), G′(c)) such that

(8.37)  𝔪_{1}(T′_{0}(c)) = 0, 𝔪_{1}(S′_{0}(c)) = 𝔪_{2}(T′_{0}(c), T_{0}(c)).

Thus, we have proved Lemma 8.35.0.

We assume Lemma 8.35.k − 1 and will prove Lemma 8.35.k.

Sublemma 8.38. 𝔐_{1}(S′_{(k−1)}) is zero on B_{k−1}C′[1]. The image of the restriction of 𝔐_{1}(S′_{(k−1)}) to B_{k}C′[1] is in B_{1}C′[1].

The proof is similar to the proof of Lemma 8.22 and is omitted.

We let Ψ be the restriction of 𝔐_{1}(S′_{(k−1)}) to B_{k}C′[1]. Sublemma 8.38 implies

Ψ ∈ Hom(B_{k}C′[1], B_{1}C′[1]).

Sublemma 8.39. d_{1} ∘ Ψ − Ψ ∘ d_{1} = 0.

Proof. This follows immediately from 𝔐_{1}(𝔐_{1}(S′_{(k−1)})) = 0 and Sublemma 8.38. □

Sublemma 8.40. There exists Φ ∈ Hom(B_{k}C_{2}, B_{1}C_{2}). such that :

Ψ = d_{1} ∘ Φ + Φ ∘ d_{1}.

Proof. We define

T_{0∗} : Hom(B_{k}C′[1], B_{1}C[1]) → Hom(B_{k}C′[1], B_{1}C′[1])

as follows. Let V ∈ Hom(B_{k}C′[1], B_{1}C[1]) and c_{0}, ⋯, c_{k} ∈ 𝔒𝔟(C′), x_{i} ∈ C′(c_{i−1}, c_{i}). We then put

T_{0∗}(V)(x_{1}, ⋯, x_{k}) = 𝔪_{2}(V (x_{1}, ⋯, x_{k}), T_{0}(c_{k})).

It follows from assumption that T_{0∗} induces an isomorphism on cohomology. Hence, by Sublemma 8.39, it suffices to show that T_{0∗}(Ψ) is a boundary. By Sublemma 8.38, we find that T_{0∗}(Ψ) is a restriction of 𝔐_{2}(Ψ, T) to B_{k}C′[1]. By A_{∞} formula of 𝔐_{k} we have

𝔐_{2}(Ψ, T) = 𝔐_{2}(𝔐_{1}(T′_{(k−1)}), T) = 𝔐_{1}(𝔐_{2}(T′_{(k−1)}), T).

By induction hypothesis

𝔐_{2}(T′_{(k−1)}, T) = Id^{G′} −𝔐_{1}(S′_{(k−1)})

on B_{k−1}C′[1]. Let −Φ be the restriction of 𝔐_{2}(T′_{(k−1)}, T) on B_{k}C′[1]. Then, by induction hypothesis, Φ ∈ Hom(B_{k}C′[1], B_{1}C′[1]). We now have

𝔐_{1}(𝔐_{2}(T′_{(k−1)}), T)) = −𝔐_{1}(Φ) = d̂_{1} ∘ Φ + Φ ∘ d̂

as required. □

If we put T′_{k} = Φ then we have

𝔐_{1}(T′_{(k)}) = 0.

We remark that Φ satisfying the conclusion of Sublemma 8.39 is not unique. Namely we may change it by any cocycle in Hom(B_{k}C′[1], B_{1}C′[1]).

We next define S′_{k}. Using induction hypothesis, we can prove that the restriction of

Id^{G′} −𝔐_{2}(T′_{(k−1)}, T) − 𝔐_{1}(S′_{(k−1)})

to B_{k}C′[1] defines an element of Hom(B_{k}C′[1], B_{1}C′[1]). We denote it by Ξ. We can prove that Ξ is a cocycle in a way similar to the proof of Sublemma 8.39. Therefore, we may choose T′_{k} so that

Ξ − T_{0∗}(T′_{k})

is a coboundary. Hence we may choose S′_{k} such that (8.36) is satisfied. The proof of Lemma 8.35.k is now complete. □

By Lemma 8.35, we obtain T′ and S′. To construct S, we proceed as follows. We apply the above construction to T′ in place of T and obtain T′′ and S′′ such that

Id^{G} −𝔐_{2}(T′, T′′) = 𝔐_{1}(S′′).

Then

T′′ ≡ 𝔐_{2}(𝔐_{2}(T, T′), T′′) ≡ 𝔐_{2}(T, 𝔐_{2}(T′, T′′)) ≡ T mod Im(𝔐_{1}).

Therefore

Id^{G′} −𝔐_{2}(T′, T) ≡ Id^{G′} −𝔐_{2}(T′, T′′) ≡ 0 mod Im(𝔐_{1}).

The proof of Proposition 8.34 is complete. □

We need here some elementary properties of homotopy equivalence.

