Applications of higher-dimensional Heegaard Floer homology to contact topology
aa r X i v : . [ m a t h . S G ] J un APPLICATIONS OF HIGHER-DIMENSIONAL HEEGAARD FLOERHOMOLOGY TO CONTACT TOPOLOGY
VINCENT COLIN, KO HONDA, AND YIN TIANA
BSTRACT . The goal of this paper is to set up the general framework of higher-dimensional Heegaard Floer homology, define the contact class, and use it to givean obstruction to the Liouville fillability of a contact manifold and a sufficientcondition for the Weinstein conjecture to hold. We discuss several classes ofexamples including those coming from analyzing a close cousin of symplecticKhovanov homology and the analog of the Plamenevskaya invariant of trans-verse links. C ONTENTS
1. Introduction 12. Preliminaries 43. Definition of higher-dimensional Heegaard Floer (co-)homologygroups 104. A ∞ -operations 195. Reformulation of d HF ( W, β, φ ; h ; J ♦ ) NTRODUCTION
Let ( W, β, φ ) be a Weinstein domain of dimension n , where β is the Liouville -form and φ is the compatible Morse function, and let h ∈ Symp(
W, ∂W, dβ ) bea symplectomorphism on W that restricts to the identity on ∂W .The goal of this paper is to define the hat version d HF ( W, β, φ ; h ) of the higher-dimensional Heegaard Floer homology groups and give some applications. Date : June 11, 2020.2000
Mathematics Subject Classification.
Primary 57M50; Secondary 53D10,53D40.
Key words and phrases.
Contact structures, Khovanov homology, Higher-dimensional HeegaardFloer homology.VC supported by ERC Geodycon and ANR Quantact. KH supported by NSF Grants DMS-1406564 and DMS-1549147. YT supported by NSFC 11601256 and 11971256.
Higher-dimensional Heegaard Floer homology groups and the contactclass.
The situation can be summarized in the following diagram: f W completion −−−−−−−→ f W ∧ cylinder −−−−−→ b X = b X f W = R × [0 , × f W ∧ capping x W completion −−−−−−−→ c W cylinder −−−−−→ b X W = R × [0 , × c W .
Here f W is obtained by capping off the pairwise disjoint Lagrangian disks a , . . . , a κ that are the unstable submanifolds of the critical handles of W (i.e., a “Lagrangianbasis”) and f W ∧ and c W are completions of f W and W . Let e a , . . . , e a κ be La-grangian spheres obtained by capping off a , . . . , a κ . The top row can be viewedas working on the Heegaard surface and the bottom row can be viewed as workingon the page of an open book decomposition.The group d HF ( W, β, φ ; h ) is defined using b X , an auxiliary almost complexstructure J ♦ , and the collection { e a , . . . , e a κ } , in a manner analogous to Lipshitz’cylindrical reformulation of Heegaard Floer homology [Li]; this will be done inSection 3. In Section 5 we present an equivalent description in terms of b X W whichis analogous to the description of Heegaard Floer homology in terms of an openbook decomposition due to Honda-Kazez-Mati´c [HKM2] and uses { a , . . . , a κ } .The higher-dimensional Heegaard Floer homology groups satisfy the following: Theorem 1.1.1. d HF ( W, β, φ ; h ) is invariant under: (I1) trivial Weinstein homotopies; (I2) changes of almost complex structure J ♦ ; and (I3) isotopies of h in Symp(
W, ∂W, dβ ) . This is proven in Section 6.
Remark . The higher-dimensional Heegaard Floer groups are not expected tobe invariant under handleslides (I4), although they are in some cases (cf. Theo-rem 1.2.1 below).In Section 7 we define the p -twisted contact class c p ( W, β, φ ; h ) ∈ d HF ( W, β, φ ; h ) for p ∈ Z corresponding to the open book decomposition ( W, β, φ ; h ) . Althoughhandleslide invariance does not hold, the contact class can nevertheless shed lighton the corresponding contact structure ( M, ξ ) as follows: Theorem 1.1.3.
If the contact class c p ( W, β, φ ; h ) ∈ d HF ( W, β, φ ; h ) vanishes for an open book ( W, β, φ ; h ) supporting ( M, ξ ) , then (1) ( M, ξ ) is not Liouville fillable; and (2) ( M, ξ ) satisfies the Weinstein conjecture. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 3
In particular the theorem provides a convenient method for verifying that a con-tact structure is not
Liouville fillable. In Section 8.1 we define higher-dimensionalanalogs of right-veering surface diffeomorphisms (called strongly right-veering and weakly right-veering symplectomorphisms, where strongly right-veering im-plies weakly right-veering) and prove the following:
Theorem 1.1.4.
A contact structure ( M, ξ ) supported by an open book decompo-sition whose pages are Liouville and whose monodromy h is not strongly right-veering is not Liouville fillable and satisfies the Weinstein conjecture. Let ( W, β, φ ; h ) be an open book decomposition for ( M, ξ ) . It is unknownwhether ( M, ξ ) is overtwisted if h is not strongly right-veering. It is even notknown whether ( M, ξ ) is overtwisted whenever h is a product of negative symplec-tic Dehn twists. For example, it is currently not known how to use the techniquesof Casals-Murphy-Presas [CMP] or the overtwisted oranges of [HH] to prove suchcontact structures are overtwisted.1.2. Variant of symplectic Khovanov homology.
Another class of non-Liouvillefillable examples comes from analyzing a close cousin of symplectic Khovanovhomology and the analog of the Plamenevskaya invariant of transverse links [Pl].Let W = W κ − be the -dimensional Milnor fiber of the A κ − singularity andlet f W = W κ − be the Milnor fiber of the A κ − singularity obtained by cappingoff the κ Lagrangian thimbles a , . . . , a κ emanating from the κ critical points.(There are κ thimbles as opposed to a Lagrangian basis with κ − elements, buttechnically there is no difference.) Let h σ be the monodromy on W which descendsto a braid σ on the -disk D with κ singular points via the standard Lefschetzfibration π : W → D . Also let b σ be the braid closure of σ .The variant of symplectic Khovanov homology of interest is the F [ A ] J ~ , ~ − ] -module Kh ♯ ( b σ ) := d HF ( W ; h σ ) , where F is a field (e.g., F = Z / or Q ), F [ A ] is a group ring over the abelian group A described in Section 9.1, and we are using the κ thimbles a , . . . , a κ . Theorem 1.2.1. Kh ♯ ( b σ ) is a link invariant, i.e., is independent of the choice ofthimbles { a , . . . , a κ } and invariant under positive and negative Markov stabiliza-tions. We will briefly discuss the relationship between Kh ♯ ( b σ ) and Kh symp ( b σ ) in Sec-tion 9.6, where Kh symp ( b σ ) is the cylindrical formulation of symplectic Khovanovhomology which was shown by Mak-Smith [MS] to be the same as the originalSeidel-Smith definition [SS]. In a forthcoming paper, we will investigate a gener-alization of Kh ♯ ( b σ ) and Kh symp ( b σ ) to braids σ in surface bundles over S .Next we consider the contact class c ( W ; h σ ) ∈ Kh ♯ ( b σ ) = d HF ( W ; h σ ) . Theorem 1.2.2.
The contact class c ( W ; h σ ) is an invariant of σ up to positiveMarkov stabilizations. VINCENT COLIN, KO HONDA, AND YIN TIAN
The contact class is denoted ψ ♯ ( b σ ) by analogy with the Plamenevskaya invariant ψ ( b σ ) of a transverse link [Pl].There is a large literature devoted to the Plamenevskaya transverse link invariant.In particular, Plamenevskaya [Pl] showed that: Theorem 1.2.3 (Plamenevskaya [Pl]) . If the braid closure b σ , viewed as a trans-verse link in the standard contact R , admits a braid representative which is notright-veering, then ψ ( b σ ) = 0 . There are also many examples of braids σ satisfying the following:(B1) σ is right-veering;(B2) the invariant ψ ( b σ ) vanishes; and(B3) the open book corresponding to σ lifted to the branched double cover car-ries a tight contact structure.The first examples are due to Baldwin-Plamenevskaya [BP, Example 7.8] and thereare others e.g., due to Hubbard-Lee [HL].In this paper we analyze the Baldwin-Planenevskaya examples σ BP,ℓ , whichare -braids of the form σ − ℓ σ σ σ , where ℓ ≥ and σ i is a half twist aboutstrands i and i + 1 . Let h BP,ℓ := h σ BP,ℓ ∈ Symp(
W, ∂W ) be the correspondingsymplectomorphism. In Section 10 we prove the following: Theorem 1.2.4. ψ ♯ ( b σ BP,ℓ ) = 0 for ℓ ≥ . Hence the contact manifolds supportedby ( W ; h BP,ℓ ) are not Liouville fillable and satisfy the Weinstein conjecture. Note that by Acu [Ac], since W = W κ − is planar, all contact structures sup-ported by an open book decomposition with page W κ − satisfy the Weinstein con-jecture, so the latter result is not new. It should be noted that we have not deter-mined whether the contact manifolds supported by ( W ; h BP,ℓ ) are tight. Question . Is ( W ; h σ ) tight if σ satisfies (B1) and (B3)? Acknowledgments.
VC thanks Baptiste Chantraine and Paolo Ghiggini for theirinterest and support. KH is grateful to Yi Ni and the Caltech Mathematics De-partment for their hospitality during his sabbatical in 2018. KH also thanks JohnBaldwin and Otto van Koert for helpful correspondence and Tianyu Yuan for point-ing out some errors. 2. P
RELIMINARIES
Liouville and Weinstein cobordisms and homotopies.
In this subsectionwe review Liouville and Weinstein cobordisms and homotopies. The reader isreferred to [CE] for more details.
Definition 2.1.1. A Liouville cobordism ( W, β, Y ) is a triple consisting of: • a compact n -dimensional manifold W with boundary ∂W = ∂ + W ⊔ ∂ − W ; • a -form β on W such that dβ is symplectic; and IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 5 • a vector field Y which satisfies i Y dβ = β , is positively transverse to ∂ + W ,and is negatively transverse to ∂ − W .The -form β is called the Liouville -form and the vector field Y the Liouvillevector field . A
Liouville domain ( W, β, Y ) is a Liouville cobordism with ∂ − W = ∅ .We will sometimes omit Y or write ( W, dβ ) . Definition 2.1.2. (1) A
Weinstein cobordism ( W, β, Y, φ ) is quadruple consisting of Liouvillecobordism ( W, β, Y ) and a Morse function φ : W → R (called a compat-ible Morse function ) such that Y is gradient-like for φ and ∂W is a unionof regular level sets of φ .(2) A Weinstein domain ( W, β, Y, φ ) is a Weinstein cobordism with ∂ − W = ∅ .(3) If the critical points of φ : W → R are Morse or birth-death type, then ( W, β, Y, φ ) is a generalized Weinstein cobordism/domain , as appropriate. In this paper we assume that Weinstein domains are connected, unless statedotherwise.
Definition 2.1.3. A trivial Weinstein cobordism is ( W = [ a, b ] × Z n − , β = e σ β , Y = ∂ σ , φ ( σ, x ) = σ ) , where ( σ, x ) are coordinates on [ a, b ] × Z and β is a contact form on Z .Given a Weinstein domain ( W, β, Y, φ ) with connected boundary, there existsa collar neighborhood [ − ε + φ ( ∂W ) , φ ( ∂W )] × ∂W which is a trivial Weinsteincobordism, i.e., β = e σ β | ∂W , Y = ∂ σ , and φ ( σ, x ) = σ . Definition 2.1.4.
Let W be a compact, oriented n -dimensional manifold.(1) A Liouville homotopy on W is a -parameter family of Liouville domainstructures ( β t , Y t ) , t ∈ [0 , , on W .(2) A Weinstein homotopy on W is a -parameter family of generalized Wein-stein domain structures ( β t , Y t , φ t ) , t ∈ [0 , , on W . Definition 2.1.5. (1) The completion ( c W , b β ) of a Liouville domain ( W, β ) is obtained by gluing ([0 , ∞ ) × ∂W, e σ β | ∂W ) to ( W, β ) along ∂W . Here σ denotes the [0 , ∞ ) -coordinate on [0 , ∞ ) × ∂W .(2) The completion ( c W , b β, b φ ) of a Weinstein domain ( W, β, φ ) additionallyrequires that the extension b φ of φ be proper and have strictly positive de-rivative with respect to the σ -coordinate.Homotopies of Liouville or Weinstein domain can easily be extended to Liou-ville or Weinstein homotopies of their completions. Definition 2.1.6.
Let [( W, β, φ )] be the equivalence class of all Weinstein struc-tures ( W, β ′ , φ ′ ) that are Weinstein homotopic to a Weinstein domain ( W, β, φ ) . (In VINCENT COLIN, KO HONDA, AND YIN TIAN particular we are assuming that φ ′ is a Morse function.) The genus of [( W, β, φ )] ,denoted by g ( W, β, φ ) or g ([( W, β, φ )]) , is / of the minimum number of critical(= n -dimensional) handles of φ ′ over all ( W, β ′ , φ ′ ) ∈ [( W, β, φ )] . Definition 2.1.7. (1) A Weinstein structure ( W, β, Y, φ ) is generic if, for all pairs ( p, q ) of crit-ical points of φ , the ascending submanifold of p and the descending sub-manifold of q , both with respect to Y , intersect transversely.(2) A Weinstein homotopy ( W, β t , Y t , φ t ) , t ∈ [0 , , is trivial if φ t is Morseand ( W, β t , Y t , φ t ) is generic for all t ∈ [0 , .In particular, if ( W, β, φ ) is generic, then there are no trajectories of Y betweentwo critical points of index n .Let ( W, β, φ ) ∈ [( W, β, φ )] and g = g ([ W, β, φ ]) . If ( W, β, φ ) is generic, then φ has κ ≥ g critical handles and there is a collection { a , . . . , a κ } of properlyembedded, pairwise disjoint Lagrangian disks with Legendrian boundary on ∂W that represent the cocores of the κ critical handles and that are the n -dimensionalunstable manifolds of the Liouville vector field Y . We will refer to { a , . . . , a κ } as the basis of Lagrangian disks for ( W, β, φ ) . We also state the following useful lemma (cf. [CE, Proposition 11.8]), which isan easy consequence of the Moser technique:
Lemma 2.1.8.
