Development in symplectic Floer theoryicm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_50.pdf · Development in symplectic Floer theory 1065 Theorem 2.2. If ϕ ∈ Ham(M,ω) has

Development in symplectic Floer theory

Kaoru Ono∗

Abstract. In the middle of the 1980s, Floer initiated a new theory, which is now called theFloer theory. Since then the theory has been developed in various ways. In this article we reportsome recent progress in Floer theory in symplectic geometry. For example, we give an outlineof a proof of the flux conjecture, which states that the Hamiltonian diffeomorphism group isC1-closed in the group of symplectomorphisms for closed symplectic manifolds. We also give abrief survey on the obstruction–deformation theory for Floer theory of Lagrangian submanifoldsand explain some of its applications.

Mathematics Subject Classification (2000). Primary 53D40; Secondary 53D35.

Keywords. Symplectic manifold, Hamiltonian systems, Lagrangian submanifolds, Floer coho-mology, A∞-structure.

1. Introduction

In [9]–[14], Andreas Floer initiated “∞2 -dimensional” (co)homology theory, which is

now called Floer theory. He invented this theory to proveArnold’s conjecture for fixedpoints of Hamiltonian diffeomorphism and, under certain assumptions, its analoguefor Lagrangian intersections. Roughly speaking, the conjecture states that there is anon-trivial topological lower bound for the number of fixed points of a Hamiltoniandiffeomorphism. It is one of his conjectures which stimulated recent developmentsin symplectic geometry. This theory was soon adapted in Donaldson theory and heconstructed the instanton homology theory. A lot of work has been done since andFloer theory has been developed in various directions. In this article, we will describesome recent development of Floer theory in symplectic geometry.

In these decades, symplectic geometry has been much developed. In particu-lar, Gromov revealed many significant phenomena based on his theory of pseudo-holomorphic curves [18] and revolutionized the study in this area. Hamiltonian dy-namics is one of main sources of symplectic geometry. The existence of periodictrajectories is a basic problem and there are many works on this subject up to now. Infact, the existence of periodic trajectories reflects so-called symplectic rigidity phe-nomena. Since trajectories of a Hamiltonian system are characterized by the leastaction principle, the variational method can be applied to the existence of periodic

∗The author is partly supported by Grant-in-Aid for Scientific Research Nos. 14340019 and 17654009, JapanSociety for the Promotion of Science.

Proceedings of the International Congressof Mathematicians, Madrid, Spain, 2006© 2006 European Mathematical Society

1062 Kaoru Ono

trajectories. Namely, closed trajectories are critical points of the action functional as-sociated to the Hamiltonian system. Floer combined the variational framework withthe theory of pseudo-holomorphic curves to construct an analogue of Morse theoryfor the action functional.

In the first part of this article, we discuss Floer theory for Hamiltonian systemsand present some applications including the flux conjecture. In the second part, wediscuss Floer theory for Lagrangian submanifolds. In general, Floer cohomologymay not be defined for a pair of Lagrangian submanifolds. We briefly describe theobstruction to defining Floer cohomology as well as the filteredA∞-algebra associatedto a Lagrangian submanifold. We also present some applications, e.g., Lagrangianintersections, non-triviality of the Maslov class, etc. Although there will be someoverlaps with Y. G. Oh’s contribution to this proceedings, we will try to put differentemphases on the theory in this lecture.

2. Floer theory for symplectomorphisms

2.1. Review on the construction. In this section, we briefly review the constructionof Floer cohomology for symplectomorphisms, especially Hamiltonian diffeomor-phisms, which was initiated in [14] and developed in e.g., [21], [36], [16], [29]. Let(M,ω) be a symplectic manifold. In this article, we assume thatM is compact with-out boundary for simplicity. Denote by Xh the Hamiltonian vector field of h definedby

i(Xh)ω = dh.

ForH = {ht }t∈R, we integrate the time-dependent vector fieldXht to obtain the one-parameter family {ϕHt } of diffeomorphisms. We call such {ϕHt } a time-dependentHamiltonian flow. A diffeomorphism ϕ of M is called a Hamiltonian diffeomor-phism, when ϕ is the time-one map of {ϕHt } for some H . We may assume thatht+1 = ht . Denote by Ham(M,ω) resp. Symp(M,ω) the group of Hamiltonian dif-feomorphisms resp. the group of symplectomorphisms which are diffeomorphismspreserving ω. Clearly, Ham(M,ω) ⊂ Symp(M,ω). Hamiltonian diffeomorphismsare fundamental in symplectic geometry and enjoy some distinguished properties,e.g., existence of fixed points (see Arnold’s conjecture below), simplicity [1], exis-tence of a biinvariant distance on (the universal covering group of) Ham(M,ω), calledHofer’s distance, etc. Now we recall the following:

Conjecture 2.1 (Arnold’s conjecture). For ϕ ∈ Ham(M,ω) there are as many fixedpoints ofϕ as the smallest number of critical points of smooth functions onM , namely,

# Fix(ϕ) ≥ min{#Crit(f ) | f ∈ C∞(M)}.If all the fixed points of ϕ are non-degenerate, i.e. 1 is not an eigenvalue of dϕ at any

Development in symplectic Floer theory 1063

fixed point, then

# Fix(ϕ) ≥ min{#Crit(f ) | f is a Morse function on M}.This conjecture has been verified for closed oriented surfaces, the torus, complex

projective spaces, etc. A weaker version of the conjecture is formulated by replacingthe lower bounds with the cup-length and the sum of Betti numbers, respectively. Wecall it homological Arnold conjecture.

Let ϕ be a symplectomorphism of (M,ω) such that all fixed points are non-degenerate. Following [6], we introduce the twisted loop space

Pϕ = {σ : [0, 1] → M | ϕ(σ(1)) = σ(0)},and define a closed 1-form, in a formal sense, on Pϕ by

αϕ(ξ) =∫ 1

0ω(ξ, σ̇ ) dt for ξ ∈ TσPϕ.

Clearly, fixed points of ϕ are in one-to-one correspondence with zeros of αϕ . We takethe smallest covering space π : P̃ϕ → Pϕ such that (1) π∗αϕ is exact, i.e., there existsa primitive function Aϕ for αϕ , and (2) the integer valued Maslov index μ is welldefined on Crit(Aϕ) = π−1(Zero(αϕ)). From now on we call such a covering spacethe Floer covering space. Pick an almost complex structure J = {Jt } compatiblewith ω such that ϕ∗J1 = J0. Then the gradient of Aϕ is formally written as

grad Aϕ(σ ) = −J σ̇,and gradient flow lines are regarded as solutions of the following equation

∂u

∂τ+ Jt (u)

∂u

∂t= 0

for

u = u(τ, t) : R × [0, 1] → M such that ϕ(u(τ, 1)) = u(τ, 0).