Proposition-Definition 8.41. An A_{∞} functor F : C_{1} → C_{2} induces A_{∞} functors F_{∗} : 𝔉𝔲𝔫𝔨(C, C_{1}) → 𝔉𝔲𝔫𝔨(C, C_{2}), F^{∗} : 𝔉𝔲𝔫𝔨(C_{2}, C) → 𝔉𝔲𝔫𝔨(C_{1}, C) such that (F_{∗})_{0}(G) = F ∘ G, (F^{∗})_{0}(G) = G ∘ F.

Proof. Let T^{i} ∈ 𝔉𝔲𝔫𝔨(G^{i−1}, G^{i}) be pre natural transformations such that 𝔡𝔢𝔤′T^{i} = t_{i}. We put

(8.42)  \begin{aligned}((F_{∗})_{k}(T^{1}, ⋯, T^{k}))(x) = ∑_{a}(−1)^{ϵ_{a}} & F(Ĝ^{0}(x_{a}^{(1)}), T̂^{1}(x_{a}^{(2)}), \\ & ⋯, T̂^{k}(x_{a}^{(2k)}), Ĝ^{k}(x_{a}^{(2k+1)})). \end{aligned}

where

ϵ_{a} = ∑_{j=1}^{k} ∑_{i=1}^{2j−1} t_{j} deg′ x_{a}^{(i)}.

We can prove that (8.42) defines an A_{∞} functor in a way similar to the proof of Theorem 7.55. (We omit the detail.)

We next define

\begin{aligned}((F^{∗})_{1}(T^{1}))(x) & = T^{1}(F̂(x)), \\ ((F^{∗})_{k}(T^{1}, ⋯, T^{k}))(x) & = 0 k > 2. \end{aligned}

It is easy see that F^{∗} is an A_{∞} functor. □

The following two corollaries are immediate consequences.

Corollary 8.43. If F : C_{1} → C_{2} is homotopic to F′ : C_{1} → C_{2} and G : C_{2} → C_{3} is homotopic to G′ : C_{2} → C_{3}, then G ∘ F is homotopic to G′ ∘ F′.

Corollary 8.44. If F : C_{1} → C_{2} and G : C_{2} → C_{3} are homotopy equivalences then G ∘ F is a homotopy equivalence.

We are now in the position to complete the proof of Proposition 8.9. We have constructed F′. By using Proposition 8.34, we can prove that F ∘ F′ is homotopic to identity.

We next prove that F′ ∘ F is homotopic to identity. For this purpose, we apply the same argument to F′ and obtain F′′ such that F′ ∘F′′ is homotopic to identity. It follows that F′′ is homotopic to F. Hence by Corollary 8.43, F′ ∘ F is homotopic to identity. The proof of Proposition 8.9 is complete. □

We continue the proof of Theorem 8.6. In this section we complete the proof in the case of differential graded category and postpone the proof of the general case to the next section.

Our next goal is the proof of Lemma 8.45, which is another case of Theorem 8.6. We define the notion of full subcategory of A_{∞} category in an obvious way.

Lemma 8.45. Let C_{2} be an A_{∞} category such that 𝔪_{k} = 0 for k ≥ 3. Let C_{1} be a full subcategory of C_{2} and F : C_{1} → C_{2} be the inclusion. We assume (8.7.2). Then F : C_{1} → C_{2} is a homotopy equivalece.

Proof. Let c ∈ 𝔒𝔟(C_{2}) − 𝔒𝔟(C_{1}). We choose F′_{0}(c) ∈ 𝔒𝔟(C_{1}) which is homotopy equivalent to c. If c ∈ 𝔒𝔟(C_{1}) we put F′_{0}(c) = c.

By the choice of F′_{0}(c), there exists

T′_{0}(c) ∈ C_{2}(c, F′_{0}(c)), T_{0}(c) ∈ C_{2}(F′_{0}(c), c)

and

S′_{0}(c) ∈ C_{2}(c, c), S_{0}(c) ∈ C_{2}(F′_{0}(c), F′_{0}(c))

such that

\begin{aligned}e_{F_{0}(c)} − 𝔪_{2}(T_{0}(c), T′_{0}(c)) & = 𝔪_{1}(S_{0}(c)) \\ e_{F′_{0}(c)} − 𝔪_{2}(T′_{0}(c), T_{0}(c)) & = 𝔪_{1}(S′_{0}(c)). \end{aligned}

For c ∈ 𝔒𝔟(C_{1}) we put T_{0}(c) = T′_{0}(c) = Id^{c} and S_{0}(c) = S′_{0}(c) = 0.

We next define F′_{k}. We first put

x ∘ y = (−1)^{deg x}𝔪_{2}(x, y), d(x) = 𝔪_{1}(x).

Then, using the fact that 𝔪_{k} = 0 for k ≥ 3, we have

\begin{aligned}d(x ∘ y) & = dx ∘ y + (−1)^{deg x+1}x ∘ dy, \\ (x ∘ y) ∘ z & = x ∘ (y ∘ z), \\ e ∘ x & = x = x ∘ e. \end{aligned}

In view of the second formula, we may write

x_{1} ∘ x_{2} ∘ ⋯ ∘ x_{k}

etc. Now let x_{i} ∈ C_{2}(c_{i−1}, c_{i}). We put

F′_{k}(x_{1}, ⋯, x_{k}) = T′_{0}(c_{0}) ∘ x_{1} ∘ S′_{0}(c_{1}) ∘ x_{2} ∘ ⋯ ∘ S′_{0}(c_{k−1}) ∘ x_{k} ∘ T′_{0}(c_{k}).