Let ( W, β t ) , t ∈ [0 , , be a Liouville homotopy. (1) If β t = β , t ∈ [0 , , on a neighborhood of ∂W , then there exists a -parameter family of diffeomorphisms g t : W ∼ → W , t ∈ [0 , , with g = id and g t | ∂W = id such that g ∗ t β t − β is exact for all t ∈ [0 , . (2) If we do not assume that β t is fixed along ∂W , then there exists a -parameter family of diffeomorphisms b g t between the completions ( c W , b β ) and ( c W , b β t ) such that b g ∗ t b β t − b β is exact for all t ∈ [0 , and is equal tozero outside a compact set. Lemma 2.1.8(2) says that homotopic Liouville domains have isomorphic com-pletions.Finally, if ( W, β ) is a Liouville domain, then we denote the group of symplec-tomorphisms of ( W, dβ ) that restrict to the identity on ∂W by Symp(
W, ∂W, dβ ) .2.2. Capping of ( W, β, φ ; h ) . In this subsection we explain how to extend a We-instein domain ( W, β, φ ) to a Weinstein domain ( f W = H ∪ W, e β, e φ ) , called the capping of ( W, β, φ ) , such that ∂ − H = ∂ + W .Let N ′ ( ∂ + W ) be a collar neighborhood [ − ε, × ∂ + W ⊂ W of ∂ + W = { } × ∂ + W with coordinates ( σ, x ) such that φ : [ − ε, × ∂W → R is given by ( σ, x ) σ and β = e σ β | ∂W , i.e., the Weinstein cobordism is trivial.The manifold H , called a cap for ( W, β, φ ) , is obtained from the trivial Weinsteincobordism N ( ∂ + W ) := [0 , ε ] × ∂ + W by attaching κ critical handles H , . . . , H κ IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 7 along { ε } × ∂a , . . . , { ε } × ∂a κ . The pair ( e β | H , e φ | H ) is chosen so that the Y -descending submanifolds of the critical points of e φ | H have boundary ∂a , . . . , ∂a κ ,where Y is the Liouville vector field of e β . We write e a , . . . , e a κ ⊂ f W (calledthe capping of a , . . . , a κ ) for the pairwise disjoint embedded Lagrangian spheresobtained by capping off a , . . . , a κ and write e a = e a ∪ · · · ∪ e a κ .Let f : f W → R be a smooth function which satisfies the following:(1) f has small C -norm;(2) f is supported on a small neighborhood of e a which is identified with aneighborhood U of the -section of π : T ∗ e a → e a ; let U i be the componentof U which is a neighborhood of e a i ;(3) f = π ∗ f i on a smaller neighborhood V i ⊂ U i of e a i , where f i : e a i → R is a Morse function with two critical points: a maximum on int( a i ) and aminimum x i on e a i − a i ;(4) f = σ when restricted to V i ∩ ([ − ε/ , ε/ × ∂W ) .Let X f be the Hamiltonian vector field of f given by df = i X f d e β . We then modify h ∪ id | H ∈ Symp( f W , ∂ f W , d e β ) by composing with the Hamiltonian diffeomor-phism which is the time- flow along X f . The result of the composition is writtenas e h ∈ Symp( f W , ∂ f W , d e β ) . Note that (4) implies that e h ( a i ) ⊂ W and that e h ( ∂a i ) is a positive pushoff of ∂a i with respect to the Reeb vector field for β | ∂W . Alsoview the minimum x i of f i as an intersection point of e a i and e h ( e a i ) .See Figure 1 for the handle H i , the Lagrangian submanifolds e a i and e h ( e a i ) , andthe intersection point x i . a i e a i − a i e h ( e a i ) x i H i W F IGURE Remark . The Lagrangian submanifolds e a i and e h ( e a i ) are automatically exact,since they are spheres of dimension ≥ . Remark . We make a local calculation to clarify a potential sign confusion.Let q , . . . , q n , p , . . . , p n be local coordinates on f W near e a i = { p = · · · = p n = 0 } such that β = − P j p j dq j and let f i : e a i → R be given by f i ( q ) = ε P j q j near the minimum x i = { q = · · · = q n = 0 } ∈ e a i . Then df i = ε P j q j dq j .Since d e β = P j dq j dp j , we have X f i = − ε P i q i ∂ p i . This means that the graph VINCENT COLIN, KO HONDA, AND YIN TIAN of − df i (not df i ) is obtained from e a i by flowing in the direction of X f i . Hencethe point x i , viewed as an intersection point of the Lagrangian Floer cohomologygroup d CF ( e h ( e a i ) , e a i ) , is a “top generator” that we call a component of the contactclass ; see Section 7 below.2.3. Open book decompositions and contact structures.
In this subsection wereview the foundational work of Giroux [Gi1] and Giroux-Mohsen [GM, Mo] re-lating contact structures and open book decompositions.2.3.1.
Open book decompositions. An open book decomposition of a closed man-ifold M consists of a pair ( K, θ ) where: • K is a codimension two submanifold of M with trivial normal bundle; • θ : M \ K → S is a fibration equal to the normal angular coordinate in atrivialized neighborhood N ( K ) = K × D of K .The submanifold K is the binding of the open book decomposition. The compact-ification of any fiber of θ by K is called a page .An open book decomposition can equivalently be described by a pair ( W, h ) ,where W is a page ( ∂W = ∅ ) and h ∈ Diff(
W, ∂W ) , the set of diffeomorphismsof W that restrict to the identity on ∂W . The manifold M is then obtained as the relative mapping torus of ( W, h ) , i.e., M ≃ W × [0 , / ∼ h , where ∼ h identifies ( h ( x ) , ∼ h ( x, for all x ∈ W and ( y, t ) ∼ h ( y, t ′ ) for all y ∈ ∂W , t, t ′ ∈ [0 , .2.3.2. General case.
Given ( W, β ; h ) with h ∈ Symp(
W, ∂W, dβ ) , we constructa relative mapping torus M = M ( W,β ; h ) (which is slightly different from the oneabove) and a contact form α ( W,β ; h ) on M as follows: First construct the mappingtorus N ( W,β ; h ) = ([0 , × W ) / ∼ , where ( t, x ) are coordinates on [0 , × W and (1 , x ) ∼ (0 , h ( x )) for all x ∈ W .We then let M = M ( W,β ; h ) = ( ∂W × D ) ∪ N ( W,β ; h ) / ∼ , where ∼ identifies ∂W × S with ∂N ( W,β ; h ) via ( x, θ ) ∼ ( θ/ π, x ) . Here ( r, θ ) are polar coordinates on D and S = ∂D . By Lemma 2.1.8 there exists acontact form α ( W,β ; h ) = dt + β t on N ( W,β ; h ) such that β = β , β t are Liouville,and β t = β on ∂W . Moreover there exists an extension of the form α ( W,β ; h ) = β | ∂W + f ( r ) dθ to ∂W × D such that f ( r ) = r near r = 0 and f ′ ( r ) > for r > . Notation 2.3.1.
We write N ( W,β ; h ) for the mapping torus and M ( W,β ; h ) for therelative mapping torus.We now make the following slightly ad hoc definition: IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 9
Definition 2.3.2.
A contact manifold ( M, ξ ) admits or is supported by an openbook decomposition ( W, β ; h ) if ( M, ξ ) is contactomorphic to ( M ( W,β ; h ) , ker α ( W,β ; h ) ) .Then ∂W × { } is the binding and ( { t } × W ) ∪ ( ∂W × { r = t/ π } ) are thepages of the open book. Theorem 2.3.3 (Giroux [Gi1]) . Let M be closed oriented (2 n + 1) -dimensionalmanifold and ξ a cooriented contact structure on M . Then ( M, ξ ) admits an openbook decomposition ( W, β, φ ; h ) , where ( W, β, φ ) is a connected n -dimensionalWeinstein domain and h ∈ Symp(
W, ∂W, dβ ) . Moreover, ( W, β, φ ; h ) can betaken to be of Donaldson type . Let α be a contact form on M and J an almost complex structure on ξ = ker α which is dα | ξ -compatible. Let g be a Riemannian metric such that the Reeb vectorfield R α is of unit length and orthogonal to ξ and such that g | ξ is compatible with J and dα | ξ . Then ( M, α ) admits an open book decomposition of Donaldson type if there exist constants
C, η > and approximate holomorphic functions s k : M → C , k ≥ , k ∈ Z , such that: • for all p ∈ M , | s k ( p ) | ≤ C , | ds k − iks k α | ≤ C √ k , and | ∂ ξ s k | ≤ C ; • if | s k ( p ) | ≤ η , then | ∂ ξ s k | ≥ η √ k ;for k sufficiently large. Here ∂ ξ s k and ∂ ξ s k are the J -linear and J -antilinearcomponents of ds k | ξ . The binding is s − k (0) and the pages are (arg ◦ s k ) − ( θ ) .Let ( W, β, φ ) be a generic Weinstein manifold, h ∈ Symp(
W, ∂W, dβ ) , a = a ∪ · · · ∪ a κ be the basis of Lagrangian disks, and c be a properly embeddedLagrangian disk in W whose Legendrian boundary is disjoint from ∂ a . We thendefine the positive stabilization S + c ( W, β, φ ; h ) := ( W ′ , β ′ , φ ′ ; h ′ ) of ( W, β, φ ; h ) along c as follows: Let ( W ′ , β ′ , φ ′ ) be the Weinstein domain ob-tained from ( W, β, φ ) by attaching a symplectic n -handle W to W along the Leg-endrian sphere ∂c . Let γ be the Lagrangian sphere obtained by gluing c to theLagrangian core of W and let τ γ be the positive symplectic Dehn twist along γ .We then set(2.3.1) h ′ := ( h ∪ id | W ) ◦ τ γ . The negative stabilization S − c ( W, β, φ ; h ) is defined similarly, where τ γ is replacedby τ − γ in the definition of h ′ .Let a be the cocore of W . Then we write a ′ = a ∪ a and α ′ = α ∪ α , where α is obtained by capping off a . Theorem 2.3.4 (Giroux-Mohsen [GM]) . Any two Donaldson open book decom-positions ( W i , β i , φ i ; h i ) , i = 0 , , of ( M, ξ ) can be taken to one another by asequence of positive stabilizations, conjugations, and Weinstein homotopies. One problem with this definition is that we are not specifying how large k needs to be. We want k ≫ such that Theorem 2.3.4 holds for any Donaldson open books corresponding to k and k ′ ≥ k . Dimension three.
When dim M = 3 , the binding K is a link in M , thepage W is a (symplectic) surface with boundary, and the Liouville form β plays noessential role.In [Gi1], Giroux showed that Theorem 2.3.4 can be improved to the following: Theorem 2.3.5.
Any two open book decompositions ( W ; h ) and ( W ; h ) of ( M, ξ ) can be taken to one another by a sequence of positive stabilizations, conju-gations, and homotopies.Remark . It is still unknown whether in higher dimensions any two open bookdecompositions ( W i , β i , φ i ; h i ) , i = 0 , , of ( M, ξ ) can be taken to one another bya sequence of positive stabilizations, conjugations, and homotopies. It is plausiblein light of the recent advances in convex hypersurface theory [HH] that this is trueif ( W i , β i ) are Weinstein.If we extend the equivalence classes of open book decomposition with negativestabilizations, they only remember a part of the homotopy class of the supportedcontact structure as plane field: the associated spin c -structure. Theorem 2.3.7 ([GG]) . The equivalence classes of open book decomposition in M modulo positive and negative stabilizations, conjugations, and isotopies are inone-to-one correspondence with spin c -structures on M . By the Alexander trick, given an open book decomposition ( K, θ ) of M , anylink L in M can be isotoped to a link which is disjoint from the binding and trans-verse to the fibers of θ . We say that such a link is in braid position with respect to ( K, θ ) . 3. D EFINITION OF HIGHER - DIMENSIONAL H EEGAARD F LOER ( CO -) HOMOLOGY GROUPS
Let ( W, β, φ ) be a generic Weinstein domain, h ∈ Symp(
W, ∂W, dβ ) , and ( f W , e β, e φ ) be a capping of ( W, β, φ ) . Also let a = ⊔ κi =1 a i be the basis of La-grangian disks for ( W, β, φ ) .The goal of this section is to define the higher-dimensional Heegaard Floer hat(co-)homology groups d HF ( W, β, φ ; h ; J ♦ ) , where J ♦ is an auxiliary almost com-plex structure on b X = R × [0 , × f W ∧ and f W ∧ is the completion of f W . This willbe done in a manner analogous to the combination of works of Lipshitz [Li] andHonda-Kazez-Mati´c [HKM2] as explained in [CGH1].3.1. Symplectic fibration.
In Sections 3.1–3.8 as well as in Section 4 we work inthe following slightly more general setting: Let (Σ , β Σ ) be a Liouville domain and ( b Σ , b β Σ ) its completion. Consider the fibration π = π R × [0 , : b X = R × [0 , × b Σ → R × [0 , , given by the projection onto the first two factors. Also let π b Σ be the projection to b Σ and π [0 , × b Σ be the projection to [0 , × b Σ . IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 11
Let ( s, t ) be coordinates for R × [0 , . Then π is a symplectic fibration withrespect to the symplectic form b Ω = ds ∧ dt + d b β Σ on b X and the symplectic form ds ∧ dt on R × [0 , . Let α = ⊔ κi =1 α i , α ′ = ⊔ κi =1 α ′ i be κ -component exact Lagrangian submanifolds of Σ . Then the submanifolds L i = R × { } × α i , L i = R × { } × α ′ i are Lagrangian submanifolds of ( b X, b Ω) . We also write L = L , α = ⊔ i L i , L = L , α ′ = ⊔ i L i . Almost complex structures.