We set

CF∗(ϕ, J )={ ∑

i

ai σ̃i | ai ∈ Q, σ̃i ∈ Crit(Aϕ) satisfy the following condition:

#{i | ai �= 0, Aϕ(γ̃i) < c} is finite for any c ∈ R}.

The grading is given by the Maslov index μ on Crit(Aϕ). The coboundary operatorδ = δϕ,J is defined by counting gradient flow lines connecting the critical pointsσ̃± of Aϕ such that μ(σ̃+) − μ(σ̃−) = 1. Note that the covering transformationgroupGM,ϕ of π : P̃ϕ → Pϕ naturally acts on the Floer complex. In fact, this action

1064 Kaoru Ono

extends to the so-called Novikov ring associated to ϕ ∈ Symp(M,ω), which is acertain completion of the group ring of GM,ϕ . To make this construction rigorous,we need to study compactness properties, transversality, etc. for the moduli space ofsolutions of the J -holomorphic curve equation above. We can achieve these points asin [16], [29], see also [28], [41], [45] based on the notion of stable maps [23], [24]. Theresulting cohomology is the Floer cohomology HF∗(ϕ, J ), which is a module over theNovikov ring associated to ϕ ∈ Symp(M,ω). We also find that Floer cohomology isinvariant under Hamiltonian deformations of ϕ. In the case that M is 2-dimensional,Seidel noticed that the Floer cohomology is invariant under a class of deformationswhich contains all Hamiltonian deformations [43].

When ϕ ∈ Symp0(M,ω), i.e., there is a path ϕt , 0 ≤ t ≤ 1, such that ϕ0 = id andϕ1 = ϕ, we can formulate the Floer theory on the loop space LM of M rather thanthe twisted loop space Pϕ . (From now on, we call ϕt with ϕ0 = id a based path.)Namely, we identify them by

σ(t) ∈ Pϕ → γ (t) = (ϕt )−1(σ (t)) ∈ LM.

In particular, whenϕ ∈ Ham(M,ω)we choose a based pathϕt in Ham(M,ω). DenotebyH the time-dependent Hamiltonian function which generates ϕt . Then fixed pointsof ϕ are in one-to-one correspondence with 1-periodic orbits of the time-dependentflow ϕt , which are characterized as zeros of the following closed 1-form αH on theloop space LM of M:

αH (ξ) =∫ 1

0ω(ξ(t), γ̇ (t)−XHt (γ (t))) dt,

where γ ∈ LM and ξ ∈ TγLM , i.e., a section of γ ∗TM . Write J ′t = (ϕt )∗Jt . (Note

that J ′0 = J ′

1. ) Then gradient flow lines are solutions of the following equation:

∂u

∂τ+ J ′(u)

(∂u

∂t−XHt (u)

)= 0,

for u = u(τ, t) : R × S1 → M . Denote by p : L̃M → LM the Floer covering spaceof LM and by AH : L̃M → R the action functional, i.e., dAH = p∗αH . Considerthe graded module generated by the critical points of AH with the grading given byμ,which is known as the Conley–Zehnder index in this setting. Then take its completionwith respect to the filtration {γ̃ ∈ Crit(AH ) | AH (γ̃ ) > c} for c ∈ R. We denote itby CF∗(H). The coboundary operator δ = δH,J is defined by counting the numberof connecting orbits joining critical points. In this case Floer cohomology can becomputed as follows:

HF∗(H, J ) ∼= H ∗+n(M; Q)⊗�ω,

where �ω is the Novikov ring of (M,ω).As a corollary we have the following result ([16], [29]).


Theorem 2.2. If ϕ ∈ Ham(M,ω) has only non-degenerate fixed points then

# Fix(ϕ) ≥∑k

rank Hk(M; Q).

More precisely, we can find that the number of fixed points which correspondto contractible 1-periodic orbits of any based Hamiltonian path is at least the sumof Betti numbers in this theorem. As a consequence, we also find that there alwaysexists a contractible 1-periodic orbit for any time-dependent periodic Hamiltoniansystem. For a based path {ψt } in Symp0(M,ω) we can define the Floer cohomology,which we may call the Floer–Novikov cohomology, in a way similar to the case ofHamiltonian diffeomorphisms. Under the ±-monotonicity assumption we have asimilar computation for ϕ ∈ Symp0(M,ω) using Novikov cohomology of the fluxof ϕt in place of the ordinary cohomology of M , see [27].

2.2. Application to the flux conjecture. The flux of a based path {φt }0≤t≤1 inSymp0(M,ω) is defined to be

F̃lux(φt ) =∫ 1

0[i(Xt )ω] dt ∈ H 1(M; R),

where Xt is the family of symplectic vector fields generating φt . The flux de-pends only on the homotopy class of paths with fixed end points φ0 = id and φ1,

and induces a homomorphism from the universal covering group S̃ymp0(M,ω) ofSymp0(M,ω) toH 1(M; R). Denote by �ω, which is called the flux group, the image

of Ker(

S̃ymp0(M;ω) → Symp0(M; R)) ∼= π1(Symp0(M; R)) under F̃lux. It is

known that the path φt above can be homotoped to a path in Ham(M;ω) keepingthe end points fixed if and only if F̃lux(φt ) = 0. F̃lux descends to a homomorphismFlux : Symp0(M;ω) → H 1(M; R)/�ω. The group Ham(M; R) is also known tobe the kernel of this homomorphism, see [1]. Hence, it is a basic question in order tounderstand Ham(M,ω) ⊂ Symp0(M,ω) how �ω is embedded in H 1(M; R).

Conjecture 2.3 (Flux conjecture). �ω is discrete in H 1(M; R).

This conjecture is equivalent to that Ham(M,ω) is C1-closed in Symp0(M,ω).There are various cases in which the flux conjecture is verified. For example, if[ω] ∈ H 1(M; Q) the conjecture clearly holds. A less trivial case is that (M,ω) isof Lefschetz type, i.e., ∧[ω]n−1 : H 1(M; R) → H 2n−1(M; R) is an isomorphism.It was Lalonde, McDuff and Polterovich [25], [26] who noticed that the affirmativeanswer to the homological Arnold conjecture can be used to prove the flux conjecture.Among other things, they proved the following:

Theorem 2.4. If c1(M) : π2(M) → Z is trivial or its minimal positive value (theminimal Chern number) is at least 2n = dimRM , then the flux conjecture holds.