We remark that

d(S′_{0}(c)) = e_{F_{0}(c)} − T′_{0}(c) ∘ T_{0}(c).

It follows from definition that

d(F′_{k}(x_{1}, ⋯, x_{k}))
= ∑_{i}(−1)^{deg x_{1}+··· deg x_{i−1}+i−1}F′_{k}(x_{1}, ⋯, dx_{i}, ⋯, x_{k})

+ ∑_{i}(−1)^{deg x_{1}+··· deg x_{i}+i}F′_{k}(x_{1}, ⋯, x_{i} ∘ x_{i+1}, ⋯, x_{k})

+ ∑_{i}(−1)^{deg x_{1}+··· deg x_{i−1}+i−1+1}F′_{i}(x_{1}, ⋯, x_{i}) ∘ F′_{k−i}(x_{i+1}, ⋯, x_{k}).

Therefore

𝔪_{1}(F′_{k}(x_{1}, ⋯, x_{k}))

+ ∑_{i}(−1)^{deg x_{1}+··· deg x_{i−1}+i−1}F′_{i}(x_{1}, ⋯, x_{i}) ∘ F′_{k−i}(x_{i+1}, ⋯, x_{k})

= ∑_{i}(−1)^{deg x_{1}+··· deg x_{i−1}+i−1}F′_{k}(x_{1}, ⋯, 𝔪_{1}(x_{i}), ⋯, x_{k})

+ (−1)^{deg x_{1}+··· deg x_{i−1}+i−1}F′_{k}(x_{1}, ⋯, 𝔪_{2}(x_{i}, x_{i+1}), ⋯, x_{k}).

It follows that F′_{k}, k = 0, ⋯ is an A_{∞} functor.

The composition F′∘F is an identity functor. We are going to show that F ∘F′ is homotopic to identity. For this purpose, we are going to construct T : F ∘F′ → 1_{C_{2}} satisfying the assumption of Proposition 8.34. We already constructed T_{0}. We remark that

(F ∘ F′)(x_{1}, ⋯, x_{k}) = F′_{k}(x_{1}, ⋯, x_{k}).

Hence the condition T : F ∘ F′ → Id^{C_{2}} to be an A_{∞} functor can be written as

(8.46)

∑_{i}(−1)^{deg x_{1}+··· deg x_{i−1}+i−1+1}T_{0}(c_{0}) ∘ x_{1} ∘ S′_{0}(c_{1}) ∘ x_{2} ∘ S′_{0}(c_{2}) ∘ ⋯

⋯ ∘ x_{i} ∘ T′_{0}(c_{i}) ∘ T_{k−i}(x_{i+1}, ⋯, x_{k}) = 𝔪_{1}(T_{k}(x_{1}, ⋯, x_{k}))

+ ∑_{i}(−1)^{deg x_{1}+··· deg x_{i−1}+i−1}T_{k}(x_{1}, ⋯, 𝔪_{1}(x_{i}), ⋯, x_{k})

+ ∑_{i}(−1)^{deg x_{1}+··· deg x_{i−1}+i−1}T_{k}(x_{1}, ⋯, 𝔪_{2}(x_{i}, x_{i+1}), ⋯, x_{k}).

We put

T_{k}(x_{1}, ⋯, x_{k}) = T_{0}(c_{0}) ∘ x_{1} ∘ S′_{0}(c_{1}) ∘ x_{2} ∘ ⋯ ∘ x_{k} ∘ S′_{0}(c_{k}).

(8.46) can be checked easily.

Lemma 8.45 now follows from Proposition 8.34. □

Now we show :

Proposition 8.47. Theorem 8.6 holds if 𝔪_{k} = 0 for k > 2 in C_{1}, C_{2}.

Proof. Let C′_{2} be the full subcategory such that 𝔒𝔟(C′_{2}) is the image of F_{0}. Lemma 8.41 implies that the inclusion C′_{2} → C_{2} is a homotopy equivalence.

For each c ∈ 𝔒𝔟(C′_{2}) we take and fix α(c) ∈ 𝔒𝔟(C_{1}) such that F_{0}(α(c)) = c. Let C′_{1} be the full subcategory of C_{1} such that 𝔒𝔟(C′_{1}) is the image of α. Proposition 8.9 implies that the restriction of F to C′ induces a homotopy equivalence C′_{1} → C′_{2}. Therefore, using Corollary 8.43 and 8.44, it suffices to show that C′_{1} → C_{1} is a homotopy equivalence to complete the proof of Proposition 8.47. This follows from Lemma 8.45 and the following Lemma 8.48. □

Lemma 8.48. We assume (8.7). If F_{0}(b) = F_{0}(b′) then b is homotopy equivalent to b′.