Consider the symplectization end b Σ − Σ onwhich b β Σ = e σ β , where σ is the (0 , ∞ ) -coordinate and β := β Σ | ∂ Σ is a contactform on ∂ Σ . Let ξ = ker β and let R = R β be the Reeb vector field for β .Let J b Σ be an almost complex structure on b Σ which is tamed by d b β Σ and is adapted to β on the symplectization end b Σ − Σ , i.e., • J b Σ ( ∂ σ ) = R , J b Σ ( R ) = − ∂ σ , and J b Σ ( ξ ) = ξ . • dβ ( v, J b Σ ( v )) > for all nonzero v ∈ ξ .The space of such almost complex structures J b Σ will be denoted by J b Σ = J b Σ , b β Σ .Let J = J R × [0 , × J b Σ be a product almost complex structure on b X where J R × [0 , ( ∂ s ) = ∂ t and J b Σ ∈ J b Σ . Let J ♦ be a C ∞ -small perturbation of J whichsatisfies the following:(J1) J ♦ is s -invariant;(J2) J = J ♦ on a neighborhood of R × { , } × b Σ and on R × [0 , × ( b Σ − Σ) ;and(J3) b Ω( v, J ♦ v ) > for all nonzero tangent vectors v , i.e., b Ω is J -positive.We will refer to J as a ( b Σ , b β Σ ) -compatible almost complex structure on b X andto J ♦ as a perturbed ( b Σ , b β Σ ) -compatible almost complex structure on b X . Thespace of ( b Σ , b β Σ ) -compatible almost complex structures on b X of class C ∞ willbe denoted by J and the space of perturbed ( b Σ , b β Σ ) -compatible almost complexstructures on b X of class C ∞ by J ♦ .3.3. Moduli spaces.Definition 3.3.1. A κ -tuple of intersection points of α and α ′ is a κ -tuple y = { y , . . . , y κ } , where y i ∈ α i ∩ α ′ σ ( i ) for some permutation σ of { , . . . , κ } . Wedenote the set of κ -tuples of intersection points of α and α ′ by S = S α , α ′ .Let M J ♦ ( y , y ′ ) be the moduli space of holomorphic maps u : ( ˙ F , j ) → ( b X, J ♦ ) , where we range over all ( ˙ F , j ) such that ( F, j ) is a compact Riemann surfacewith boundary, p + and p − are disjoint sets of boundary punctures of F , and ˙ F = F − p + − p − , and: (1) each component of ∂F − p + − p − maps to some L i or L i and each L i and L i , i = 1 , . . . , κ , is used exactly once;(2) u maps the neighborhoods of the punctures of p + (resp. p − ) asymptoti-cally to strips over the Reeb chords of y (resp. y ′ ) at the positive end (resp.negative end).Here we refer to the s → + ∞ (resp. s → −∞ ) end as the positive end (resp. negative end ). An element u ∈ M J ♦ ( y , y ′ ) will be referred to as a curve in b X ora multisection of π : b X → R × [0 , from y to y ′ . Remark . By the definition of J ♦ , the function σ ◦ π b Σ ◦ u is subharmonic forall u ∈ M J ♦ ( y , y ′ ) , where σ is the (0 , ∞ ) -coordinate for the end b Σ − Σ . Thismeans that Im( u ) ⊂ X := R × [0 , × Σ .3.4. Fredholm index.
We now compute the Fredholm index of u : ˙ F → b X in M J ♦ ( y , y ′ ) . What we call the Fredholm index in this paper takes into considera-tion the variations of complex structures on the domain ˙ F . The setup is similar tothat of [CGH1, Section 4.4.2].Let ˇ X = [ − , × [0 , × Σ be the compactification of X = R × [0 , × Σ ,obtained by attaching [0 , × Σ at the positive and negative ends, and let ˇ L = [ − , × { } × α , ˇ L = [ − , × { } × α ′ be the compactifications of L and L . We then define Z α , α ′ = ˇ L ∪ ˇ L ∪ ( {− , } × [0 , × ( α ∩ α ′ )) . A holomorphic map u from y to y ′ can be compactified to a continuous map ˇ u : ( ˇ F , ∂ ˇ F ) → ( ˇ X, Z α , α ′ ) , where ˇ F is obtained from ˙ F by performing a real blow-up at its boundary punc-tures.Given u from y to y ′ , we define its Maslov index µ ( u ) as follows: We constructa (not necessarily oriented) real rank n subbundle L of ˇ u ∗ T Σ on ∂ ˇ F . The bundle L is given by ˇ u ∗ T α and ˇ u ∗ T α ′ along ∂ ˙ F and we extend L to ∂ ˇ F − ∂ ˙ F by rotatingfrom T y α ′ to T y α via e J b Σ t , t ∈ [0 , π ] , where y ∈ y ∪ y ′ , each component of ∂ ˇ F − ∂ ˙ F is given an oriented parametrization by [0 , π ] , and we assume withoutloss of generality that J b Σ ( T y α ′ ) = T y α . Then µ ( u ) is the Maslov index of L withrespect to any trivialization of ˇ u ∗ T Σ on ˇ F .The following is a generalization of [CGH1, Equation (4.4.4)] and its proof willbe omitted. We often write χ ( u ) = χ ( F ) . Lemma 3.4.1.
The Fredholm index of u is (3.4.1) ind( u ) = ( n − χ ( u ) + µ ( u ) + (2 − n ) κ, where dim( W ) = 2 n and dim( b X ) = 2 n + 2 . Strictly speaking, u is guaranteed to be a multisection of π : b X → R × [0 , only when J ♦ = J . IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 13
Splittings of generators and moduli spaces.
In this subsection we discussthe splittings of generators and moduli spaces for α , α ′ in general.3.5.1. Generators.
Let(3.5.1) h α , α ′ ⊂ H ([0 , × Σ , ( { } × α ) ∪ ( { } × α ′ ); Z ) be the subset of classes [ δ ] which admit representatives δ such that ∂ δ = P κi =1 δ i − P κi =1 δ i , where δ i is a point on { } × α i and δ i is a point on { } × α ′ i . Let H α , α ′ : S α , α ′ → h α , α ′ be the map which sends y = { y , . . . , y κ } to the homology class [[0 , × y ] . Then H α , α ′ gives a splitting S α , α ′ = ` [ δ ] ∈ h α , α ′ S α , α ′ , [ δ ] . Note that M J ♦ ( y , y ′ ) = ∅ only if H α , α ′ ( y ) = H α , α ′ ( y ′ ) .We will suppress “ α , α ′ ” from the notation when it is understood.3.5.2. Moduli spaces.
Suppose y , y ′ ∈ S α , α ′ , [ δ ] for some [ δ ] . Let A y , y ′ α , α ′ ⊂ H ([0 , × Σ , ( { } × α ) ∪ ( { } × α ′ ) ∪ ([0 , × y ) ∪ ([0 , × y ′ ); Z ) be the subspace spanned by classes [ T ] which admit representatives T such that(3.5.2) ∂T = P κi =1 ([0 , × y i ) − P κi =1 ([0 , × y ′ i ) − P κi =1 w i + P κi =1 w i , where w i is an arc in { } × α i from y i to y ′ i and w i is an arc in { } × α ′ i . A class [ T ] satisfying Equation (3.5.2) is said to be from y to y ′ .There is a splitting M J ♦ ( y , y ′ ) = ` B ∈A y , y ′ α , α ′ M BJ ♦ ( y , y ′ ) , where the superscript B in M BJ ♦ ( y , y ′ ) is the modifier “ u ∈ M J ♦ ( y , y ′ ) is inthe class B ∈ A y , y ′ α , α ′ .” More generally, if ∗ is a modifier, we write M ∗ J ♦ ( y , y ′ ) toindicate the subset of M J ♦ ( y , y ′ ) satisfying property ∗ . Similarly, if χ is the modifier “ χ ( u ) = χ ”, then there is a further decomposition M BJ ♦ ( y , y ′ ) = ` χ ∈ Z ,χ ≤ k M B,χJ ♦ ( y , y ′ ) . Since π ◦ u is a κ -fold branched cover of R × [0 , when J = J ♦ , it follows that χ ( u ) ≤ κ for J ♦ arbitrarily close to J by Gromov compactness. Complete set of capping surfaces.
Definition 3.5.1. A complete set of capping surfaces consists of the following data: • For each [ δ ] ∈ h α , α ′ , a representative δ = ⊔ κi =1 δ i consisting of κ disjointembedded arcs. • For each pair ([ δ ] , y ) consisting of [ δ ] ∈ h α , α ′ and y ∈ S α , α ′ , [ δ ] , a rep-resentative T δ , y satisfying Equation (3.5.2) with [0 , × y i and [0 , × y ′ i replaced by δ i and [0 , × y i .We will often denote a complete set of capping surfaces by { T δ , y } .Any two classes in A y , y ′ α , α ′ from y to y ′ differ by an element of(3.5.3) A α , α ′ := H ([0 , × Σ , ( { } × α ) ∪ ( { } × α ′ ); Z ) . If we choose a complete set { T δ , y } of capping surfaces, then for each pair y , y ′ ∈S α , α ′ , [ δ ] there is a unique class A ∈ A y , y ′ α , α ′ given by T δ , y + A = T δ , y ′ . We thenwrite M BJ ♦ ( y , y ′ ) instead as M B − AJ ♦ ( y , y ′ ) so that B − A ∈ A α , α ′ .3.6. Regularity.Lemma 3.6.1.
Suppose J ∈ J . (1) If u ∈ M A,χJ ( y , y ′ ) , then no irreducible component of u lies on a fiber of π . (2) Every irreducible component of u is simply-covered.Proof. (1) Arguing by contradiction, suppose e u is a component of u such that π ◦ e u maps to a point x ∈ R × [0 , . If x is an interior point, then π b Σ ◦ e u is a closed curve,which is not possible since ( b Σ , b β ) is an exact symplectic manifold. Similarly, if x is a boundary point, then π b Σ ◦ e u is compact with Lagrangian boundary on either α or α ′ , which contradicts the exactness of α and α ′ .(2) By (1), for every component e u of u , π ◦ e u is a deg ≥ branched cover of R × [0 , . The positive ends are asymptotic to a subset of y and the negative endsare asymptotic to a subset of y ′ . Since each element of y and y ′ is used exactlyonce, (2) follows. (cid:3) Lemma 3.6.2.
The moduli space M A,χJ ♦ ( y , y ′ ) is transversely cut out if J ♦ ∈ J ♦ is generic and sufficiently close to some J ∈ J .Proof. By Lemma 3.6.1 and Gromov compactness, u ∈ M A,χJ ♦ ( y , y ′ ) is simply-covered, provided J ♦ is sufficiently close to J . This implies that π [0 , × b Σ ◦ u is somewhere injective. We then apply the usual argument; see for example [Li,Proposition 3.8]. (cid:3) Let ( J ♦ ) reg ⊂ J ♦ be the dense subset of regular almost complex structures,i.e., almost complex structures J ♦ for which the moduli spaces M A,χJ ♦ ( y , y ′ ) aretransversely cut out for all y , y ′ , A , and χ . IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 15
Orientations.
We briefly discuss how to give a coherent (= compatible withgluing) system of orientations for M J ( y , y ′ ) , following [FOOO]. The key point isto pick (stable) trivializations of T L and T L and to extend them over the chords [0 , × y and [0 , × y ′ .The orientation data can be decoupled into the base direction R × [0 , and thefiber direction b Σ .We first discuss the fiber direction and orient the curves π b Σ ◦ u . The orientationdata for ( b Σ , α , α ′ ) consists of:(1) orientations on α and α ′ ;(2) a relative spin structure for the pair ( α , α ′ ) ;(3) and for each intersection point p ∈ α ∩ α ′ a capping Lagrangian path , a capping orientation , and a stable capping trivialization .Since our Lagrangians are (unions of) spheres, a relative spin structure is givenby: • a trivial R -bundle over b Σ such that ( T α ⊕ R ) | α and ( T α ′ ⊕ R ) | α ′ are trivial;and • trivializations t and t ′ for ( T α ⊕ R ) | α and ( T α ′ ⊕ R ) | α ′ .A capping Lagrangian path is a path {L p,t } ≤ t ≤ in the oriented Lagrangian Grass-mannian Lag( T p b Σ , dβ Σ ( p )) such that L p, = T p α ′ and L p, = T p α as orientedvector spaces. A stable capping trivialization is a trivialization e t p of {L p,t ⊕ R } ≤ t ≤ that agrees with t p and t ′ p that we have already chosen.For each p ∈ α ∩ α ′ , let π p : H → b Σ be the constant map to p , where H = { z | Im z ≥ } is the upper half plane, and let ξ = π ∗ p ( T p M ) be a triv-ial(ized) vector bundle over H . We define a Cauchy-Riemann tuple ( ξ, η p + , D p + ) as follows: • the real subbundle η p + ⊂ ξ is given by η p + z = L p, for z ∈ ( −∞ , , η p + z = L p,z for z ∈ [0 , , and η p + z = L p, for z ∈ (1 , + ∞ ) ; and • D p + is a fixed real linear Cauchy-Riemann operator W k +1 ,p ( H , ξ ) → W k,p ( H , ∧ , H ⊗ C ξ ) with boundary condition on η p + ; see [BH, Section3.2] for a more complete discussion.Then a capping orientation is a choice of orientation o ( D p + ) . We can similarlychoose the Cauchy-Riemann tuple ( ξ, η p − , D p − ) by swapping the roles of α and α ′ .The R × [0 , -direction is easy: The Lagrangians are ℓ i = R × { i } , i = 0 , , the“capping Lagrangian paths” at the end s = ±∞ we take to be limits of R h ∂ s i , andthe trivializations are given by ∂ s .3.8. Definition of d CF (Σ , α ′ , α ; J ♦ ) . In this subsection we define the cochaingroups d CF (Σ , α ′ , α ; J ♦ ) , where J ♦ ∈ ( J ♦ ) reg . We assume that n > . Remark . The astute reader might have noticed that we have switched theorders of α and α ′ (i.e., we switched to cohomology). The order was switched forconsistency with the A ∞ -conventions from [Se3]. Coefficient ring.