1066 Kaoru Ono

Theorem 2.5. The rank of the flux group �ω is at most the first Betti number b1(M)

of M . In particular, the flux conjecture holds if b1(M) = 1.

Remark 2.6. Note that Theorem 2.5 follows from the flux conjecture.

We give an outline of the proof of the flux conjecture. First of all, we collect somenotation and fundamental properties of the Floer–Novikov cohomology. For any basedpath {ψt } in Symp0(M,ω), we can deform it by a homotopy so that i(Xt )ω does notdepend on t and is equal to θ = F̃lux(ψt ). HereXt is the family of symplectic vectorfields generatingψt . Denote byπ : M → M the covering space ofM associated to thehomomorphism Iθ : π1(M) → R obtained by integrating θ along loops. Then thereexists H̃ = {h̃t }, a smooth family of smooth functions on M such that π∗i(Xt )ω =dh̃t . Denote by L̃θM the Floer covering space of LM for {ψt } which dependsonly on its flux θ , and by Gω,θ its covering transformation group. Then we canperform the Floer construction for AH̃ : L̃θM → R and obtain the cochain complex

(CFN∗(H̃ , J ), δ = δH̃ ,J ). The groupGω,θ naturally acts on this complex. Moreover,the action extends to the Novikov completion�ω,θ of the group ring ofGω,θ . Denoteby HFN∗({ψt }) the resulting cohomology, which is the Floer–Novikov cohomologyof {ψt } and which is a finitely generated module over �ω,θ .

We collect its fundamental properties as follows.

Theorem 2.7. For based paths {ψ(1)t } and {ψ(2)t } with F̃lux({ψ(1)t }) = F̃lux({ψ(2)t })we have a natural isomorphism

HFN∗({ψ(1)t }) ∼= HFN∗({ψ(2)t }).

Theorem 2.8. If F̃lux({ψt }) is sufficiently small we have

HFN∗({ψt }) ∼= HN∗+n(θ)⊗�θ�ω,θ .

Here HN∗(θ) is the Novikov cohomology of θ and �θ is its coefficient ring.

Secondly, we note that the Floer construction can be performed with coefficientsin a local system as in the ordinary cohomology theory, see e.g., [38], [39]. Inparticular, when the flux vanishes, i.e. {ψt } is a Hamiltonian path, we obtain the Floercohomology for based Hamiltonian paths with coefficients in a local system. LetL → M be a local system or a flat vector bundle. We denote by HFN∗({ψt };L)the Floer–Novikov cohomology of {ψt } with coefficients in L. Then Theorems 2.7and 2.8 holds with coefficients in L. We state them for reference.

Theorem 2.9. LetL → M be a flat vector bundle. For based paths {ψ(1)t } and {ψ(2)t }with F̃lux({ψ(1)t }) = F̃lux({ψ(2)t }) we have a natural isomorphism

HFN∗({ψ(1)t };L) ∼= HFN∗({ψ(2)t };L).


Theorem 2.10. If F̃lux({ψt }) is sufficiently small we have

HFN∗({ψt };L) ∼= HN∗+n(θ;L)⊗�θ�ω,θ

for any flat vector bundle L. Here HN∗(θ;L) is the Novikov cohomology of θ withcoefficients in L.

Based on the above preparation, we give an outline of the proof of the flux con-jecture. Let U ⊂ H 1(M; R) be a neighborhood of the origin consisting of θ ∈ U ,which is represented by a sufficientlyC1-small closed 1-form such that Theorems 2.8and 2.10 holds for the flux [θ ]. We may assume that U is symmetric with respect tothe origin. It is enough to show the following.

Claim. �ω ∩ U = {0}.If it is false, there is a based loop {ψt } in Symp0(M,ω) such that θ = F̃lux({ψt })

belongs to (�ω ∩U) \ {0}. Denote by {ψ−θt } the based symplectic isotopy generated

by the vector fieldX−θ which is the symplectic dual of −θ . Then {ψ ′t = ψ−θ

t �ψt } isa based symplectic isotopy, the flux of which vanishes. Hence, we can deform {ψ ′

t }up to homotopy keeping end points fixed to a Hamiltonian path {φt }. Thus we obtaina based Hamiltonian path {φt } and a based symplectic path {ψ−θ

t } with ψ−θ1 = φ1.

Since ψθt = (ψ−θt )−1,�t = φt �ψθt is a based loop in Symp0(M,ω), which induces

an isomorphism � : γ (t) ∈ LM → �t(γ (t)) ∈ LM . It is clear that � restricts toone-to-one correspondence between 1-periodic orbits of {φ−θ

t } and 1-periodic orbitsof {φt }. Note that the former are constant loops at zeros of θ , since we assumed that θis sufficiently C1-small. On the other hand, Theorem 2.2 guarantees the existenceof contractible 1-periodic orbits of {φt } as we noted there. Hence, � preserves thecomponent of LM consisting of contractible loops. We have the following:

Lemma 2.11. �∗α{φt } = α{ψ−θt }.

As a consequence, we find that � : LM → LM admits a lift �̃ : L̃−θM →L̃0M . Note also that �t preserves the homotopy class of almost complex structurescompatible with ω, hence c1(M)(u) = c1(M)[�#(u)]. Here u : S1 × S1 → M and�#(u)(s, t) = �t(u(s, t)). Therefore �̃ induces an isomorphism between the Floer–Novikov cohomology of {ψ−θ

t } and the Floer cohomology of {φt }. (� also inducesan isomorphism between the moduli spaces of gradient trajectories in the sense ofKuranishi structures, after choosing almost compatible structures appropriately.) Wecan also see that �̃ induces an isomorphism between the Novikov rings �ω,−θ and�ω = �ω,0. Namely, we find

Proposition 2.12. Let L → M be an arbitrary flat vector bundle. Then there existsc ∈ Z such that

�̃∗ : HFN∗({ψ−θt };L) ∼= HF∗+c({φt };L).

1068 Kaoru Ono

Since −θ is sufficiently C1-small, Theorem 2.10 implies that

HFN∗({ψ−θt };L) ∼= HN∗+n(−θ;L)⊗

�−θ �ω,−θ .

On the other hand, we have

HF∗({φt };L) ∼= H ∗+n(M;L)⊗�ω.