Proof. By (8.7) we have

F_{1∗} : H(C_{1}(b, b′); 𝔪_{1}) → H(C_{2}(F_{0}(b), F_{0}(b′)); 𝔪_{1})

is an isomorphism. We take [f] ∈ H(C_{1}(b, b′); 𝔪_{1}) which is mapped to [e_{F_{0}(b)}] by F_{1∗}. It is easy to see that

\begin{aligned}𝔪_{2}(f, ·) & : C_{1}(b′, a) → C_{1}(b, a) \\ 𝔪_{2}(·, f) & : C_{1}(a, b) → C_{1}(a, b′) \end{aligned}

induces isomorphism on homology. The lemma then follows from Lemma 6.24. □

We need the following proposition in the next section.

Proposition 8.49. If C_{1} is homotopy equivalent to C_{2}, then 𝔉𝔲𝔫𝔨(C, C_{1}) is homotopy equivalent to 𝔉𝔲𝔫𝔨(C, C_{2}) and 𝔉𝔲𝔫𝔨(C_{1}, C) is homotopy equivalent to 𝔉𝔲𝔫𝔨(C_{2}, C).

Proof. Let F : C_{1} → C_{2}, F′ : C_{2} → C_{1} be as in Definition 8.5 and let H,H′ be natural transformations from F ∘ F′ to identity functor and from identity functor to F ∘ F′ respectively. Let G : C → C_{2} be an A_{∞} functor. We put

\begin{aligned}ℌ_{0}(G) & = (G^{∗})_{1}(H) ∈ 𝔉𝔲𝔫𝔨((F_{∗} ∘ F′_{∗})_{0}(G), G) \\ ℌ_{k} & = 0, k > 0. (Note (G^{∗})_{k} = 0 for k > 1.) \end{aligned}

It is easy to check that ℌ is a natural transform from F_{∗} ∘ F′_{∗} to the identity functor Id_{𝔉𝔲𝔫𝔨(C,C_{2})}. We define in a similar way a natural transformation ℌ′ from Id_{𝔉𝔲𝔫𝔨(C,C_{2})} to F_{∗} ∘ F′_{∗}. We assume that

𝔐_{2}(H, H′) − Id_{Id_{𝔉𝔲𝔫𝔨(C_{1},C_{1})}} = 𝔐_{1}(T)

(Note that the confusing symbol Id_{Id_{𝔉𝔲𝔫𝔨(C_{1},C_{1})}} denotes the identity transform from the identity functor : C_{1} → C_{1} to itself.) We assume also that

𝔐_{2}(H′, H) − Id_{F∘F′} = 𝔐_{1}(T′).

We put

\begin{aligned}𝔗_{0}(G) & = (G^{∗})_{1}(T) ∈ 𝔉𝔲𝔫𝔨((F_{∗} ∘ F′_{∗})_{0}(G), G) \\ 𝔗_{k} & = 0, k > 0. \end{aligned}

And we define 𝔗′ in a similar way. Then we find

\begin{aligned}Id_{Id_{𝔉𝔲𝔫𝔨(C,C_{1})}} −𝔐̂_{2}(ℌ, ℌ′) & = 𝔐̂_{1}(𝔗), \\ Id_{(F∘F′)_{∗}} −𝔐̂_{2}(ℌ′, ℌ) & = 𝔐̂_{1}(𝔗′). \end{aligned}

Here 𝔐̂ is the A_{∞} structure on 𝔉𝔲𝔫𝔨(𝔉𝔲𝔫𝔨(C, C_{1}), 𝔉𝔲𝔫𝔨(C, C_{1})).

Thus we proved that (F ∘ F′)_{∗} is homotopic to identity. This complete the proof of the first half of the Proposition 8.49. The second half can be proved in a similar way. □

9. An A_{∞} analogue of Yoneda lemma.

In §7 we constructed an A_{∞} functor ℜ𝔢𝔭_{0}(c) : C^{o} → CH for each object c of C. The purpose of this section is to make it functorial. Moreover we prove that c ↦ ℜ𝔢𝔭_{0}(c) defines a homotopy equivalence between the A_{∞} category C and one of representable A_{∞} functors : C^{o} → CH. Namely we prove the following Theorem 9.1. We let ℜ𝔢𝔭(C^{o}, CH) be the full subcategory of 𝔉𝔲𝔫𝔨(C^{o}, CH) such that 𝔒𝔟(ℜ𝔢𝔭(C^{o}, CH)) is the set of all representable A_{∞} functors. We define a full subcategory 𝔇ℜ𝔢𝔭(C^{o}, CH) of 𝔉𝔲𝔫𝔨(C^{o}, CH) such that 𝔒𝔟(𝔇ℜ𝔢𝔭(C^{o}, CH)) is the set of all derived representable A_{∞} functors.

Theorem 9.1. There exists a homotopy equivalences of A_{∞} functors ℜ𝔢𝔭 : C ≅ ℜ𝔢𝔭(C^{o}, CH), 𝔇ℜ𝔢𝔭 : DC ≅ 𝔇ℜ𝔢𝔭(C^{o}, CH), such that ℜ𝔢𝔭_{0}(c) is an in Definition 7.28.

Remark 9.2. The first half of Theorem 9.1 was proved in [Fu4] §12, over ℤ_{2} coefficient. In this article we are going to put precise sign in its proof. We also improve the presentation of the proof in [Fu4].