We first describe the coefficient ring Λ J ℏ K . Since we have alreadychosen auxiliary orientation data in Section 3.7, the ground ring is Z . The “Planckconstant” ℏ is a variable, Λ J ℏ K is the formal power series ring in ℏ over Λ , and Λ := Λ α , α ′ is the Novikov ring with Z -coefficients over A ′ α , α ′ := A α , α ′ / torsion with respect to the symplectic form dβ . More explicitly, elements λ ∈ Λ are formalsums P A ∈A ′ α , α ′ λ A e A , where λ A ∈ Z and for each C > there are only finitely many A ∈ A ′ α , α ′ with dβ (( π b Σ ) ∗ ( A )) ≤ C . Cochain group.
The cochain group d CF (Σ , α ′ , α ; J ♦ ) is the free Λ J ℏ K -modulegenerated by S = S α , α ′ . The differential is given by:(3.8.1) d y = P y ′ ∈S ,χ ≤ κ,A ∈A ′ α , α ′ h d y , ℏ κ − χ e A y ′ i · ℏ κ − χ e A y ′ , where h d y , ℏ κ − χ e A y ′ i is the count of M ind=1 ,A,χJ ♦ ( y , y ′ ) / R . Remark . When n = 2 , we additionally impose an embeddedness conditionon the holomorphic curves (or, alternatively, count ECH index I = 1 curves). This,together with the adjunction formula, gives restrictions on χ . On the other hand,when n ≥ , the embeddedness condition is generically satisfied for ind = 1 curves. Also when n = 3 , the Fredholm index formula (Equation (3.4.1)) gives norestrictions on χ . These differences account for the slightly different form of thedifferential for n > . Grading.
In view of Equation (3.4.1) and the fact that the differential is degree-increasing, we set | ~ | = n − . Also | e A | = − c ( A ) . The elements in S are relatively graded such that if u ∈ M ind= ℓ,A,χJ ♦ ( y , y ′ ) , then ( | y ′ | + ( n − κ − χ ) − c ( A )) − | y | = ℓ. Splitting.
The splitting S = ` [ δ ] ∈ h α , α ′ S [ δ ] gives rise to the splitting d CF (Σ , α ′ , α ; J ♦ ) = ⊕ [ δ ] ∈ h α , α ′ d CF (Σ , α ′ , α ; J ♦ ; [ δ ]) . It remains to show that M ind=1 ,A,χJ ♦ ( y , y ′ ) / R is compact and that d = 0 . Lemma 3.8.3. d = 0 .Proof. Let M ′ := M ind=2 ,A,χJ ♦ ( y , y ′ ) / R . We claim that ∂ M ′ = ` ( M ind=1 ,A ,χ J ♦ ( y , y ′′ ) / R ) × ( M ind=1 ,A ,χ J ♦ ( y ′′ , y ′ ) / R ) , where the union is over all y ′′ ∈ S , χ + χ − κ = χ , and A , A ∈ A ′ α , α ′ suchthat A + A = A . Here ∂ M ′ is constructed using the usual SFT compactnesstheorem [BEHWZ]. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 17
First suppose that J = J ♦ and J ∈ ( J ♦ ) reg . By Lemma 3.6.1(1), M ′ and ∂ M ′ have no components that lie in fibers π − ( pt ) . Let u ∞ ∈ ∂ M ′ be the SFT limit ofa sequence u i : ˙ F i → b X , i = 1 , , . . . , in M ′ ; without loss of generality we mayassume that the topological types of all the ˙ F i are the same. Observe that u i hasimage in X = R × [0 , × Σ , since the maximal principle holds for π b Σ ◦ u i .We claim that u ∞ cannot have any interior nodes that are obtained in the limitby pinching a closed curve in ˙ F i . Suppose for simplicity that the domain of u ∞ isobtained by pinching a single closed curve in ˙ F i . Then ind( u ∞ ) = ind( u i ) + 2( n − by Equation (3.4.1). On the other hand, the matching/incidence condition at thenode forces ind( u ∞ ) > n − : Assuming that two components of u ∞ arematched, in order for π [0 , × Σ ◦ u ∞ to have an intersection point in [0 , × Σ ,we require(3.8.2) ind( u ∞ ) − ≥ dim([0 , × Σ) − n − . Here the − on the left-hand side comes from quotienting out the R -translations. Ifthere is only one component of u ∞ , then a slightly different argument also yieldsInequality (3.8.2). Hence ind( u ∞ ) > n − and ind( u i ) > , a contradiction.It remains to consider the limit of pinching an arc c i in ˙ F i . It is not possible for c i to connect α q to α r , where q = r , since α q and α r are disjoint (and similarly α ′ q to α ′ r , where q = r ). On the other hand, if c i connects α q to itself, then π ◦ u ∞ maps a boundary component of the domain ˙ F ∞ of u ∞ identically to a point x on ∂ ( R × [0 , . This implies that π ◦ u ∞ maps a component of ˙ F ∞ to x ,contradicting Lemma 3.6.1(1). Finally, if an arc c i connects α q to α ′ r , then thispinching corresponds to stretching in the s -direction.It follows that u ∞ is an l -level building v ∪ · · · ∪ v l , where each v j is a degree κ multisection of π . By Lemma 3.6.2 and the assumption of the regularity of J , l = 2 and ind( v ) = ind( v ) = 1 .Next suppose that J ♦ ∈ ( J ♦ ) reg is sufficiently close to J ∈ J . Then M ′ and ∂ M ′ have no components that are close to lying on a fiber π − ( pt ) . The rest of theargument that shows that u ∞ is an l -level building v ∪ · · · ∪ v l , where each v j isclose to being a degree κ multisection of π , is identical. (cid:3) Lemma 3.8.4.
For all A , χ , y , and y ′ , M ind=1 ,A,χJ ♦ ( y , y ′ ) / R is compact.Proof. Similar to that of Lemma 3.8.3 and is left to the reader. (cid:3)
We write d HF (Σ , α ′ , α ; J ♦ ) for the cohomology of d CF (Σ , α ′ , α ; J ♦ ) . Remark . It is not hard to see that two cochain groups d CF (Σ , α ′ , α ; J ♦ ) corresponding to distinct complete sets of capping surfaces are isomorphic cochaincomplexes.3.9. Definition of d CF ( W, β, φ ; h ; J ♦ ) . The higher-dimensional Heegaard Floercochain group d CF ( W, β, φ ; h ; J ♦ ) is a special case of d CF (Σ , α ′ , α ; J ♦ ) , where: • (Σ , β Σ , φ Σ ) is the capping ( f W , e β, e φ ) of ( W, β, φ ) , • α is the capping e a of the basis a of Lagrangian disks for ( W, β, φ ) , • α ′ = e h ( α ) , where e h is obtained from h ∪ id H and H is the cap for W ,by composing with a small Hamiltonian diffeomorphism as in Section 2.2,and • the coefficient ring has been further specialized.We write d HF ( W, β, φ ; h ; J ♦ ) for the cohomology of d CF ( W, β, φ ; h ; J ♦ ) .We often write d CF ( h ( a ) , a ) for d CF ( W, β, φ ; h ; J ♦ ) .We first discuss the splittings of generators and moduli spaces in the special casewhen α = e a and α ′ = e h ( e a ) .3.9.1. Generators.
Let S = S e a , e h ( e a ) . Recall there is a unique point x i ∈ e a i ∩ e h ( e a i ) which is contained in the handle H i . We define two maps H ′ and ι . The map H ′ : S → H ([0 , × W, ( { } × a ) ∪ ( { } × h ( a )); Z ) , sends y = { y , . . . , y κ } to the homology class [[0 , × η ( y )] , where η is definedas follows: η maps y i ∈ e a i ∩ e h ( e a σ ( i ) ) to the nearby point η ( y i ) ∈ a i ∩ h ( a σ ( i ) ) if y i = x i (recall that e h is close to but not exactly the same as h ∪ id | H ) and tosome point on ∂a i if y i = x i . Let M = M ( W,β ; h ) be the relative mapping torus of ( W, β ; h ) . The map ι : Im( H ′ ) → H ( M ; Z ) , is defined as follows: Let δ be a representative of [ δ ] ∈ Im( H ′ ) with “initialpoints” δ i on { } × h ( a i ) for i = 1 , . . . , κ and “terminal points” δ i on { } × a i for i = 1 , . . . , κ . Viewing δ as a chain in M , we define ι ([ δ ]) to be the homologyclass of the “closure”, obtained from δ by adjoining arcs from δ i to δ i (which wecall augmenting arcs ) along { } × a i for each i . Note that the resulting homologyclass is independent of the choice of augmenting arcs, since the a i are simply-connected. We then set H ′′ = ι ◦ H ′ .The map H ′′ gives a splitting S = ` Γ ∈ H ( M ; Z ) S Γ , which is analogous to the splitting into Spin c -structures in dimension . Moduli spaces.
Let A y , y ′ ⊂ H ([0 , × W, ( { } × a ) ∪ ( { } × h ( a ))) ∪ ([0 , × η ( y )) ∪ ([0 , × η ( y ′ )); Z ) be the subspace spanned by classes [ T ] that admit representatives T such that ∂T = P κi =1 ([0 , × η ( y i )) − P κi =1 ([0 , × η ( y ′ i )) − P κi =1 w i + P κi =1 w i , where w i is an arc in { } × a i from y i to y ′ i and w i is similarly defined.Next let A y , y ′ := H ( M, e H ( y ) ∪ e H ( y ′ ); Z ) , In general, Spin c -structures on M are classified by H ( M ; Z ) ⊕ H ( M ; Z / Z ) . When dim M = 2 n + 1 = 3 , this can be identified with H ( M ; Z ) and hence Spin c -structures are inone-to-one correspondence with H ( M ; Z ) . For n > , there is no such identification. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 19 where e H ( y ) refers to the “closure” of [0 , × η ( y ) , as described in Section 3.9.1,representing the homology class H ′′ ( y ) .There is a map κ = κ e H ( y ) : A y , y ′ → A y , y ′ , defined as follows: View the representative T as a surface in M . Then the arcs w i and w i , together with the augmenting arcs for η ( y ) and η ( y ′ ) , form closed curvesin ∪ i ( { } × a i ) which bound disks D i ⊂ { } × a i since the a i are contractible.Then κ ([ T ]) is represented by T ∪ ( ∪ i D i ) . By the contractibility of a i , κ ([ T ]) does not depend on the choice of D i and κ e H ( y ) and κ e H ( y ) coming from differentchoices of augmenting arcs can be naturally identified.The map κ gives rise to the splitting M J ♦ ( y , y ′ ) = ` B ∈A y , y ′ M BJ ♦ ( y , y ′ ) , where the superscript B is the modifier “ u ∈ M J ♦ ( y , y ′ ) is in the class B ∈A y , y ′ .”3.9.3. Definition of d CF ( W, β, φ ; h ; J ♦ ) . We now choose a complete set of cap-ping surfaces { T δ, y } in [0 , × W , defined in a manner analogous to that of Sec-tion 3.5.3, and take their “closures” { e T δ, y } in M .Using this, we can define d CF ( W, β, φ ; h ; J ♦ ) as in Section 3.8 with Λ J ℏ K -coefficients, where Λ is the Novikov ring with Z -coefficients over H ( M ; Z ) .4. A ∞ - OPERATIONS
In this section we explain how to modify the discussion from Section 3 when thebase of the symplectic fibration is a disk with m + 1 boundary punctures instead of R × [0 , and obtain the A ∞ -operations.Let D = {| z | ≤ } ⊂ C and let D m = D − { p , . . . , p m } , where p , . . . , p m ∈ ∂D , arranged in counterclockwise order around ∂D . Let ∂ i D m , i = 0 , . . . , m , bethe component of ∂D m which is the counterclockwise open arc from p i to p i +1 ,where the subscripts are taken modulo m +1 . Also let e i ⊂ D m be a “neighborhoodof p i ” which we view as a strip-like end [0 , ∞ ) × [0 , with coordinates ( s i , t i ) . p p p p m − p m α α α m − α m F IGURE
2. The base D m of the symplectic fibration b X m = D m × b Σ . The labels α i indicate the boundary condition L i = ∂ i D m × α i . Consider the symplectic fibration π m : ( b X m = D m × b Σ , b Ω m = ω + d b β Σ ) → ( D m , ω ) , where ω is an area form on D which restricts to ds i ∧ dt i on each strip-like end e i .For i = 0 , . . . , m , let E i = π − m ( e i ) = [0 , ∞ ) × [0 , × b Σ be the “ends in the horizontal direction” of b X m . Also let α i = ⊔ κj =1 α ij be acompact exact κ -component Lagrangian submanifold of ( b Σ , b β ) such that α i ⋔ α i +1 for all i = 0 , . . . , m . We then set L i = ∂ i D m × α i , L ij = ∂ i D m × α ij . Next let j m be the standard complex structure on D m ; we may assume that j m ( ∂ s i ) = ∂ t i on each e i . Also let J m = j m × J b Σ be a product almost complexstructure on b X m , and J ♦ m be a C ∞ -small perturbation of J m such that (J1)–(J3)from Section 3.2 hold on each end E i , with R replaced by [0 , ∞ ) and ( s, t ) replacedby ( s i , t i ) .For each i = 0 , . . . , m , let y i = { y i , . . . , y iκ } be a κ -tuple of intersection points y ij ∈ α i − j ∩ α iσ i ( j ) , j = 1 , . . . , κ , for some permutation σ i .Let M J ♦ m ( y , . . . , y m ) be the moduli space of holomorphic maps u : ( ˙ F , j ) → ( b X m , J ♦ m ) , where we range over all ( ˙ F , j ) such that ( F, j ) is a compact Riemann surfacewith boundary, p , . . . , p m are disjoint sets of boundary punctures of F , and ˙ F = F − ∪ i p i , and:(1) each component of ∂F − ∪ i p i maps to some L ij and each L ij is usedexactly once;(2) u maps the neighborhoods of the punctures of p i asymptotically to stripsover the Reeb chords of y i on E i .Note that ˙ F has ( m + 1) κ punctures.Let ˇ X m be the compactification of D m × Σ , obtained by attaching [0 , × Σ to each end [0 , × [0 , ∞ ) × Σ , let ˇ L i be the compactification of L i , and let ˇ u : ˇ F → ˇ X m be the compactification of u ∈ M J ♦ m ( y , . . . , y m ) , defined as inSection 3.4. The Maslov index µ ( u ) is defined as in Section 3.4 via the bundle L :for each component of ∂ ˙ F , L is given by some ˇ u ∗ T α ij and, on the components of ∂ ˇ F − ∂ ˙ F , L is given by rotating from T y ij α i − j to T y ij α iσ i ( j ) via e J b Σ t , t ∈ [0 , π ] ,where i = 1 , . . . , m and j = 1 , . . . , κ , and by rotating from T y mj α mj to T y mj α σ m ( j ) via e − J b Σ t , t ∈ [0 , π ] , where j = 1 , . . . , κ . Without loss of generality we areassuming that J b Σ ( T y ij α i − j ) = T y ij α iσ i ( j ) for all i = 0 , . . . , m , j = 1 , . . . , κ .The following lemma generalizes Lemma 3.4.1 and its proof is left to the reader. Lemma 4.0.1.