Now we choose L → M as the flat real line bundle Lεθ associated to � ∈π1(M) → exp(ε

∫�∗θ) ∈ R∗. Then the pull back π∗Lεθ of Lεθ by π : M → M ,

which is used to define the Novikov cohomology of ±θ , becomes trivial as a flatbundle. Hence HN∗(−θ;Lεθ ) is isomorphic to HN∗(−θ; R) after forgetting themodule structure over the Novikov ring. On the other hand, for ordinary cohomol-ogy, we have the jumping phenomenon at ε = 0, i.e., since θ is not an exact 1-form,H 0(M;Lεθ ) = 0 for ε �= 0 while H 0(M; R) = R for ε = 0. Based on this observa-tion, we can derive a contradiction. Hence the flux conjecture is proved.

Theorem 2.13. The flux conjecture holds for any closed symplectic manifolds.

Remark 2.14. The action of Hamiltonian loops on Floer cohomologies was studiedby Seidel [44]. Viterbo [47] developed the theory of generating functions and exploredapplications to symplectic invariants. Y. G. Oh is the first to apply the Floer theoreticalframework to Hofer’s geometry [34], [35], partly inspired by the work of Chekanov [3]to be mentioned later. Seidel’s work also stimulated progress in the study of Hofer’sgeometry, e.g., Entov’s work [7] and Schwarz [42]. Oh generalized Schwarz’s resultto closed symplectic manifolds which are not necessarily symplectically aspherical,cf. Oh’s contribution to this proceedings. Based on this generalization, Entov andPolterovich constructed in [8] an R-valued quasi-homomorphism from (the universalcovering group of) Ham(M,ω).

There are different kinds of development from those mentioned in this section.For example, Viterbo applied the Floer cohomology to a problem in real algebraicgeometry and proved that hyperbolic manifolds cannot be realized as the real part of“sufficiently positively curved” complex projective manifolds; cf. [22].

3. Floer theory for Lagrangian submanifolds

3.1. Fundamental construction. Let L0, L1 be closed embedded Lagrangian sub-manifolds in a closed symplectic manifold (P,�). We assume that L0 and L1 in-tersect transversely. Consider the path space P (L0, L1) = {γ : [0, 1] → P |γ (0) ∈ L0, γ (1) ∈ L1} and define the action 1-form α = αL0,L1 by

αL0,L1(ξ) =∫ 1

0�(ξ(t), γ̇ (t)) dt for ξ = {ξ(t)} ∈ TγP (L0, L1).


Then αL0,L1 is a “closed 1-form”. In fact, a local primitive function around γ0 is givenby

AlocL0,L1

(γ ) =∫

[0,1]×[0,1]w∗�,

where w : [0, 1] × [0, 1] → P such that w(s, i) ∈ Li for i = 0, 1, w(0, t) = γ0(t)

and w(1, t) = γ (t). As long as the image of w is contained in a small neighborhoodof γ0, Aloc

L0,L1is well defined.

Before going further, we clarify the relation to the case of symplectomorphisms.Let φ be a symplectomorphism of (M,ω). Then its graph �φ is a Lagrangian sub-manifold in (M ×M,−ω⊕ω). Denote by� the diagonal subset, which is the graphof the identity. Then we have the following identification:

G : LφM → P (�φ,�); σ(t) →(σ

(1 − t

2

), σ

(t

2

)),

which satisfies G∗α�φ,� = αφ . In this way, the construction in this section is ageneralization of the one in the previous section.

Pick a compatible almost complex structure J to equip P (L0, L1)withL2-metric.Then the locally gradient flow line for αL0,L1 is described by u : R×[0, 1] → P withu(τ, i) ∈ Li for i = 0, 1, which satisfies

∂u

∂τ+ J (u)

∂u

∂t= 0.

Existence of the limits limτ→±∞ u(τ, t) ∈ L0 ∩L1 is equivalent to the condition thatthe energy E(u) is finite. Note also that the zeros of αL0,L1 are exactly the constantpaths at L0 ∩ L1.

In [9]–[13], Floer realized the idea of constructing an analogue of Morse complexfor the action functional under the assumption that π2(P, Li) = 0 and that L1 is aHamiltonian deformation of L0. In this situation the action functional admits a prim-itive function on P (L0, L1) and the grading of L0 ∩ L1, called the Maslov–Viterboindex μ = μL0,L1 , is well defined with values in Z. Define CF∗(L1, L0) by theZ/2Z-module freely generated by L0 ∩L1. Counting gradient flow lines connectingcritical points of AL0,L1 , we define the coboundary operator δ : CF∗(L1, L0) →CF∗+1(L1, L0) by

δ〈p〉 =∑

#M(p, q)〈q〉,where q runs over L0 ∩ L1 such that μ(q) = μ(p) + 1, and M(p, q) is the modulispace of gradient flow lines, which we call connecting orbits, of AL0,L1 from p to q.Under the above assumption, for a generic choice of J , the moduli space M(p, q) isshown to be compact if μ(q)−μ(p) = 1. If μ(q)−μ(p) = 2, M(p, q)may not becompact, but its end is described as the union of M(p, r)×M(r, q) over r ∈ L0 ∩L1

1070 Kaoru Ono

such that μ(r) − μ(p) = 1. Hence we find that δ � δ = 0 and obtain the Floercomplex (CF∗(L1, L0), δ). We denote by HF∗(L1, L0) the resulting cohomology. Itis also shown that the Floer cohomology is invariant under Hamiltonian deformationof Lagrangian submanifolds. If L1 is a sufficiently small Hamiltonian deformationof L0, L1 is regarded as the graph of a C2-small Morse function on L0 in T ∗L0. TheMorse gradient trajectories appear as Floer connecting orbits. Although, in general,there may exist non-small Floer connecting orbits, the assumption that π2(P, Li) = 0excludes such a possibility. Hence HF∗(L1, L0) is isomorphic to H ∗(L0; Z/2Z) upto a shift in the grading. It is worth mentioning that Hofer [19], [20] developed anidea similar to Floer’s and established the Lagrangian intersection property under theassumption that π2(P, L) = 0.

Without the assumption that π2(P, Li) = 0, there arise some problems in theabove argument. As we explain below, δ � δ may not vanish1, in general. It wasY. G. Oh [30], [31], [32] who extended Floer’s construction to the case that Li aremonotone and their minimal Maslov number is at least 3. (He also computed Floercohomology for some cases, e.g., RPn ⊂ CPn.) In general, the difficulties are causedby J -holomorphic discs with boundary onLi as well as J -holomorphic spheres whicharise as “bubbles” from sequences of connecting orbits with bounded energies. Asin the case of symplectomorphisms, the bubbling-off of J -holomorphic spheres isexpected to occur in real codimension 2 and does not cause any essential difficulty,which can be handled by Kuranishi structures. However, the bubbling-off of J -holo-morphic discs occurs in real codimension 1 and we cannot avoid it, in general. Ifwe restrict ourselves to some portion of P (L0, L1), on which the range of the actionfunctional is sufficiently narrow, then there do not appear effects from J -holomorphicdiscs and J -holomorphic spheres. In fact, Chekanov [3] gave an alternative proof forthe non-degeneracy of Hofer’s distance on Ham(M,ω) based on such an idea.