Remark 9.3. In the case of usual category, the first half of Theorem 9.1, which is Yoneda’s lemma, is well known.

We remark that actually ℜ𝔢𝔭(C^{o}, CH) is a differential graded category, since all the higher compositions are zero. Therefore Theorem 9.1 implies the following :

Corollary 9.4. Any A_{∞} category is homotopy equivalent to a differential graded category.

Proof of Theorem 9.1. We already defined ℜ𝔢𝔭_{0}. We will define ℜ𝔢𝔭_{k} : B_{k}(C) → B_{1}(ℜ𝔢𝔭(C^{o}, CH)). Let c_{0}, ⋯, c_{k} ∈ 𝔒𝔟(C), x_{i} ∈ C(c_{i−1}, c_{i}). We need to define a natural transformation

(9.5)  ℜ𝔢𝔭_{k}(x_{1}, ⋯, x_{k}) : ℜ𝔢𝔭_{0}(c_{0}) → ℜ𝔢𝔭_{0}(c_{k}).

Let b_{0}, ⋯, b_{ℓ} ∈ 𝔒𝔟(C), y_{i} ∈ C^{o}(b_{i−1}, b_{i}) = C(b_{i}, b_{i−1}). To define (9.5) we need to define

ℜ𝔢𝔭_{k}(x_{1}, ⋯, x_{k})_{ℓ}(y_{1}, ⋯, y_{ℓ}) ∈ Hom(ℜ𝔢𝔭_{0}(c_{0})_{0}(b_{0}), ℜ𝔢𝔭_{0}(c_{k})_{0}(b_{ℓ})).

Let z ∈ ℜ𝔢𝔭_{0}(c_{0})_{0}(b_{ℓ}) = C(b_{0}, c_{0}). We put x = x_{1} ⊗ ⋯ ⊗ x_{k}, y = y_{1} ⊗ ⋯ ⊗ y_{ℓ}. We use the notations x^{op} = x_{k} ⊗ ⋯ ⊗ x_{1}, ϵ(x) = ∑_{i<j}(deg x_{i} + 1)(deg x_{j} + 1) etc. introduced in §7. Now we define

Definition 9.6.

(9.7)  (ℜ𝔢𝔭_{k}(x))_{ℓ}(y)(z) = (−1)^{ϵ(y)+(deg′ x)(deg′ y+deg′ z)}𝔪_{k+ℓ+1}(y^{op}, z, x).

We remark that the right hand side of (9.7) is in ℜ𝔢𝔭_{0}(c_{k})(b_{ℓ}) = C(b_{ℓ}, c_{k}). We also remark that

𝔡𝔢𝔤′(ℜ𝔢𝔭_{k}(x))_{ℓ}(y) = deg′ x + deg′ y.

Lemma 9.8. (9.7) defines an A_{∞} functor.

Proof. By A_{∞} formula of 𝔪 we have

(9.9)

0 = ∑_{a}(−1)^{deg′ y{(3) \atop a}}𝔪(y_{a}^{(3)op}, 𝔪(y_{a}^{(2)op}), y_{1}^{(1)op}, z, x)

+ ∑_{a,b}(−1)^{deg′ y{′′op \atop a}}𝔪(y_{a}^{′′op}, 𝔪(y_{a}^{′op}, z, x′_{b}), x′′_{b})

+ (−1)^{deg′ y+deg′ z}𝔪(y^{op}, z, d̂x).

We will rewrite each term of (9.9). The first term is equal to :

(9.10)  (−1)^{ϵ_{1}}ℜ𝔢𝔭(x)(y_{a}^{(1)}, 𝔪^{o}(y_{a}^{(2)}), y_{a}^{(3)})(z) = (−1)^{ϵ_{2}}ℜ𝔢𝔭(x)(d̂^{o}y)(z)

where d̂^{op} is the coderivation induced by 𝔪^{o} on C^{o} and

ϵ_{1} = deg′ y_{a}^{(3)} + ϵ(y_{a}^{(2)}) + 1 + (deg′ x)(deg′ y + deg′ z + 1) + ϵ(y_{a}^{(1)}, 𝔪^{o}(y_{a}^{(2)}), y_{a}^{(3)})

= deg′ y_{a}^{(1)} + (deg′ x)(deg′ y + deg′ z + 1) + ϵ(y).

Hence

(9.11)  ϵ_{2} = (deg′ x)(deg′ y + deg′ z + 1) + ϵ(y).

We divide the second terms of (9.9) into the following five cases.