The Fredholm index of u ∈ M J ♦ m ( y , . . . , y m ) is (4.0.1) ind( u ) = ( n − χ ( u ) + µ ( u ) + 2 κ − mκn, IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 21 where dim( W ) = 2 n and dim( b X ) = 2 n + 2 . Choose complete sets of capping surfaces { T α , α δ , y } , . . . , { T α m − , α m δ m − , y m − } , { T α , α m δ m , y m } corresponding to the pairs ( α , α ) , . . . , ( α m − , α m ) , ( α , α m ) . For each ( m + 1) -tuple δ , . . . , δ m , we choose [ T α ,..., α m δ ,..., δ m ] ∈ H ( ˇ X m , ( ⊔ mi =0 ˇ L i ) ∪ ( ⊔ mi =0 δ i )) , such that ∂T α ,..., α m δ ,..., δ m is equal to δ + · · · + δ m − − δ m plus some arcs on ⊔ mi =0 ˇ L i ,if such a homology class exists. Such a collection { T α ,..., α m δ ,..., δ m } is a complete set ofcapping surfaces with base D m .The coefficient ring for d CF (Σ , α i , α i +1 ) is Λ α i , α i +1 J ~ K , where Λ α i , α i +1 is theNovikov ring with Z -coefficients over A ′ α i , α i +1 := A α i , α i +1 / torsion with respect to the symplectic form dβ and the coefficient ring for d CF (Σ , α , α m ) is Λ α , α m . Let Λ ′ be the Novikov ring with Z -coefficients over A ′ := H ( ˇ X m , ⊔ mi =0 ˇ L i ) / torsion , which can be viewed as a module over Λ α i , α i +1 and over Λ α , α m . By tensoringwith Λ ′ J ~ K , we obtain d CF (Σ , α i , α i +1 ; Λ ′ J ~ K ) and d CF (Σ , α , α m ; Λ ′ J ~ K ) .Using the complete sets of capping surfaces, we define the map Ψ m : d CF (Σ , α m − , α m ; Λ ′ J ~ K ) ⊗· · ·⊗ d CF (Σ , α , α ; Λ ′ J ~ K ) → d CF (Σ , α , α m ; Λ ′ J ~ K ) , where the coefficient of e κ − χ e A y m in Ψ m ( y ⊗ · · · ⊗ y m − ) , with χ ≤ κ and A ∈ A ′ , is the count of M ind=0 ,A,χJ ♦ m ( y , . . . , y m ) . Lemma 4.0.2. Ψ m is a cochain map.Proof. The proof is similar to that of Lemma 3.8.3. (cid:3)
We can also make a similar construction when the points p , . . . , p m ∈ ∂D are allowed to move, i.e., when the conformal structure on D m is parametrizedby the interior of the Stasheff associahedron A m . Summing over all rigid maps u : ( ˙ F , j ) → ( b X m , J ♦ m ) when viewed in the set of maps in a varying targetparametrized by int A m gives the A ∞ -map(4.0.2) µ m : d CF (Σ , α m − , α m ; Λ ′ J ~ K ) ⊗· · ·⊗ d CF (Σ , α , α ; Λ ′ J ~ K ) → d CF (Σ , α , α m ; Λ ′ J ~ K ) . Proposition 4.0.3.
The operations µ m , m = 1 , , . . . , satisfy the A ∞ -relations.Proof. The proof is also similar to that of Lemma 3.8.3. (cid:3)
Let R κ (Σ) be the A ∞ -category whose objects are compact κ -component La-grangian submanifolds α = ⊔ j α j of Σ , whose hom sets are Hom( α , α ′ ) := d CF (Σ , α , α ′ ; Λ ′ ) , and whose A ∞ maps µ m are given by Equation (4.0.2).5. R EFORMULATION OF d HF ( W, β, φ ; h ; J ♦ ) In this section we give a reformulation of d HF ( W, β, φ ; h ; J ♦ ) where we countholomorphic curves in b X W := R × [0 , × c W instead of b X = R × [0 , × b Σ . Here c W is the completion of W . We can think of b Σ as the higher-dimensional analog ofa Heegaard surface, whereas c W is the page of an open book decomposition.5.1. The completion c W . Recall that ( W, β, φ ) has a collar neighborhood N ′ ( ∂W ) =[ − ε, × ∂W with coordinates ( σ, x ) on which β = e σ β | ∂W and φ ( σ, x ) = σ . Let ( c W , b β, b φ ) be the completion of ( W, β, φ ) , obtained by gluing ([0 , ∞ ) × ∂W, e σ β | ∂W ) , with coordinates ( σ, x ) , to ( W, β ) along ∂W and extending φ to b φ ( σ, x ) = σ . Let b h be the extension of e h | W from Section 2.2 such that b h ( σ, x ) = ( σ, e h | ∂W ( x )) on [0 , ∞ ) × ∂W . Next we set b a = a ∪ ([0 , ∞ ) × ∂ a ) . Note that ∂ a is a Legendrian submanifold of ( ∂W, β | ∂W ) and that, by Condition(4) in the definition of f in Section 2.2, [ − ε, ∞ ) × ∂ a is a cylinder over the Leg-endrian ∂ a .We are initially in a Morse-Bott situation where there is an S -family of shortReeb chords from ∂a i to e h ( ∂a i ) for each i , since e h ( ∂a i ) is a positive Reeb pushoffof ∂a i on ∂W . Here we are measuring the length of a Reeb chord c using the action A β ∂W ( c ) = R c β ∂W . We then perturb e h ( ∂ a ) (without changing its name) sothat for each i = 1 , . . . , κ there are two short Reeb chords from ∂a i to e h ( ∂a i ) . Thelonger (resp. shorter) of the two will be denoted by ˆ x i (resp. ˇ x i ); see Figure 3. ˆ x i ˇ x i ∂a i e h ( ∂a i ) F IGURE
3. Legendrian ∂a i and its pushoff e h ( ∂a i ) in the La-grangian projection. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 23
Generators.
We modify the set of generators S as follows: Given y = { y , . . . , y κ } ∈ S , let b y be the result of replacing all the contact class intersec-tion points x i by ˆ x i in y . We then set b S = { b y | y ∈ S} . Let us write b y = b y ⊔ b y ,where none of the terms of b y are of the form ˆ x i and all of the terms of b y are ofthe form ˆ x i . As in Section 3.5.1, b S admits a splitting b S = ` Γ ∈ H ( M ; Z ) b S . Symplectic fibrations.
Let π R × [0 , : b X W = R × [0 , × c W → R × [0 , be a symplectic fibration with fiber c W , where ( s, t ) are coordinates on R × [0 , ,and let π c W be the projection b X W → c W . We define the Lagrangian submanifolds L , b a = R × { } × b a , L , b h ( b a ) = R × { } × b h ( b a ) . Similarly let b X R × ∂W := R × [0 , × ( R × ∂W ) → R × [0 , be a symplectic fibration whose fiber R × ∂W is a symplectization of ( ∂W, β | ∂W ) .When we stretch a holomorphic curve in b X along R × [0 , × ∂W , i.e., do a “side-ways stretching”, the sideways levels that we obtain are b X W , b X H := R × [0 , × b H ,and several levels of b X R × ∂W in between. Here b H refers to the completion of H obtained by attaching positive and negative ends. Refer to Figure 4. b a i b h ( b a i ) ˆ x i ˆ x i [0 , ∞ ) × ∂WW F IGURE
4. Schematic picture indicating the result of stretching f W along ∂W . ˆ x i indicates the possible locations of ˆ x i .5.4. Almost complex structures.
Let J c W be an almost complex structure on c W which is tamed by d b β and is adapted to β ∂W on the symplectization end c W − W .The space of such almost complex structures J c W will be denoted by J c W . Wedefine J b X W = J R × [0 , × J c W where J c W ∈ J c W and a C ∞ -small perturbation J ♦ b X W of J b X W as in Section 3.2 so that (J1)–(J3) are satisfied with b Σ replaced by c W .Let J b X W and J ♦ b X W be the space of such J b X W and J ♦ b X W . We can similarly define J R × ∂W , J b X R × ∂W = J R × [0 , × J R × ∂W , and spaces J b X R × ∂W and J ♦ b X R × ∂W . Moduli spaces.
When a curve in M J ♦ ( y , y ′ ) is stretched along b X R × ∂W ,almost all of the essential information is contained in the b X W part.Let ( s , t ) be coordinates on the infinite strip R × [0 , with the standard com-plex structure and let J b X R × ∂W = J R × [0 , × J R × ∂W ∈ J b X R × ∂W . Definition 5.5.1.
Let c be a Reeb chord of ( ∂W, β | ∂W ) . A holomorphic map v : R × [0 , → b X R × ∂W is a diagonal strip over a Reeb chord c if π R × [0 , ◦ v ( s , t ) = ( s + C, t ) , or ( − s + C, − t ) for some C ∈ R and π R × ∂W ◦ v is a trivial strip over c . Definition of M ( b y , b y ′ ) . Let J ♦ b X W ∈ ( J ♦ b X W ) reg . We define M ( b y , b y ′ ) , where b y = b y ⊔ b y and b y ′ = b y ′ ⊔ b y ′ . If b y b y ′ , then we set M ( b y , b y ′ ) = ∅ . On the other hand, if b y ⊂ b y ′ , then M ( b y , b y ′ ) := M J ♦ b XW ( b y , b y ′ − b y ) × M J ♦ b XH ( b y , b y ) , where M J ♦ b XW ( b y , b y ′ − b y ) is the set of holomorphic maps b u : ( ˙ F , j ) → ( b X W , J ♦ b X W ) , where we range over all ( ˙ F , j ) such that ( F , j ) is a compact Riemann surfacewith boundary, p + and p − are disjoint sets of boundary punctures of F , and ˙ F = F − p + − p − , and:(1) each component of ∂F − p + − p − maps to some component of L , b a or L , b h ( b a ) and each component of L , b a and L , b h ( b a ) is used at most once;(2) at the positive end s ≫ , b u maps the neighborhoods of the punctures of p + asymptotically to strips over Reeb chords [0 , × b y ;(3) at the negative end s ≪ , b u maps the neighborhoods of the puncturesof p − asymptotically to strips over Reeb chords [0 , × b y ′ or to diagonalstrips over b y ′ − b y ;and M J ♦ b XH ( b y , b y ) is a one-element set { b u } where b u is the union of trivial stripsin b X H from y to y . Note that b u has b y positive ends and b y negative endsand b u has b y positive ends and b y negative ends.As in Section 3.5.2, M ( b y , b y ′ ) admits a splitting M ( b y , b y ′ ) = ` χ ∈ Z ,A ∈A y , y ′ M A,χ ( b y , b y ′ ) , where A y , y ′ is defined analogously. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 25
Definition of the variant.
We now define the variant ( C ′ , d ′ ) := d CF ′ ( W, β, φ ; h ; J ♦ b X W ) of d CF ( W, β, φ ; h ; J ♦ ) , where C ′ is the free Λ J ℏ K -module generated by b S and Λ is the Novikov ring with Z -coefficients over H ( M ; Z ) . The differential d ′ b y isas given in Equation (3.8.1), with y and y ′ replaced by b y and b y ′ and the modulispace M ind=1 ,A,χJ ♦ ( y , y ′ ) / R replaced by M ind=1 ,A,χ ( b y , b y ′ ) / R . In particular, when b y = b y , i.e., b y = ∅ , then M ind=1 ( b y , b y ′ ) = ∅ since M ( b y , b y ′ ) is a one-elementset which is the union of trivial strips. Notation 5.6.1.
We often write d CF ( b h ( b a ) , b a ) as shorthand for d CF ′ ( W, β, φ ; h ; J ♦ b X W ) . Lemma 5.6.2. ( d ′ ) = 0 .Proof. This follows from analyzing the degenerations of M ind=2 ,A,χ ( b y , b y ′ ) / R andproperly keeping track of components which are diagonal strips over ˆ x i in b X . (cid:3) Equivalence with original definition.
The goal of this subsection is to provethe following equivalence result:
Theorem 5.7.1.
Suppose the function f in the definition of e h is sufficiently C r -small for r ≫ . Then there is an isomorphism d CF ′ ( W, β, φ ; h ; J ♦ b X W ) ≃ d CF ( W, β, φ ; h ; J ♦ ) of cochain complexes for some J ♦ ∈ ( J ♦ ) reg .Proof. This follows from stretching f W along ∂W .Let [ − ε, ε ] × ∂W be a fixed collar neighborhood of ∂W in f W with coordinates ( σ, x ) . Let J i f W ∧ ∈ J reg f W ∧ , i = 1 , , . . . , be a sequence of almost complex structureson f W ∧ that satisfy the following: • J i f W ∧ agrees with a fixed J f W ∧ ∈ J reg f W ∧ on [ − ε, ε ] × ∂W ; • on [ − ε , ε ] × ∂W , J i f W ∧ is adapted to β | ∂W with one slight modification: J i f W ∧ ( i ∂ σ ) = R where R is the Reeb vector field for β | ∂W ;We then set J i = J R × [0 , × J i f W ∧ . For questions about compactness, it suffices touse the split almost complex structures J i instead of the perturbed ones.Let u i ∈ M ind=1 ,A,χJ i ( y , y ′ ) , i = 1 , , . . . , be a sequence of holomorphic maps.We have the following compactness result: Lemma 5.7.2.