As we noticed, the bubbling-off of J -holomorphic discs is a codimension 1 phe-nomenon, hence we cannot, in general, avoid such a bubbling-off phenomenon fromthe moduli space M(p, q) even thoughμ(q)−μ(p) ≤ 2. In order to understand howδ � δ = 0 fails to hold, we study all J -holomorphic discs systematically. Fromnow on we follow our joint work with K. Fukaya, Y. G. Oh and H. Ohta [15].Firstly, we arrange elements of π2(P, L)

2, which are represented as the union of J -holomorphic discsw : (D2, ∂D2) → (P, L) and J -holomorphic spheres v : S2 → P

as β0 = 0, β1, β2, . . . such that∫βi� ≤ ∫

βi+1� and

∫βi� → +∞ as i → +∞.

This can be done with the so-called Gromov weak compactness. Denote by μ(w) theMaslov index of (w∗T P,w|∂D2)∗T L) → (D2, ∂D2). Denote by Mk+1(β) the mod-uli space of J -holomorphic discs3 which represent class β, with k+ 1 marked pointson ∂D2. Then the moduli space Mk+1(β) is of dimension n+ μ(β)+ k − 2, where

1I heard from A. Sergeev that Floer himself had (certainly) noticed this fact. This phenomenon is not only abad news. We used this fact in [37].

2More precisely, we work with the image of π2(P, L) in H 2(P, L; Z).3More precisely, we use the stable maps from the prestable Riemann surface with 1 boundary component of

genus 0.


n = dimL. In general, the transversality, i.e., the surjectivity of the linearization ofthe J -holomorphic curve equation, may not hold. In order to overcome this trouble,we use the framework of Kuranishi structure. Since we use the multi-valued perturba-tion technique, we need a compatible system of orientations on various moduli spaces.However, the moduli space of J -holomorphic discs may not be orientable4, in gen-eral. Therefore we assume the relative spin condition for Lagrangian submanifoldsas follows. Pick a triangulation of L and extend it to a triangulation of P .

Definition 3.1 (Relative spin structure). Let L be a Lagrangian submanifold. If thereis a cohomology class w ∈ H 2(P ; Z/2Z) such that w2(L) is the restriction of w toL, we callL relatively spin. Under this condition, there is an orientable vector bundleV on the 3-skeleton P (3) of P with w2(V ) = w. A relative spin structure for L isthe tuple of w, V and a spin structure on the restriction of T L ⊕ V to L ∩ P (2). Arelative spin structure on (L0, L1) is the above tuple which is chosen in common forLi , i = 0, 1.

Then we have the following:

Theorem 3.2. (1) A relative spin structure onL determines a canonical orientation onthe moduli spaces Mk+1(β), which satisfies a certain compatibility condition underthe gluing operation.

(2) A relative spin structure on (L0, L1) determines a canonical orientation on themoduli spaces M(p, q) of connecting orbits, which satisfies a certain compatibilitycondition under the gluing operation.

From now on, we assume that a Lagrangian submanifold L or a pair (L0, L1) ofLagrangian submanifolds are equipped with a relative spin structure. We work withQ-coefficients rather than Z/2Z-coefficients. Clearly, a spin structure on L gives arelative spin structure with a trivial bundle V .

We define obstruction classes forL to define Floer cohomology by inductive stepsas follows5. Start with β1, the first non-trivial case. Since the bubbling-off doesnot happen in M1(β1), the evaluation map ev0 : M1(β1) → L is a cycle with Q-coefficients. This cycle represents the first obstruction class6 o1 = o(β1). Supposethat oi = o(βi) is defined for i = 1, . . . , k and there exist Q-chains Bi in L suchthat oi = (−1)n∂Bi for i = 1, . . . , k. (We call such a system of Bi , i = 1, 2, . . .a bounding chain.) We define the next obstruction class ok+1 = o(βk+1) as follows.The moduli space Mk+1(β) may have codimension 1 boundary, hence may not be acycle. So we try to glue other (moduli) spaces along boundaries so that we finallyobtain a cycle. Consider the moduli space M�+1(β; Bi1, . . . ,Bi�) consisting of J -holomorphic discs w representing the class β such that βk+1 = β + ∑�

j=1 βij and

intersecting Bi1, . . . ,Bi� along ∂D2. The moduli space M�+1(β; Bi1, . . . ,Bi�) is

4Vin de Silva independently studied this problem in [5].5The idea of this construction was inspired by Kontsevich around 1997.6In the next subsection we adopt cohomological convention. Thus we take the Poincaré dual of ok .

1072 Kaoru Ono

described as the fiber product of the spaces with Kuranishi structures:

M�+1(β) ev1,...,ev� ×�∏

j=1

Bij .

We can assign to them an orientation so that the union M̂1(βk+1) of M1(βk+1) andall possible M�+1(β; Bi1, . . . ,Bi�) becomes a Q-virtual cycle. Note that we havethe evaluation map ev0 : M�+1(β;Pi1, . . . , Pi�) → L at the remaining marked pointafter taking the fiber product. Then ev0 : M̂1(βk+1) → L is a Q-cycle of L, whichrepresents the obstruction class ok+1 = o(βk+1). Then we can find the following:

Theorem 3.3. Suppose that a pair (L0, L1) of Lagrangian submanifolds is equippedwith a relative spin structure. If all the obstruction classes forLi , i = 0, 1 are definedand vanish, then we can revise the definition of Floer’s coboundary operator to obtainthe Floer complex (CF∗(L1, L0), δ). Moreover, the Floer cohomology HF∗(L1, L0)

is invariant under Hamiltonian deformation of Li .