Case 1 : x′_{b} ≠ 1 ∈ B_{0}C, x′′_{b} ≠ 1 ∈ B_{0}C :

The second term of (9.9) of this case is

(9.12)

∑_{a,b}(−1)^{ϵ_{3}(a,b)}𝔪(y_{a}^{′′op}, ℜ𝔢𝔭(x′_{b})(y′_{a})(z), x′′_{b})

= ∑_{a,b}(−1)^{ϵ_{4}(a,b)}ℜ𝔢𝔭(x′′_{b})(y′′_{a})(ℜ𝔢𝔭(x′_{b})(y′_{a})(z))

= ∑_{a,b}(−1)^{ϵ_{5}(a,b)}𝔪_{2}(ℜ𝔢𝔭(x′_{b})(y′_{a}), ℜ𝔢𝔭(x′′_{b})(y′′_{a}))(z)

where 𝔪_{2} in the third line is an operation in CH and

ϵ_{3}(a, b) = deg′ y′′_{a} + (deg′ x′_{b})(deg′ y′_{a} + deg′ z) + ϵ(y′_{a})
ϵ_{4}(a, b) = ϵ_{3}(a, b) + (deg′ x′′_{b})(1 + deg′ y′′_{a} + deg′ z + deg′ x′_{b} + deg′ y′_{a}) + ϵ(y′′_{a})
ϵ_{5}(a, b) = ϵ_{4}(a, b) + (deg′ x′_{b} + deg′ y′_{a} + 1)(deg′ x′′_{b} + deg′ y′′_{a}) + 1.

We calculate (using 𝔡𝔢𝔤′ℜ𝔢𝔭(x′_{b})(y′_{a}) = deg′ x′_{b} + deg′ y′_{a} + 1) that

ϵ_{5}(a, b) = ϵ(y) + deg′ x deg′ z + deg′ x deg′ y + deg′ x′′_{b} deg′ y′_{a}.

Therefore (9.12) is equal to

(9.13)  ∑_{b}(−1)^{ϵ_{6}}𝔐_{2}(ℜ𝔢𝔭(x′_{b}), ℜ𝔢𝔭(x′′_{b}))(y)(z)

where

(9.14)  ϵ_{6} = ϵ(y) + (deg′ x)(deg′ z + deg′ y) + 1.

Case 2 : x′_{b} = 1 ∈ B_{0}C, y′_{a} ≠ 1 ∈ B_{0}C :

In this case the second term of (9.9) is :

(9.15)

∑_{a}(−1)^{ϵ_{7}(a)}𝔪(y_{a}^{′′op}, ℜ𝔢𝔭_{0}(c_{0})(y′_{a})(z), x)

= ∑_{a,b}(−1)^{ϵ_{8}(a)}ℜ𝔢𝔭(x)(y′′_{a})(ℜ𝔢𝔭_{0}(c_{0})(y′_{a})(z))

= ∑_{a}(−1)^{ϵ_{9}(a)}𝔪_{2}(ℜ𝔢𝔭_{0}(c_{0})(y′_{a}), ℜ𝔢𝔭(x)(y′′_{a}))(z)

= (−1)^{ϵ_{6}}𝔐_{2}(ℜ𝔢𝔭_{0}(c_{0}), ℜ𝔢𝔭(x))(y)(z) where ϵ_{6} is as in (9.14). In fact

ϵ_{7}(a) = deg′ y′′_{a} + ϵ(y′_{a}),
ϵ_{8}(a) = ϵ_{7}(a) + (deg′ x)(y′′_{a} + y′_{a} + deg′ z + 1) + ϵ(y′′_{a})
ϵ_{9}(a) = ϵ_{8}(a) + (deg′ y′_{a} + 1)(deg′ x + deg′ y′′_{a})
≡ ϵ_{6} + deg′ x deg′ y′_{a} mod 2.

Case 3 : x′_{b} = 1 ∈ B_{0}C, y′_{a} = 1 ∈ B_{0}C :

In a similar way, we have

(9.16)  (−1)^{ϵ_{6}+deg′ x+deg′ y}ℜ𝔢𝔭(x)(y)(𝔪_{1}(z)).

In face we have

deg′ y + (deg′ x)(deg′ z + deg′ y) + ϵ(y) = ϵ_{6} + deg′ x + deg′ y.

Case 4 : x′′_{b} = 1 ∈ B_{0}C, y′′_{a} ≠ 1 ∈ B_{0}C :

In a similar way we have

(9.17)  (−1)^{ϵ_{6}}𝔐_{2}(ℜ𝔢𝔭(x), ℜ𝔢𝔭_{0}(c_{k}))(y)(z).

Case 5 : x′′_{b} = 1 ∈ B_{0}C, y′′_{a} = 1 ∈ B_{0}C :

In a similar way we have

(9.18)  (−1)^{ϵ_{6}}𝔪_{1}(ℜ𝔢𝔭(x)(y)(z)).

We remark that the sum of (9.10), (9.16) and (9.18) is

(9.19)  (−1)^{ϵ_{6}}𝔐_{1}(ℜ𝔢𝔭(x))(y)(z)

Finally the third term of (9.9) is :

(9.20)  (−1)^{ϵ_{6}+1}ℜ𝔢𝔭(d̂x)(y)(z).

Thus (−1)^{ϵ_{6}} times (9.9) implies

∑_{x′_{b}≠1, x′′_{b}≠1} 𝔐_{2}(ℜ𝔢𝔭(x′_{b}), ℜ𝔢𝔭(x′′_{b}))(y) + 𝔐_{2}(ℜ𝔢𝔭_{0}(c_{0}), ℜ𝔢𝔭(x))(y)

+ 𝔐_{2}(ℜ𝔢𝔭(x), ℜ𝔢𝔭_{0}(c_{k}))(y) + 𝔐_{1}(ℜ𝔢𝔭(x))(y)

= ℜ𝔢𝔭(d̂x)(y).