Suppose the function f in the definition of e h is sufficiently C r -smallfor r ≫ . Then u i limits to u ∞ = u ∞ ∪ u ∞ , where u ∞ ∈ M ind=1 ,A,χJ ♦ b XW ( b y , b y ′ − b y ) and u ∞ is the union of strips in b X H from ˆ x i to x i when x i ∈ b y ′ − b y and trivialstrips in b X H from x i to x i when x i ∈ b y . Proof.
Consider the sequence v i = π f W ∧ ◦ u i . Then, by the usual SFT compactnesstheorem [EGH], v i limits to a multi-level building v ∞ = v ∞ ∪ · · · ∪ v ∞ l , where v ∞ maps to c W , v ∞ , . . . , v ∞ ,l − map to R × ∂W , and v ∞ l maps to b H . The endsof v ∞ ,j a priori limit to Reeb chords of R from ∂ a to e h ( ∂ a ) and closed orbits of R .We claim that the negative ends of v ∞ ,l can only be short Reeb chords for f with sufficiently small C r -norm. If the claim fails to hold, then, since there areno positive punctures and f has arbitrarily small C r -norm, the presence of longnegative Reeb chords and closed Reeb chords contradicts the usual energy boundsinvolving β | H .The claim, together with action considerations, then implies that all the ends of v ∞ j , j = 1 , . . . , l , are short Reeb chords. By area and Fredholm index considera-tions, all the components of v ∞ j , j = 2 , . . . , l − , are trivial strips over short Reebchords, which is equivalent to saying that they can be discarded. A similar argu-ment implies that v ∞ l is a union of strips from x i to ˆ x i (here x i is an intersectionpoint and ˆ x i is a negative puncture with respect to b H ).The above description of v ∞ then implies the lemma. (cid:3) Lemma 5.7.3. ind( u i ) = 1 if and only if ind( u ∞ ) = 1 .Proof. Each component of u ∞ has index . Since the gluing/incidence conditionfor gluing a component of u ∞ to u ∞ is a codimension condition, ind( u i ) = 1 is equivalent to ind( u ∞ ) = 1 . (cid:3) Lemma 5.7.4.
The mod count of strips from ˆ x i to x i in b X H modulo translationis one.Proof. This is equivalent to the assertion that the mod count of strips from x i to ˆ x i in b H is one. Let ( e a ∩ H ) ∧ be the completion of e a ∩ H in b H , obtained byattaching cylindrical ends. By Gromov compactness, for sufficiently small f , allthe strips are contained in a neighborhood of ( e a ∩ H ) ∧ . Hence it suffices to do amodel calculation on T ∗ S n , where e a is the -section, e h ( e a ) is the graph of df , f issufficiently small, df = 0 at the north and south poles p n , p s , and we are stretchingalong the equator. By the usual argument from [Fl, Proposition 1], for any (generic)point q ∈ e a , there is a unique holomorphic strip from p n to p s that passes through q . When we stretch along the neck and view the equator as an S -family of Reeborbits, the count of strips from p n to a generic point on the S -family is , implyingthe lemma. (cid:3) In view of Lemma 5.7.4 and the definition of M ( b y , b y ′ ) from Section 5.5, thereis a bijection M ind=1 ,A,χJ ♦ ( y , y ′ ) / R ≃ M ind=1 ,A,χ ( b y , b y ′ ) / R , which implies the theorem. (cid:3) IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 27
6. P
ARTIAL INVARIANCE
In this section we prove Theorem 1.1.1, i.e., invariance of the higher-dimensionalHeegaard Floer homology groups with respect to:(I1) trivial Weinstein homotopies;(I2) changes of almost complex structure;(I3) changes of symplectomorphism within its symplectic isotopy class.We will work out (I1) and (I2) in Section 6.1; the invariance under (I3) is similarand will be omitted. We will leave the non-invariance with respect to handleslides(I4) to the reader.In Section 6.2 we also discuss invariance under elementary stabilizations (cf.Definition 6.2.1).6.1.
Invariance under (I1) and (I2).
Let ( W, β s , φ s ) , s ∈ [0 , , be a trivial We-instein homotopy with compatible Liouville vector fields Y s . Let a s = a s ∪· · ·∪ a sκ be the Lagrangian basis for ( W, β s , φ s ) . We attach a cap H s to W along the Legen-drian spheres { ∂a s , . . . , ∂a sκ } in a smooth manner with respect to s . The manifold W ∪ H s is diffeomorphic to a fixed compact manifold f W , which inherits a family ( e β s , e φ s ) of Weinstein structures and cappings α s = e a s of a s .By Lemma 2.1.8(2), there exists a -parameter family g s : ( f W ∧ , d e β ∧ ) ∼ → ( f W ∧ , d e β ∧ s ) of symplectomorphisms such that g ∗ s e β ∧ s = e β ∧ outside of a compact set. Hence,after pulling back using g s and renaming, we may assume that:(H1) each α s is a submanifold of a fixed symplectic manifold ( f W ∧ , ω = d e β ∧ ) with a cylindrical end; and(H2) the Weinstein structure ( e β ∧ s , e φ ∧ s ) depends on s only on a compact subset of f W ∧ .In particular, the Liouville vector field Y s is constant outside of the same compactset.Consider the submanifolds L ′ , α = (( −∞ , × { } × α ) ∪ ( ∪ s ∈ [0 , ( { s } × { } × α s )) ∪ ([1 , ∞ ) × { } × α ) ,L ′ , e h ( α ) = (( −∞ , × { } × e h ( α )) ∪ ( ∪ s ∈ [0 , ( { s } × { } × e h ( α s ))) ∪ ([1 , ∞ ) × { } × e h ( α )) , of b X = R × [0 , × f W ∧ which agree with L , α and L , e h ( α ) at the positive endand with L , α and L , e h ( α ) at the negative end. Lemma 6.1.1.
There exists a (closed) symplectic connection b Ω ′ on b X which satis-fies the following: (1) on each fiber b Ω ′ restricts to ω ; (2) b Ω ′ agrees with b Ω at the positive and negative ends of b X ; and (3) L ′ , α and L ′ , e h ( α ) are Lagrangian.Proof. Let H s,t , s, t ∈ [0 , , be a -parameter family of Hamiltonian functionson f W ∧ with H s,t = 0 outside of a compact region of [0 , s × [0 , t × f W ∧ . Wealso view the family H s,t as a function H on b X . Let ψ s ,t be the correspondingfamily of Hamiltonian diffeomorphisms, obtained by integrating the vector fieldcorresponding to d f W ∧ H s,t along the segment { t = t , ≤ s ≤ s } . (We aretaking ψ s ,t = id for s ≤ .) Here d f W ∧ is the differential in the f W ∧ -direction.Recall that we are using the convention that the Hamiltonian vector field X g of afunction g with respect to a symplectic form ω satisfies dg = i X g ω .We may choose H s,t such that ψ s,t satisfies the following:(A) ψ s,t = id for s ≤ and ψ s,t is independent of ( s, t ) for s ≥ ;(B) ψ s, ( α ) = α s and ψ s, ( e h ( α )) = e h ( α s ) for s ∈ [0 , .This is possible since the α s are exact.We then set(6.1.1) b Ω ′ = ds ∧ dt + ω + dH ∧ ds = ds ∧ dt + ω + d f W ∧ H s,t ∧ ds − ∂H s,t ∂t ds ∧ dt. b Ω ′ is closed. (1) and (2) are immediate. (3) uses (A) and (B) and is left to thereader. (cid:3) Let J i = J R × [0 , × J f W ∧ ,i , i = 0 , , be ( f W ∧ , e β ∧ i ) -compatible almost complexstructures on b X and let J ♦ i be their perturbations which we assume to be regular.Also let E = [1 , ∞ ) × ∂ f W ⊂ f W ∧ . (Here we are assuming that the boundary of f W is { } × ∂ f W . Hence f W ∧ − f W =[0 , ∞ ) × ∂ f W is slightly larger than E .) By (H2) we may assume without loss ofgenerality that e β ∧ s , s ∈ [0 , , is independent of s on E . Case 1.
First suppose that J ♦ = J ♦ on R s × [0 , t × E . Let J ′ be an b Ω ′ -tamealmost complex structure on b X which satisfies the following:(K1) J ′ agrees with J at the positive end ( s ≫ ) and with J at the negativeend ( s ≪ );(K2) the map π : b X → R × [0 , is ( J ′ , J R × [0 , ) -holomorphic; and(K3) J ′ is R × [0 , -invariant on R × [0 , × E ;and let ( J ′ ) ♦ be its perturbation which agrees with J ♦ = J ♦ at the positive andnegative ends and which we assume to be regular.The count of index holomorphic curves in ( b X, b Ω ′ , ( J ′ ) ♦ ) with boundary on L ′ , α and L ′ , e h ( α ) yields a map:(6.1.2) I : d CF ( W, β , φ ; h ; J ♦ ) → d CF ( W, β , φ ; h ; J ♦ ) . The compactness of the relevant moduli space is proved as in Lemma 3.8.4. Inparticular, curves cannot enter the region R × [0 , × E by the maximum principle. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 29
Reversing the roles of ( β , φ ) and ( β , φ ) , we obtain the inverse map:(6.1.3) I : d CF ( W, β , φ ; h ; J ♦ ) → d CF ( W, β , φ ; h ; J ♦ ) . Lemma 6.1.2.
The maps I and I are cochain maps.Proof. Similar to the proof of Lemma 3.8.3. (cid:3)
Lemma 6.1.3.
The maps I and I are homotopy inverses.Proof. There exists a family ( b Ω ′′ η , L ′′ , α ,η , L ′′ , b h ( α ) ,η , ( J ′′ ) ♦ η ) of cobordisms parametrizedby η ∈ [0 , such that: • ( b Ω ′′ , L ′′ , α , , L ′′ , b h ( α ) , , ( J ′′ ) ♦ ) is a trivial cobordism; • ( b Ω ′′ η , L ′′ , α ,η , L ′′ , b h ( α ) ,η , ( J ′′ ) ♦ η ) coincides with ( b Ω , L , α , L , e h ( α ) , J ♦ ) at thepositive and negative ends; and • as η → , ( b Ω ′′ η , L ′′ , α ,η , L ′′ , b h ( α ) ,η , ( J ′′ ) ♦ η ) limits to a -level building wherethe top level is the cobordism for I and the bottom level is the cobordismfor I .By the usual chain homotopy argument, I ◦I is chain homotopic to the identity.The case of I ◦ I is similar. (cid:3) Case 2.
Next suppose that e β ∧ s , s ∈ [0 , , is independent of s on all of f W ∧ . Let J f W ∧ ,s , s ∈ [0 , , be a family of almost complex structures on f W ∧ compatiblewith e β ∧ s (for any s ) from J f W ∧ , to J f W ∧ , such that: • J f W ∧ ,s = J f W ∧ , on E for all s ∈ [0 , ] ; and • J f W ∧ ,s = J f W ∧ , on f W for all s ∈ [ , .Then let J s = J R × [0 , × J f W ∧ ,s , s ∈ [0 , , and let J ♦ s be its regular perturbation.Using J s , s ∈ [ , , we can construct a cobordism which restricts to the trivialcobordism on R × [0 , × f W and hence induces a quasi-isomorphism I , / : d CF ( W, β , φ ; h ; J ♦ ) → d CF ( W, β / , φ / ; h ; J ♦ / ) , by the maximal principle. In other words, if u is a curve that is counted in I , / ,then π f W ◦ u has image inside f W . Since we can also define a quasi-isomorphism I / , : d CF ( W, β / , φ / ; h ; J ♦ / ) → d CF ( W, β , φ ; h ; J ♦ ) by Case 1, the invariance with respect to (I1) and (I2) follows.6.2. Invariance under elementary stabilizations.
Let S ± c ( W, β, φ ; h ) = ( W ′ , β ′ , φ ′ ; h ′ ) be the positive/negative stabilization of ( W, β, φ ; h ) along a Lagrangian disk c ⊂ W . Definition 6.2.1. An elementary stabilization is a stabilization where c is disjointfrom the Lagrangian basis a = a ∪ · · · ∪ a κ of W associated with the Weinsteinstructure ( β, φ ) .We may assume that the handle W = W ′ − W is attached to W along theLegendrian sphere ∂c and that W is disjoint from H . Hence we can attach thehandles of H and W separately in any order. Let H be an n -handle attached tothe boundary of the cocore of W and let H ′ = H ∪ H . We then obtain f W ′ := W ′ ∪ H ′ . The cocore of W together with the core of H form a Lagrangian sphere e a . Asbefore, we first extend h ′ from W ′ to f W ′ by the identity and then perturb theextension by the Hamiltonian flow of a function f i near e a i , for i = 0 , . . . , κ . Theresult is a symplectomorphism e h ′ ∈ Symp( f W ′ , ∂ f W ′ , dβ ′ ) . There is a unique point x (resp. z ) of e a ∩ e h ′ ( e a ) in H ′ corresponding to the minimum (resp. maximum)of f in the case of a positive (resp. negative) stabilization.Let ( J ′ ) ♦ and J ♦ be perturbed ( c W ′ , b β ′ ) - and ( c W , b β ) -compatible almost com-plex structures on b X f W ′ = R × [0 , × ( f W ′ ) ∧ and b X = R × [0 , × ( f W ) ∧ . Wethen have the following: Theorem 6.2.2 (Invariance under elementary stabilizations) . (1) If S + c ( W, β, φ ; h ) = ( W ′ , β ′ , φ ′ ; h ′ ) is an elementary stabilization, thenthe map Θ + : d CF ( W, β, φ ; h ; J ♦ ) → d CF ( W ′ , β ′ , φ ′ ; h ′ ; ( J ′ ) ♦ ) y
7→ { x } ∪ y , is a quasi-isomorphism. It takes the contact class [ x ] to the contact class [ { x } ∪ x ] . (2) If S − c ( W, β, φ ; h ) = ( W ′ , β ′ , φ ′ ; h ′ ) is an elementary stabilization, thenthe map Θ − : d CF ( W, β, φ ; h ; J ♦ ) → d CF ( W ′ , β ′ , φ ′ ; h ′ ; ( J ′ ) ♦ ) y
7→ { z } ∪ y , is a quasi-isomorphism.Proof. We prove (1); (2) is similar. Note that e a does not intersect any e h ′ ( e a i ) for i = 1 , . . . , κ since h ′ was defined to be ( h ∪ id | W ) ◦ τ γ (not the other way around).Hence the only intersection between e a and ∪ κi =0 e h ′ ( e a i ) is x .We will see in Lemma 7.1.2 that, for a good choice of almost complex structure ( J ′ ) ♦ , the only curve with x at the positive end is a trivial strip. Hence any curve u from { x } ∪ y to { x } ∪ y ′ consists of a trivial strip from x to itself and othercurves from y to y ′ that do not involve x . (cid:3) IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 31
7. T
HE CONTACT CLASS
The goal of this section is to define the p -twisted contact class c p ( W, β, φ ; h ) ∈ d HF ( W, β, φ ; h ; J ♦ ) of a Weinstein open book decomposition ( W, β, φ ; h ) for p ∈ Z and study its prop-erties. Although d HF ( W, β, φ ; h ; J ♦ ) is not invariant under handleslides, there arenevertheless surprising applications of the contact class.One of the main properties of the contact class is Theorem 1.1.3, which givesa convenient method for certifying that certain contact manifolds are not Liouvillefillable. Theorem 1.1.3 provides large classes of contact manifolds that are notLiouville fillable.7.1.