Our construction depends on the choice of bounding chains for Li, i = 0, 1. Theinvariance under Hamiltonian deformations also requires a subtle argument. Namely,we must describe the relation of bounding chains under Hamiltonian deformation.These points are clarified in terms of the filteredA∞-algebras associated toLi , whichwe discuss in the next subsection. We may weaken the assumption that the obstructionclasses vanish. One of them is the deformation using Q-cycles in P . It may also hap-pen that the effects of J -holomorphic discs with boundary onLi , i = 0, 1 cancel eachother. When all non-vanishing obstruction classes for Li are of top dimension, i.e.,dimL, then they are multiples of the fundamental class ofL. We call the coefficient ofthe fundamental cycle as the potential function of Li . If the potential function of Li ,i = 0, 1, coincide, they cancel each other in the construction of the Floer complex,hence the Floer cohomology. This is an extension of Oh’s discovery that the Floercomplex can be constructed for monotone Lagrangian submanifolds with minimalMaslov numbers are at least 2. Although we can define the Floer complex, hence theFloer cohomology under the assumption that all obstruction classes vanish, it is verydifficult to compute it in general.

However, whenL1 is a Hamiltonian deformation ofL0, we can construct a certainspectral sequence with E2-term being the ordinary cohomology with coefficients inthe Novikov ring, which converges to the Floer cohomology, see Theorem 3.10 below.

3.2. The filtered A∞-algebras associated to Lagrangian submanifolds. Basedon [15], we describe the framework of the Floer theory for Lagrangian submanifolds.We generalize the idea of the construction of obstruction classes, which we men-tioned in the previous subsection, and construct the filtered A∞-algebras associatedto Lagrangian submanifolds. We also include some applications at the end of thissubsection.


We introduce the universal Novikov ring which we use from now on. Let R bea commutative ring with the unit. In this note, we mostly use the case that R = Q.Let T and e be formal generators of degree 0 and 2, respectively. Set

�nov ={ ∞∑i=0

aiTλi eμi | ai ∈ R, λi ∈ R, μi ∈ Z, lim

i→∞ λi = ∞}.

If R is a field, the degree 0-part of �nov is also a field. We also set

�0,nov ={ ∞∑i=0

aiTλi eμi ∈ �nov | λi ≥ 0

}.

These rings �nov and �0,nov are complete with respect to the decreasing filtrationby λ for T λ. The Novikov rings, we mentioned before, are subrings of �nov.

Now we shall present a rough idea of the construction of the A∞-operations. Let(fi : Pi → L), i = 1, . . . , k, be chains in L. We often abbreviate them as Pi . Takethe fiber product

Mk+1(β;P1, . . . , Pk) = Mk+1(β)ev1,...,evk ×f1,...,fk

k∏j=1

Pj .

We can give an orientation to these spaces with Kuranishi structure in such a way thatthe following construction works. Define a chain (Mk+1(β;P1, . . . , Pk), ev0) in Lby taking the remaining marked point, i.e., ev0 : Mk+1(β;P1, . . . , Pk) → L. Fork ≥ 2, we set

mk,β(P1, . . . , Pk) = (Mk+1(β;P1, . . . , Pk), ev0).

In the other cases, we set

m1,0(P ) = (−1)n∂P,

m1,β(P ) = (M2(β;P), ev0), when β �= 0

m0,β(1) = (M1(β), ev0), when β �= 0.

In the last line, 1 is the unit of R ⊂ �nov, which is regarded as an element inB0C(L;�0,nov)[1] below. We also set m0,0(1) = 0. If we study the structure ofcompactifications of the moduli spaces Mk+1(β;P1, . . . , Pk) in a heuristic way, weexpect to obtain certain algebraic relations among these operations, the so-calledA∞-relations. However, when we perform this construction in a rigorous way, weencounter several problems, e.g., transversality of the moduli spaces, transversalityfor taking the fiber product, etc. So we have to clarify which class of chains of L wedeal with and how to take the (multi-valued) perturbation for achieving transversality,etc. Here, we give some flavor of the argument. For details see [15].

1074 Kaoru Ono

First of all, we forget the effect of non-trivial J -holomorphic discs and consideronly the contribution from β = 0 (classical case). Naively, m2,0(P1, P2) shouldbe P1 ∩ P2 up to sign. However, when P1 = P2, the transversality does not hold.Thus we are forced to perturb M3,0(0;P1, P2) to define m2,0(P1, P2). (It is alsonecessary to take a suitable countable family of chains which spans a subcomplex ofthe chain complex of L. We also assume that its cohomology is isomorphic to theordinary cohomology of L.) It causes a discrepancy between m2,0(m2,0(P1, P2), P3)

and m2,0(P1,m2,0(P2, P3)). Namely, m2,0 does not satisfy the associativity. Never-theless, the above discrepancy is described using m3,0(P1, P2, P3), which is definedby the perturbation of M4,0(0;P1, P2, P3), as follows.

m2,0(m2,0(P1, P2), P3)− (−1)degP1m2,0(P1,m2,0(P2, P3))

= −{m1,0 � m3,0(P1, P2, P3)+ m3,0(m1,0(P1), P2, P3)

− (−1)degP1m3,0(P1,m1,0(P2), P3)

+ (−1)degP1+degP2(m3,0(P1, P2,m1,0(P3))}.Here we define the degree of P by degP = n − dim P and work with the coho-mological framework rather than the homological framework from now on. A seriesof similar formulae successively hold in higher order. We call these relations theA∞-relations. We can show that this algebraic gadget, the A∞-algebra, obtained bythe chain level intersection theory is “equivalent” to the de Rham homotopy theoryin the realm of A∞-algebras.

Next we include the effect from non-trivial J -holomorphic discs. Then we firsttake a suitable countably generated subcomplex C∗(L) of the (co)chain complex7

and (multi-valued) perturbations of the moduli spaces Mk+1(β;P1, . . . , Pk) to definemk,β(P1, . . . , Pk). We assign the degree to P ⊗T λeμ ∈ C∗(L;�nov) by degP +2μ.We shift the degree as C(L;�nov)[1]∗ = C∗+1(L;�nov). Then we can easily seethat

mk,β ⊗ T∫β �eμ(β)/2 :

k⊗C(L;�nov)[1]∗ → C(L;�nov)[1]∗

shifts the degree by +1, in other words, they are operations of degree +1. Write

mk =∑β

mk,β ⊗ T∫β eμ(β)/2.

Write

BC[1]∗ =∞⊕k=0

BkC[1]∗

and

BkC[1]∗ = BkC(L;�0,nov)[1]∗ =k⊗C(L;�0,nov)[1]∗,

7More precisely, we consider the quotient complex by identifying chains, which give the same current.


the bar construction of C∗ = C∗(L;�0,nov). It is a free tensor coalgebra generatedby the graded free module C[1]∗. We extend mk to m̂k : BC[1]∗ → BC[1]∗ as acoderivation and define d̂ = ∑

m̂k . Then we find the following:

Theorem 3.4. d̂ � d̂ = 0.