(We remark that we need over all minus sign in the definition of 𝔐_{k} to show this formula.) The proof of Lemma 9.8 is complete. □

Using Lemma 9.8 and Proposition 8.9, it suffices to check (8.7.1) for ℜ𝔢𝔭 to complete the proof of Theorem 9.1. Namely we need to show

ℜ𝔢𝔭_{1} : C(c_{1}, c_{2}) → 𝔉𝔲𝔫𝔠(ℜ𝔢𝔭_{0}(c_{1}), ℜ𝔢𝔭_{0}(c_{2}))

induces an isomorphism on homology. We define

Π : 𝔉𝔲𝔫𝔠(ℜ𝔢𝔭_{0}(c_{1}), ℜ𝔢𝔭_{0}(c_{2})) → C(c_{1}, c_{2})

by

Π(T) = (−1)^{𝔡𝔢𝔤′T} T_{0}(c_{1})(e_{c_{1}}).

(Note e_{c_{1}} ∈ C(c_{1}, c_{1}) = ℜ𝔢𝔭_{0}(c_{1})_{0}(c_{1}). Hence T_{0}(c_{1})(e_{c_{1}}) ∈ ℜ𝔢𝔭_{0}(c_{2})_{0}(c_{1}) = C(c_{1}, c_{2}).)

Lemma 9.21. Π is a chain map. Proof. Let T ∈ 𝔉𝔲𝔫𝔠(ℜ𝔢𝔭_{0}(c_{1}), ℜ𝔢𝔭_{0}(c_{2})). We have

\begin{aligned}Π(𝔐_{1}(T)) & = (−1)^{𝔡𝔢𝔤′T} (𝔐_{1}(T))_{0}(c_{1})(e_{c_{1}}) \\ & = (−1)^{𝔡𝔢𝔤′T +1}𝔪_{1}(T_{0}(c_{1})(e_{c_{1}})) = 𝔪_{1}(Π(T)). \end{aligned}

(We remark that overall minus sign in the definition of 𝔐_{1} is essential here.) □ We have

(Π ∘ ℜ𝔢𝔭_{1})(x) = (−1)^{deg′ x}(ℜ𝔢𝔭_{1}(x)_{0}(c_{1}))(e_{c_{1}}) = 𝔪_{2}(e_{c_{1}}, x) = x.

So, to complete the proof of Theorem 9.1, it suffices to show that ℜ𝔢𝔭_{1} ∘ Π is homotopic to identity. We define an operator

H : 𝔉𝔲𝔫𝔠(ℜ𝔢𝔭_{0}(c_{1}), ℜ𝔢𝔭_{0}(c_{2})) → 𝔉𝔲𝔫𝔠(ℜ𝔢𝔭_{0}(c_{1}), ℜ𝔢𝔭_{0}(c_{2}))

of degree +1 by

(H(T))_{k}(y)(z) = (−1)^{deg′ z deg′ y+deg′ z+deg′ y+𝔡𝔢𝔤′T} T_{k+1}(z, y)(e_{c_{1}}).

Here y ∈ B_{k}C^{j}(b_{1}, b_{k}), z ∈ C(b_{1}, c_{1}) = ℜ𝔢𝔭_{0}(c_{1})(b_{1}). (Then T_{k+1}(y, z)(e_{c_{1}}) ∈ ℜ𝔢𝔭_{0}(c_{2})(b_{k}). Lemma 9.22.

(9.23)  T − (ℜ𝔢𝔭_{1} ∘ Π)(T) = (𝔐_{1} ∘ H + H ∘ 𝔐_{1})(T).

Proof. We have

(−1)^{𝔡𝔢𝔤′T} 𝔐_{1}(H(T))(y)(z))

= 𝔪_{1}(H(T)(y)(z)) + (−1)^{𝔡𝔢𝔤′T} H(T)(d̂^{op}y)(z) + (−1)^{𝔡𝔢𝔤′T +deg′ y}H(T)(y)(𝔪_{1}(z))

+ ∑_{a} 𝔪_{2}(H(T)(y′_{a}), ℜ𝔢𝔭_{0}(y′′_{a})))(z)

+ ∑_{a}(−1)^{(𝔡𝔢𝔤′T +1) deg′ y′_{a}}𝔪_{2}(ℜ𝔢𝔭_{0}(y′_{a}), H(T)(y′′_{a}))(z)

= (−1)^{ϵ_{1}}𝔪_{1}(T (z, y)(e))

+ ∑_{a}(−1)^{ϵ_{2}}T (z, y_{a}^{(1)}, 𝔪(y_{a}^{(2)op}), y_{a}^{(3)})(e)

+ (−1)^{ϵ_{3}}T (𝔪_{1}(z), y)(e)

+ ∑_{a}(−1)^{ϵ_{4}}𝔪(y_{a}^{′′op}, T (z, y′_{a})(e))

+ ∑_{a}(−1)^{ϵ_{5}}T (𝔪(y_{a}^{′op}, z), y′′_{a})(e)