Definition of the contact class.
Recall the functions f : f W → R and f i : e a i → R from Section 2.2. Let x i ∈ e a i − a i be the minimum of the Morse function f i , viewed as a point of e a i ∩ e h ( e a i ) and satisfying e h ( x i ) = x i . Definition 7.1.1.
The contact class of the open book decomposition ( W, β, φ ; h ) is(7.1.1) c ( W, β, φ ; h ) = x = { x , . . . , x κ } ∈ d CF ( W, β, φ ; h ; J ♦ ) . More generally, for p ∈ Z , the p -twisted contact class of ( W, β, φ ; h ) is c p ( W, β, φ ; h ) = ~ p x . Lemma 7.1.2.
For all p ≥ , the p -twisted contact class c p ( W, β, φ ; h ) is a cyclefor a certain choice of almost complex structure J ♦ ∈ ( J ♦ ) reg and sufficiently C -small f : f W → R . Hence the contact class can be viewed as an element of d HF ( W, β, φ ; h ; J ♦ ) .We often abuse notation and refer to both the cycle and the homology class as the“contact class”.Before proving Lemma 7.1.2 we review some notions from [CE]. Recall thattwo functions φ , φ : Σ → R on a manifold Σ are target equivalent if there is anincreasing diffeomorphism g : R ∼ → R such that φ = g ◦ φ .According to [CE, Theorem 1.1(a)], given a Weinstein domain (Σ , β , φ ) , thereexists a Morse function φ which is target equivalent to φ and a complex structure J b Σ on b Σ such that:(P1) φ is plurisubharmonic with respect to J b Σ on Σ − N ( ∂ Σ) , where N ( ∂ Σ) is a small collared neighborhood of ∂ Σ ;(P2) (Σ , β , φ ) is Weinstein homotopic to (Σ , β , φ ) through a homotopy whichfixes φ , such that β := − dφ ◦ J b Σ on Σ − N ( ∂ Σ) ;(P3) J b Σ is adapted to β | ∂ Σ on b Σ − Σ ;(P4) x i , i = 1 , . . . , κ , are critical points of φ with φ -value and all the othercritical points of φ have φ -value < .Let e h ∈ Symp( f W , ∂ f W , d e β ) be the symplectomorphism from Section 2.2 with e β = e β and sufficiently C -small f : f W → R . Let J be the space of ( f W ∧ , e β ∧ ) -compatible almost complex structures on b X = b X f W . Proof of Lemma 7.1.2.
We show that there exist J ♦ ∈ ( J ♦ ) reg and f : f W → R such that no nonconstant J ♦ -holomorphic map is asymptotic to x at the positiveend.We may assume that e a ⊂ { φ ≤ } . If the C -norm of f is sufficiently small,then we may also assume that e h ( e a ) ⊂ { φ ≤ } .First consider the case J ♦ = J = j R × [0 , × J f W ∧ , where the pair ( φ , J f W ∧ ) satisfies (P1)–(P4) with Σ = f W .We claim that each x i there exists a neighborhood N ( x i ) and a coordinate chart ϕ i : N ( x i ) → R nq ,...,q n ,p ,...,p n , such that ϕ i ( x i ) = 0 , the symplectic form is P j dq j dp j , J f W ∧ ( ∂ q j ) = ∂ p j , and ϕ i ( e a i ) ⊂ { p = · · · = p n = 0 } . Indeed the proof of [CE, Theorem 1.1(a)] —and in particular Step 1 of [CE, Proposition 13.12] — gives such a neighborhood N ( x i ) . We also take f ( q , . . . , q n , p , . . . , p n ) = − ε P nj =1 q j on N ( x i ) for sufficiently small ε > . Then ϕ i ( e h ( e a i )) is contained in the linearsubspace spanned by ∂ q j − ε∂ p j , j = 1 , . . . , n .Arguing by contradiction, suppose there exists a nonconstant map u : ( ˙ F , j ) → ( b X, J ) which is asymptotic to x at the positive end. Since J is a product, the map u f W ∧ := π f W ∧ ◦ u is holomorphic. By Remark 3.3.2, Im( u f W ∧ ) ⊂ Σ . We may assume that u f W ∧ is nonconstant near x i . Consider the restriction u f W ∧ ,i of u f W ∧ to N ( x i ) . Let π j : R n → R be the projection to the q j p j -plane and let v j := π j ◦ u f W ∧ ,i . Thechoices in the previous paragraph were made so that π j is holomorphic and that π j ( e a i ) and π j ( e h ( e a i )) are real lines spanned by ∂ q j and ∂ q j − ε∂ p j , respectively.The holomorphic map v j can be viewed as a map ( −∞ , σ × [0 , τ → C which is dominated by a term of the form c j e ( θ + m j π )( σ + iτ ) as σ → −∞ , where c j ∈ R − { } , < θ < π is the angle corresponding to the vector − ∂ q j + ε∂ p j ,and m j is a nonnegative integer. In other words, if v j is nonconstant, then theimage of v j sweeps out a large (= angle > π ) sector. The asymptotic descriptionof v j then implies that Im( u f W ∧ ) ∩ { φ > } 6 = ∅ . Now, since u f W ∧ maps ∂ ˙ F to e a ∪ e h ( e a ) ⊂ { φ ≤ } , the map φ ◦ u f W ∧ attains a maximum in the interior of ˙ F ,contradicting the maximum principle.Now consider the case where J ♦ = J = j R × [0 , × J f W ∧ on the subset R × [0 , × ( f W ∧ − W ) . Observe the following:(1) there are no other intersection points of e a i ∩ e h ( e a j ) in f W ∧ − W besidespoints of x ; and(2) if v is a component of u ∈ M J ♦ ( y , y ′ ) and one of the positive ends of v limits to x i , then v is a trivial strip R × [0 , × { x i } . The letter j is used to denote a complex structure on ˙ F and as a subscript; we hope this will notcreate any confusion. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 33
Hence every component of u ∈ M J ♦ ( y , y ′ ) nontrivially intersects R × [0 , × W ,provided the component is not a trivial strip over some x i . This means that thereexists an almost complex structure J ♦ ∈ ( J ♦ ) reg such that J ♦ = J on R × [0 , × ( f W ∧ − W ) ; in other words, J ♦ just needs to be generic on R × [0 , × W to attainregularity. The contact class x is a cycle for such a J ♦ , proving the lemma. (cid:3) From now on we assume that J ♦ ∈ ( J ♦ ) reg and f are chosen so that Lemma 7.1.2holds. Connected sums.
Let ( W , β , φ ; h ) and ( W , β , φ ; h ) be supportingopen book decompositions for the contact manifolds ( M , ξ ) and ( M , ξ ) . Weassume the Morse functions φ i , i = 0 , , have been normalized so that ∂W i is theregular level set φ − i (1) .Let ( W, β, φ ) be the Weinstein domain obtained by attaching a Weinstein -handle H to ( W , β , φ ) ⊔ ( W , β , φ ) along small balls that are disjoint fromthe boundaries of the bases a and a of Lagrangian disks for ( W , β , φ ) and ( W , β , φ ) and then a collar neighborhood N , such that φ is an extension of φ ∪ φ to W , φ ( ∂W ) = 3 , and φ − ([2 , H ∪ N . Also let h be the extensionof h ∪ h by the identity on H ∪ N .The open book ( W, β, φ ; h ) is an adapted open book for the connected sum ( M M , ξ ξ ) , where ξ = ξ ξ is obtained by gluing the complement of twostandard Darboux balls respectively in ( M , ξ ) and ( M , ξ ) along their bound-aries.Let ( f W , e β, e φ ; e h ) be the capping of ( W, β, φ ; h ) as usual. The capping ( f W , e β, e φ ) is trivial Weinstein homotopic to ( f W ′ , e β ′ , e φ ′ ) , which is obtained from ( f W , e β , e φ ) ⊔ ( f W , e β , e φ ) by first capping off the Lagrangian bases a and a so that ( e φ ′ ) − (2) = ∂ f W ∪ ∂ f W , then attaching a -handle H ′ connecting f W and f W , and finally attachinga collar neighborhood N ′ such that ( e φ ′ ) − ([2 , H ′ ∪ N ′ . The symplecto-morphism e h is homotopic to e h ′ , which is the extension of e h ∪ e h to f W ′ by theidentity.The Heegaard Floer homology groups are well-behaved under this connectedsum operation: Lemma 7.2.1.
With ( W, β, φ ; h ) as above, d HF ( W, β, φ ; h ) ≃ d HF ( W , β , φ ; h ) ⊗ d HF ( W , β , φ ; h ) . The contact class c ( W, β, φ ; h ) vanishes if and only if one of the contact classes c ( W , β , φ ; h ) or c ( W , β , φ ; h ) vanishes.Proof. Let α , viewed as a Lagrangian in f W ′ , be the union of cappings e a and e a in ( f W , e β , e φ ) and ( f W , e β , e φ ) of the Lagrangian bases a and a . Since e h ′ | H ′ ∪ N ′ = id , α ∪ e h ′ ( α ) ⊂ { e φ ′ ≤ } and is disjoint from H ′ ∪ N ′ . No J ♦ -holomorphic curve u that is counted in d CF ( f W ′ , e h ′ ( α ) , α ; J ♦ ) , for a generic J ♦ close to a product J , has a projection π f W ′ ◦ u to f W that enters the handle H ′ bythe maximum principle and Gromov compactness. Hence d CF ( f W ′ , e h ′ ( α ) , α ) ≃ d CF ( W , β , φ ; h ) ⊕ d CF ( W , β , φ ; h ) . Since ( f W , e β, e φ ) and ( f W ′ , e β ′ , e φ ′ ) are Weinstein homotopic through a trivial Wein-stein homotopy, Theorem 1.1.1(I1) implies the first statement of the lemma. Sincethe maps in Theorem 1.1.1 take the contact class to the contact class, the secondstatement follows. (cid:3) Question . If ( M , ξ ) and ( M , ξ ) have non-vanishing contact class for allsupporting Weinstein open book decompositions, then does ( M M , ξ ξ ) alsohave a non-vanishing contact class for all supporting Weinstein open book decom-positions? In particular is ( M M , ξ ξ ) tight?7.3. Cobordisms.
In this subsection we define several cobordisms and describethe effect of the induced Floer cohomology maps on the contact classes.Let ( M = M ( W,β ; h ) , ξ = ξ ( W,β ; h ) ) be the contact manifold supported by theopen book decomposition ( W, β ; h ) . As before, ( f W , e β ) is the capping off of W and f W ∧ is the completion of f W . Let h ⋆ be the extension of h to f W or f W ∧ by theidentity. The cobordism ( X + , Ω + , J + ) . The cobordism ( X + , Ω + ) interpolates from thesymplectization of f W ∧ × [0 , at the positive end to the symplectization of ( M, ξ ) with a Lagrangian boundary condition L + e a ⊂ ∂X + at the negative end. This isthe higher-dimensional analog of the cobordism from [CGH3] that was used toconstruct an isomorphism from the plus version of the Heegaard Floer homologyof a -manifold to its embedded contact homology. We only give a brief sketchsince the constructions from [CGH3] extend to higher dimensions with minimalchange.Let W = W and W / = f W ∧ − W .First we construct fibrations π : X → B and π : X → D with fibersdiffeomorphic to f W ∧ and W / . Here B = ([0 , ∞ ) × R / Z ) − B c + with coordi-nates ( s, t ) and B c + is [2 , ∞ ) × [1 , with the corners rounded. We then glue X and X and smooth a boundary component M ⊓ of X ∪ X to obtain M . Finallywe attach the negative end X = ( −∞ , × M to get X + .The fibration π : X → B is a subset of [0 , ∞ ) × N ( f W ∧ ,h ⋆ ) , where N ( f W ∧ , e h ) = ( f W ∧ × [0 , / ( x, ∼ ( h ⋆ ( x ) , . It has a symplectic form Ω = π ∗ ds ∧ dt + dβ . The fibration π : X → D istopologically trivial and its symplectic form Ω is the split form dβ | W / + ω D .We fiberwise glue ( X , Ω ) and ( X , Ω ) along W / × { } × R / Z and W / × ∂D with identity fiberwise gluing maps. We then round corners to obtainthe concave contact boundary ( M, ξ ) with a compatible open book presentationand glue to it the negative symplectization ( X , Ω ) = ( −∞ , × M . This com-pletes the construction of ( X + , Ω + ) . IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 35
The Lagrangian L + e a ⊂ ∂X + is obtained by placing a copy of e a over ( s, t ) =(3 , and parallel transporting it along ∂X + using the symplectic connection Ω + .It agrees with L , e a on π − ([3 , ∞ ) × { } ) and with L ,h ⋆ ( e a ) on π − ([3 , ∞ ) ×{ } ) . (Strictly speaking, we want to use L , e h ( e a ) where e h is obtained from h ⋆ bycomposing with a small Hamiltonian isotopy as in Section 2.2. To do this, we needto slightly modify the symplectic connection Ω + by adding a Hamiltonian term asin Equation (6.1.1).)The almost complex structure J + on ( X + , Ω + ) is(i) compatible with ( f W ∧ , e β ∧ ) at the positive end;(ii) of the form J + = j B ∪ D × J W / on ( B ∪ D ) × W / ; and(iii) adapted to a contact form on the symplectization end X .(ii) ensures that, as in Lemma 7.1.2, the only holomorphic curves that limit to thecontact class x at the positive end are constant horizontal sections over the once-punctured disk B ∪ D . Lefschetz cobordism . A Lefschetz cobordism is a Lefschetz fibration over R s × [0 , t whose fiber is ( f W ∧ , e β ∧ ) and which has p singular fibers, located at ( s, t ) =(1 , / , . . . , ( p, / and corresponding to the vanishing cycles S , . . . , S p of W .We will not be explicit about the construction of the almost complex structure andthe symplectic connection except to say that at the positive and negative ends theyneed to coincide with J ♦ ∈ ( J ♦ ) reg and b Ω . The only additional ingredient is theconstruction near the singular fibers, which is carried out in Seidel [Se1]. Such acobordism induces a map d CF ( h ( a ) , a ) → d CF ( τ − S p ◦ · · · ◦ τ − S ( h ( a )) , a ) , by a count of pseudo-holomorphic multisections. Here τ S denotes the symplecticDehn twist along the Lagrangian sphere S .The following two lemmas are proved in the same way as Lemma 7.1.2 and theirproofs will be omitted. Lemma 7.3.1.