The filtered A∞-relations are the formulae which express the above equality interms of mk . We call (BC(L;�0,nov)[1], d̂) the filteredA∞-algebra associated to theLagrangian submanifoldL. So far, this object depends on the choice of the compatiblealmost complex structure, the countably generated subcomplexC(L), various (multi-valued) perturbations, etc. We can define the notion of (gapped filtered) A∞-algebramorphisms, homotopy equivalences, homotopy units, etc., and find the following:

Theorem 3.5. (1) The homotopy type of the filtered A∞-algebra

(BC(L;�0,nov)[1], d̂)depends only on the embedding of the Lagrangian submanifold L ⊂ (P,�). Thefundamental cycle of L is a homotopy unit.

(2) A symplectomorphismψ of (P,�) induces a homotopy equivalence ψ̂ betweenthe filtered A∞-algebras associated to L and ψ(L).

In fact, by the algebraic theory of the (filtered) A∞-algebras, we can derive theA∞-algebra structure, resp. the filtered A∞-algebra structure on H ∗(L), resp.H ∗(L;�0,nov). One of the advantages to work in the framework of (filtered) A∞-algebras is that quasi-isomorphisms have homotopy inverses8. This is not true in thecategory of differential graded algebras.

In general, m0(1) may not vanish. From the A∞-relation we have

m1 � m1(P ) = −(m2(m0(1), P )+ (−1)degP+1m2(P,m0(1))),

which means that m1 � m1 does not necessarily vanish. This is the obstruction todefine the Floer cohomology, which we discussed in the previous subsection.

Let b ∈ C(L;�0,nov)[1]0 with positive energy, i.e, b contains only terms with T λ

with λ > 0 and set

mbk(P1, . . . , Pk) =

∑mk+�(b, . . . , b, P1, b, . . . , b, Pi, b, . . . , b, Pk, b, . . . , b),

where � is the number of b’s appearing above in all possible ways and the sum is takenover all possibilities. We define d̂b using mb

k instead of mk . Then d̂b also satisfies theA∞-relation d̂b � d̂b = 0. Write eb = 1 + b + b ⊗ b + b ⊗ b ⊗ b + · · · . Then wefind the following:

Theorem 3.6. If there exists b ∈ C(L;�0,nov)[1]0 which satisfies d̂(eb) = 0, thenwe have mb

0(1) = 0, hence mb1 � mb

1 = 0.

8We do not claim any priority in the unfiltered case.

1076 Kaoru Ono

We call the equation d̂(eb) = 0 the Maurer–Cartan equation for the filteredA∞-algebra. If there is a solution b for the Maurer–Cartan equation, the complex(C(L;�0,nov),m

b1) and its extension (C(L;�nov),m

b1) are the Bott–Morse Floer

complex in the case that L = L0 = L1. We denote the resulting cohomologygroups by HF∗((L, b);�0,nov) and HF∗((L, b);�nov), respectively. For a boundingchain Bi in the previous subsection we set

b =∑

Bi ⊗ T∫β �eμ(β)/2.

Then b is a solution of the Maurer–Cartan equation. There is a notion of the gaugeequivalence relation among solutions of the Maurer–Cartan equation. We can seethat the Floer cohomologies are isomorphic for gauge equivalent b and b′. Notethat the filtered A∞-morphism maps a solution of the Maurer–Cartan equation forthe source to a solution of the Maurer–Cartan equation for the target. Hence, for asymplectomorphism ψ of (P,�), ψ∗(b), the B1-component of ψ̂(eb) is a solution ofthe Maurer–Cartan equation for ψ(L) if b is a solution for L. With respect to themwe have the following:

ψ̂ : HF∗((L, b);�0,nov) ∼= HF∗((ψ(L), ψ∗(b));�0,nov).

Now we consider a pair (L0, L1) of Lagrangian submanifolds. By counting Floerconnecting orbits intersecting k chains inL1 and � chains inL0, we define the operation

nk,� : BkC(L1;�0,nov)[1] ⊗ C(L1, L0;�0,nov)⊗ B�C(L0;�0,nov)[1]−→ C(L1, L0;�0,nov).

Using the filtered A∞-algebra structures on L1 and L0 as well as nk,�, we obtain thecoderivation d̂(L1,L0) on

BC(L1;�0,nov)[1] ⊗ C(L1, L0;�0,nov)⊗ BC(L0;�0,nov)[1].

We have d̂(L1,L0) � d̂(L1,L0) = 0. We call (BC(L1;�0,nov)[1]⊗C(L1, L0;�0,nov)⊗BC(L0;�0,nov)[1], d̂(L1,L0)) the filteredA∞-bimodule associated to the pair (L0, L1).More precisely, we say that it is a left C(L1;�0,nov), right C(L0;�0,nov) filteredA∞-bimodule. Similar to the case of the filtered A∞-algebras, we obtain the follow-ing:

Theorem 3.7. (1) For a pair (L0, L1) of Lagrangian submanifold equipped with arelative spin structure as a pair, the filtered A∞-bimodule is uniquely defined up tohomotopy equivalences.

(2) A pair of Hamiltonian diffeomorphisms φi , i = 0, 1, induces a homotopyequivalence between the filtered A∞-bimodules with coefficients in�nov of (L0, L1)

and (φ0(L0), φ1(L1)).


If there exists solutions bi of the Maurer–Cartan equations for Li , we can revisethe Floer coboundary operator as follows:

δb1,b0(p) =∑

nk,�(b1, . . . b1, p, b0, . . . , b0).

Then we have the following:

Theorem 3.8. Let bi be solutions of the Maurer–Cartan equations for Li , i = 0, 1.Then δb1,b0 � δb1,b0 = 0 holds.

We denote the resulting cohomology by HF∗((L1, b1), (L0, b0);�0,nov) and itscoefficient extension to �nov by HF∗((L1, b1), (L0, b0);�nov) Then we have thefollowing:

Corollary 3.9. Let bi be solutions of the Maurer–Cartan equation for Li and φiHamiltonian diffeomorphisms of (P .�). Then (φ1, φ0) induces an isomorphism

HF∗((L1.b1), (L0, b0);�nov) ∼= HF∗((φ1(L1), φ1∗(b1)), (φ0(L0), φ0∗(b0));�nov).

In a similar way to the case of filtered A∞-algebras, if bi is gauge equivalentto b′

i , i = 0, 1, the corresponding Floer cohomologies are isomorphic. Suppose thatm0(1) = c[L] for some c ∈ �0,nov. We set c = PO(L), the potential function.If PO(L0) = PO(L1) we can modify the above construction to obtain the Floercomplex. For example, if L0 is a Lagrangian submanifold such that m0(1) = c[L0]andL1 is a Hamiltonian deformation ofL0, then we can obtain the Floer cohomologyfor (L0, L1).