FLOER HOMOLOGY AND MIRROR SYMMETRY II 73

Here

ϵ_{1} = deg′ y deg′ z + deg′ y + deg′ z,
ϵ_{2} = 𝔡𝔢𝔤′T + (deg′ y + 1)(deg′ z) + deg′ y + 1 + deg′ z + deg′ y_{a}^{(1)} + ϵ(y_{a}^{(2)}),
ϵ_{3} = 𝔡𝔢𝔤′T + deg′ y + (deg′ y)(deg′ z + 1) + deg′ y + deg′ z + 1,
ϵ_{4} = (𝔡𝔢𝔤′T + 1 + deg′ y′_{a})(deg′ y′′_{a}) + deg′ z deg′ y′_{a} + ϵ(y′′_{a}) + deg′ y′_{a} + deg′ z,
ϵ_{5} = (𝔡𝔢𝔤′T + 1)(deg′ y′_{a}) + (deg′ y′_{a} + 1)(𝔡𝔢𝔤′T + deg′ y′′_{a} + 1) + ϵ(y′_{a})
+ deg′ y′′_{a}(deg′ z + deg′ y′_{a} + 1) + deg′ y′′_{a} + deg′ z + deg′ y′_{a} + 1.

We also have

(9.24)

(−1)^{𝔡𝔢𝔤′T +1}H(𝔐_{1}(T))(y)(z)

= (−1)^{deg′ y+deg′ z+deg′ z deg′ y}𝔪_{1}(T (z, y)(e))

+ ∑_{a}(−1)^{ϵ_{6}}T (𝔪(y_{a}^{′op}, z), y′′_{a})(e)

+ ∑_{a}(−1)^{ϵ_{7}}T (z, y_{a}^{(1)}, 𝔪(y_{a}^{(2)op}), y_{a}^{(3)})(e)

+ (−1)^{deg′ y+deg′ z+deg′ z deg′ y} ∑_{deg y′′_{a}≠0} 𝔪_{2}(T (z, y′_{a}), ℜ𝔢𝔭(y′′_{a}))(e)

+ (−1)^{deg′ y+deg′ z+deg′ z deg′ y}𝔪_{2}(T_{0}, ℜ𝔢𝔭_{0}(z, y))(e)

+ (−1)^{deg′ y+deg′ z+deg′ z deg′ y+𝔡𝔢𝔤′T deg′ z}𝔪_{2}(ℜ𝔢𝔭_{0}(z), T (y))(e).

Here

ϵ_{6} = deg′ y + deg′ z + deg′ z deg′ y + deg′ z deg′ y′_{a} + ϵ(y′_{a}) + 𝔡𝔢𝔤′T + 1 = ϵ_{5},
ϵ_{7} = deg′ y + deg′ z + deg′ z deg′ y + 𝔡𝔢𝔤′T + 1 + deg′ z + deg′ y_{a}^{(1)} + ϵ(y^{(2)}) = ϵ_{2}.

The fourth term of (9.24) is :

(−1)^{ϵ_{8}}𝔪(y_{a}^{′′op}, T (z, y′_{a})(e)),

where ϵ_{8} = deg′ y + deg′ z + deg′ z deg′ y + (𝔡𝔢𝔤′T + deg′ z + deg′ y′_{a}) deg′ y′′_{a} + ϵ(y′′_{a}) = ϵ_{4}. The fifth term of (9.24) is

(−1)^{ϵ_{9}}𝔪(y^{op}, z, T (e)) = ℜ𝔢𝔭_{1}(T (e))(y)(z) = (−1)^{𝔡𝔢𝔤′T} ((ℜ𝔢𝔭_{1} ∘ Π)(T))(y)(z), where ϵ_{9} = deg′ y + deg′ z + deg′ z deg′ y + (𝔡𝔢𝔤′T)(deg′ y + deg′ z) + deg′ y deg′ z + ϵ(y). Finally the sixth term of (9.24) is :

(−1)^{deg′ y+deg′ z+deg′ z deg′ y+𝔡𝔢𝔤′T deg′ z+(deg′ z+1)(𝔡𝔢𝔤′T +deg′ y)}T (y)(𝔪_{2}(e, z)),

= (−1)^{𝔡𝔢𝔤′T +1}T (y)(z).

The lemma follows. □

The proof of the first half of Theorem 9.1 is now complete. We omit the proof of the second half. (Which is analogous to the argument of the last section of [Fu7].) □

We are now in the position to complete the proof of Theorem 8.6. We consider the following Diagram.

Diagram 9.1.

We can easily see that the diagram commutes. Moreover

𝔉_{∗} : ℜ𝔢𝔭(C_{1}^{o}, CH) → ℜ𝔢𝔭(C_{2}^{o}, CH)

satisfies the assumption of Theorem 8.6, by the proof of Proposition 8.49. Hence by Proposition 8.47 it is a homotopy equivalence. Moreover by Theorem 9.1 the vertical allows are homotopy equivalence. Therefore by Corollary 8.44 𝔉 is a homotopy equivalence. The proof of Theorem 8.6 is now complete. □

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Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa, Kyoto, 606-8224, JAPAN

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