It is not easy to compute the Floer cohomology HF∗((L1, b1), (L0, b0)), evenwhen L1 = φ(L0) and b1 = φ∗(b0) for some Hamiltonian diffeomorphism φ.In such a case we find that it is isomorphic to the Bott–Morse Floer cohomologyHF∗((L0, b0);�nov). Using the energy filtration, we have a spectral sequence asfollows.

Theorem 3.10. There is a spectral sequence with E2-term being H ∗(L;�0,nov) andconverging to HF∗((L0, b0);�0,nov).

We can also use a cycle in the ambient space P to deform the filtered A∞-algebraassociated to L. Pick a cycle b in P . Consider the moduli space of stable maps withone boundary component. In addition to the k + 1 boundary marked points put �interior marked points. Take the fiber product

Mk+1,�(β;P1, . . . , Pk) = Mk+1,�(β)×∏k L×∏� P

( k∏i=1

Pi

)×

( �∏b).

Summing up these moduli spaces for all �, we obtain the deformed operation mbk . The

corresponding d̂b gives a deformation of the filtered A∞-algebra structure. We canalso discuss the Maurer–Cartan equation for the deformed structure, gauge equiva-lences, etc. Thanks to this larger class of deformations, we have the following:

1078 Kaoru Ono

Theorem 3.11. Let L be a relatively spin Lagrangian submanifold. If the embeddingL ⊂ P induces an injectionH ∗(L; Q) → H ∗(P ; Q), there is a�+

0,nov-cycle9 b of P

such that the deformed Maurer–Cartan solution d̂b(eb) = 0 has a solution.

The following theorem is a direct consequence.

Theorem 3.12. Let L be a relatively spin Lagrangian submanifold. Suppose that theembeddingL ⊂ P induces an injection on homology with rational coefficients. Then,for any Hamiltonian diffeomorphism φ of (P,�), we have

#L ∩ φ(L) ≥∑p

rank Hp(L; Q).

Note that the graph of a Hamiltonian diffeomorphism satisfies the above assump-tion, hence Theorem 3.12 is a generalization of Theorem 2.2. Although the completecomputation is difficult, there are cases where we have the non-vanishing result.

Theorem 3.13. Let L be a relatively spin Lagrangian submanifold. Suppose thatthere is a �+

0,nov-cycle b in P and b ∈ C(L;�0,nov)[1]0 such that d̂b(eb) = 0.Suppose also that the Maslov index of any J -holomorphic disc with boundary on Lis non-positive. Then, after adding correction terms which are of positive energy,the cycle [pt] and the cyle [L] become linearly independent, non-trivial cohomologyclasses in HF∗((L, b);�nov).

Here we denote by b the pair (b, b). When the Maslov class μ vanishes for L, allobstruction classes belong to H 2(L; Q). Hence we obtain the following:

Theorem 3.14. Let L be a relatively spin Lagrangian submanifold with vanishingMaslov class such thatH 2(L; Q) = 0. Then, for any Hamiltonian diffeomorphism φ,L ∩ φ(L) �= ∅. Moreover, there is p ∈ L ∩ φ(L) with Viterbo–Maslov index 0.

Thomas and Yau [46] used this theorem to establish the uniqueness of specialLagrangian homology spheres. From an opposite viewpoint, if L is a relatively spinLagrangian submanifold with vanishing second rational cohomology and admits aHamiltonian diffeomorphism φ such that L ∩ φ(L) = 0, then the Maslov class μLdoes not vanish. For instance, we have the following:

Theorem 3.15. Let L be a Lagrangian submanifold in the symplectic vector space(R2n, ωcan). If H 2(L; Q) = 0 then μL �= 0. Moreover, its minimal Maslov numberis at most n+ 1.

Some results in a similar spirit were also obtained by Biran and Cieliebak [2].Y. G. Oh obtained a more precise upper bound for the minimal Maslov number forLagrangian tori up to a certain dimension [33]. Once we know that there existsa Hamiltonian diffeomorphism φ of (P,�) such that L ∩ φ(L) = ∅, either some

9�+0,nov = { ∑

aiTλi eμi ∈ �nov|λi > 0

}.


obstruction class does not vanish, or some differential in the spectral sequence inTheorem 3.10 is non-trivial. In each case we obtain the existence of non-trivial J -holomorphic discs with boundary on L. Thus, for example, we can find that anyLagrangian submanifolds in symplectic vector spaces are not exact. Namely, therestriction of the Liouville form λ = ∑

pidqi to L is not an exact 1-form on L

(Gromov).Finally we discuss an analogue of the flux conjecture for Lagrangian submanifolds.

Denote by Lag(L) the space of all Lagrangian submanifolds which are Lagrangianisotopic to L with C1-topology. Consider the quotient Lag(L)/Ham(P,�) by theobvious action of Ham(P,�). The question is whether Lag(L)/Ham(P,�) is Haus-dorff or not. This is false in general. In fact, Chekanov’s example in [4] provides acounterexample. In his example the Maslov class is non-zero. As an application ofour theory [15] we find the following result which is an analogue to Theorem 2.4 (thecase that the Chern number is 0).

Theorem 3.16. Let L be a relatively spin Lagrangian submanifold L with vanishingMaslov class. Suppose that the (deformed) Maurer–Cartan equation for L has asolution. If L′ = φ(L), for some φ ∈ Ham(P,�), is sufficiently C1-close to L,then L′ is regarded as the graph of an exact 1-form on L via Weinstein’s tubularneighborhood theorem.

We expect that Lag(L)/Ham(P,�) is Hausdorff under the above assumption.Finally, we make a remark that if L is a so-called semi-positive Lagrangian sub-

manifold, we can work with Z/2Z-coefficients rather than Q-coefficients. We do notneed the relative spin condition in this case. There is also an approach to the Floer co-homology with Z-coefficients [17]. There are also applications in relation to “mirrorsymmetry” which we do not discuss here.

Acknowledgement. I would like to thank my collaborators, K. Fukaya, Y. G. Oh,H. Ohta in [15] and H. V. Le in [27].

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Department of Mathematics, Hokkaido University, Sapporo, 060-0810, JapanE-mail: [email protected]

Development in symplectic Floer theoryicm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_50.pdf · Development in symplectic Floer theory 1065 Theorem 2.2. If ϕ ∈ Ham(M,ω) has

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