The only pseudoholomorphic curve in ( X + , Ω + , J + ) that limits tothe contact class x at the positive end is the union of κ constant sections x i × B + . Lemma 7.3.2.
The induced map d CF ( h ( a ) , a ) → d CF ( τ − S p ◦ · · · ◦ τ − S ( h ( a )) , a ) of a Lefschetz cobordism with vanishing cycles S , . . . , S p maps the contact classto the contact class. We now come to the proof of Theorem 1.1.3:
Proof of Theorem 1.1.3. (1) If ( M, ξ ) is Liouville fillable, then we can attach thefilling to the negative end of ( X + , Ω + ) to obtain an exact symplectic cobordism ( X + , Ω + ) from f W ∧ × [0 , to ∅ . If (*) P i c i d y i = x with c i ∈ Z , then wecan glue the holomorphic curves in R × [0 , × f W ∧ that are involved in (*) to the constant sections x × B + in ( X + , Ω + ) to get an index one family of curves.Algebraically over Z , there is no possible breaking for this family other than theone we started from, contradicting the compactness of moduli spaces in ( X + , Ω + ) .(2) We show that every Reeb vector field R for ξ admits a finite collection ofperiodic orbits whose sum is homologous to zero in M . Consider the cobordism ( X + , Ω + ) and argue as in (1) with the understanding that J + is adapted to thesymplectization at the negative end. Again, we can glue the holomorphic curves in R × [0 , × f W ∧ that are involved in (*) to the constant sections x × B + in ( X + , Ω + ) to get an index one family of curves. Algebraically over Z , there is no possiblebreaking for this family other than a breaking at −∞ , involving periodic orbits ofthe Reeb vector field R as well as pseudoholomorphic curves in the symplectizationof ( M, ξ ) without negative ends. Those provide a bounding chain for the orbits of R in M . (cid:3)
8. E
XAMPLES OF NON -L IOUVILLE - FILLABLE CONTACT STRUCTURES
Right-veering symplectomorphisms.
In this subsection we give two higher-dimensional generalizations of a right-veering surface diffeomorphism and discussthe non-Liouville-fillability of contact manifolds which admit open book decom-positions with non-right-veering monodromy.Let S be a bordered surface , i.e., a compact connected oriented surface withnonempty boundary, and let Diff + ( S, ∂S ) be the set of orientation-preserving dif-feomorphisms of S that restrict to the identity on ∂S . According to [HKM], anelement h ∈ Diff + ( S, ∂S ) is right-veering if h takes every arc a ⊂ S with bound-ary on ∂S to itself or “to the right”, after isotopy.Let ( W, β ) be a Liouville domain. Definition 8.1.1 (Locally right-veering) . A symplectomorphism h ∈ Symp(
W, ∂W, dβ ) is locally right-veering if there is a collar neighborhood [ − ǫ, s × ∂W of ∂W = { } × ∂W such that h = φ − s on { s } × ∂W , s ∈ [ − ǫ, , where φ s is the time- s flow of the Reeb vector field of β | ∂W . In what follows we assume that h ∈ Symp(
W, ∂W, dβ ) is locally right-veering. Given an exact Lagrangian submanifold a ⊂ W with Legendrian boundary ∂a ⊂ ∂W , we write HF ( W, h ( a ) , a ) for the Lagrangian Floer cohomology ofthe pair ( h ( a ) , a ) subject to a clean intersection condition along ∂a = ∂h ( a ) . Let x a be the “contact class”, i.e., the top generator corresponding to the Morse-Bottfamily ∂a .The following is a straightforward higher-dimensional generalization of a right-veering surface diffeomorphism: Definition 8.1.2 (Right-veering) . Let h ∈ Symp(
W, ∂W, dβ ) be locally right-veering.(1) h is strongly right-veering if there exists no exact Lagrangian submani-fold a ⊂ W with Legendrian boundary ∂a ⊂ ∂W such that [ x a ] = 0 ∈ HF ( W, h ( a ) , a ) (i.e., h sends a to the left ). IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 37 (2) h is weakly right-veering if there exists no regular Lagrangian disk a ⊂ W with Legendrian boundary ∂a ⊂ ∂W such that [ x a ] = 0 ∈ HF ( W, h ( a ) , a ) .If we want to specify the coefficient ring R for the Floer homology groups, we saystrongly or weakly R -right-veering. By a regular Lagrangian disk (cf. [EGL]) we mean a Lagrangian disk whichcan be completed to a Lagrangian basis with respect to some ( W, β ′ , φ ′ ) that isLiouville homotopic to ( W, β ) . Note that strongly right-veering implies weaklyright-veering. Question . Is there a difference between strongly right-veering and weaklyright-veering?
Remark . Allowing individual Lagrangians in the definition of d HF , it is im-mediate that d HF ( h ( a ) , a ) = HF ( h ( a ) , a ) , where the left-hand side is the Hee-gaard Floer group using the cylindrical reformulation and the right-hand side is theusual Floer cohomology group.We first prove the following warm-up theorem: Theorem 8.1.5. If ( M, ξ ) is overtwisted, then ξ admits a supporting open book de-composition ( W, β, φ ; h ) for which h is not weakly right-veering and c ( W, β, φ ; h ) =0 in d HF ( W, β, φ ; h ) . In this paper “overtwisted” means overtwisted in the sense of Borman-Eliashberg-Murphy [BEM]. Theorem 8.1.5 and Theorem 1.1.3 imply the well-known fact thatovertwisted contact structures are not Liouville fillable and satisfy the Weinsteinconjecture.
Proof.
According to Casals-Murphy-Presas [CMP], an overtwisted contact struc-ture is supported by an open book decomposition ( W ′ , β ′ , φ ′ ; h ′ ) which is a neg-ative stabilization of some Weinstein open book ( W, β, φ ; h ) . Using the notationfrom Section 2.3, we have a basis a = a ∪ · · · ∪ a κ , a Lagrangian disk c ⊂ W with Legendrian boundary which is disjoint from ∂ a , a handle W attached along ∂c , and γ = c ∪ c ′ , where c ′ is the core of W . Then h ′ = ( h ∪ id W ) ◦ τ − γ . We let a be the cocore of W . Then ( W ′ , β ′ , φ ′ ) has basis a ′ = a ∪ · · · ∪ a κ .We now define a special Hamiltonian deformation of α = e a in a given Wein-stein neighborhood N ( α ) ≃ T ∗ α equipped with the projection π : T ∗ α → α .To that end, we consider a Morse function f : α → R with four critical points x , x ′ , y and z , where x , x ′ have index , y is an index saddle point betweenthem, and z has index n . We may assume that:(1) x , y , and z are contained in α − a ;(2) the trajectory of ∇ f from x to y is entirely contained in α − a ;(3) x ′ = α ∩ γ and the intersection T ∗ α ∩ γ is the fiber of T ∗ α over x ′ .We define F : ( f W ′ ) ∧ → R to be π ∗ f on T ∗ α and on a slightly biggerneighborhood and let φ ǫ be the time- ǫ flow of the Hamiltonian vector field X F . The monodromy e h ′ on f W ′ sends α to τ γ ( φ − ǫ ( α )) . In particular the intersection e h ′ ( α ) ∩ α consists of three points: x , y , and z , where the pair x and y canbe eliminated by a Hamiltonian isotopy.Let y = { y , x , . . . , x κ } and x = { x , . . . , x κ } . We claim that ± d y = x when ǫ > is sufficiently small and J ♦ is suitably chosen. Indeed, by Lemma 7.3.1, fora well-chosen J ♦ there is no nonconstant connected holomorphic curve that has acontact coordinate at the positive end. Hence any curve from y is a collection of κ trivial strips from x , . . . , x κ to itself, together with an index curve from y to a point in e h ′ ( α ) ∩ α , that must therefore be x . We have now reduced thecalculation to a standard Floer homology calculation of dy in CF ( e h ′ ( α ) , α ) .Since x and y can be canceled by a Hamiltonian isotopy, HF ( e h ′ ( α ) , α ) isgenerated by z and ± dy = x . In other words, h ′ sends a to the left. Thisimplies the claim and hence the theorem. (cid:3) Proof of Theorem 1.1.4.
Follows immediately from the proofs of Theorems 1.1.3and 8.1.5. Note that in the proof of Theorem 1.1.3 it does not matter whether wetake a full Lagrangian basis or just a single exact Lagrangian. (cid:3)
The following theorem gives examples of symplectomorphisms that are notweakly right-veering.
Theorem 8.1.6. If h ∈ Symp(
W, ∂W, dβ ) is a product of negative symplecticDehn twists, then h is not weakly R -right-veering. We remark that it is currently not known whether the corresponding contactmanifold ( M, ξ ) is overtwisted. Proof.
In this proof the Floer homology groups are over R .Since the proof of Lemma 7.3.2 also applies to a single exact Lagrangian a ,it suffices to show that if h is a single negative symplectic Dehn twist τ − S alonga Lagrangian sphere S , then there exist a regular exact Lagrangian a such that [ x a ] = 0 ∈ HF ( h ( a ) , a ) .We use the fact that, since the basis a = { a , . . . , a κ } generates the wrappedFukaya category of W by [CDGG], there exists i ∈ { , . . . , κ } for which theLagrangian Floer homology HF ( S, a i ) = 0 . (Note that since S is compact, thereis no difference between the wrapped and unwrapped groups involving S .) Up toreordering, we assume a = a has this property.Next, switching to the Heegaard decomposition perspective, we show that theelement [ x a ] = [ x ] ∈ HF ( τ − S ( e a ) , e a ) is zero: This is a consequence of Seidel’sexact triangle [Se1]: HF ( e a ′ , e a ) L ∗ / / HF ( τ − S ( e a ′ ) , e a ) u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ HF ( S, e a ) ⊗ HF ( e a ′ , S ) µ i i ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 39 where e a ′ is the small Hamiltonian deformation of e a as given in Section 2.2. Notethat HF ( S, a ) and HF ( S, e a ) are canonically isomorphic. In this triangle, the map L ∗ is the one given by the Lefschetz cobordism over the strip with one singularfiber and whose vanishing cycle is S . Hence it maps the contact class in HF ( e a ′ , e a ) to the contact class in HF ( τ − S ( e a ′ ) , e a ) by the analog of Lemma 7.3.2. The latter iszero if the contact class in HF ( e a ′ , e a ) is in the image of µ , as defined in Section 4.From now on we make crucial use of R -coefficients. For e a ′ sufficiently close to e a , the intersection points e a ′ ∩ S and e a ∩ S are canonically identified and we refer toeither by { y , . . . , y s } . This gives us an identification CF ( e a ′ , S ) ≃ CF ( S, e a ) as vector spaces. We consider the standard inner products on CF ( e a ′ , S ) and CF ( S, e a ) for which { y , . . . , y s } is an orthonormal basis and identify both withan inner product space ( V, h , i ) . The differentials on CF ( e a ′ , S ) and CF ( S, e a ) canbe written as adjoints d and d ∗ on V satisfying(8.1.1) h d ∗ y i , y j i = h y i , dy j i . Taking an orthogonal decomposition ker d = Im d ⊕ (Im d ) ⊥ , there exists a nonzero y ∈ (Im d ) ⊥ ⊂ V such that dy = 0 since HF ( S, e a ) ≃ HF ( e a ′ , S ) = 0 . By Equa-tion (8.1.1), y ∈ (Im d ) ⊥ implies that d ∗ y = 0 .We now claim that µ ( y, y ) is a positive multiple of x . Writing y = P si =1 a i y i ,where a i ∈ R , we show that (i) µ ( y i , y i ′ ) = 0 for i = i ′ , (ii) µ ( y i , y i ) countsa single thin triangle T i which corresponds to a Morse gradient trajectory on e a from y i to x a , viewed as the top generator of CF ( e a ′ , e a ) , and (iii) there exists anorientation system such that µ ( y i , y i ) = ± , where all the signs are same. Herewe are considering upward gradient trajectories with respect to a Morse function f on e a whose maximum is the generator x a and whose minimum = y , . . . , y s is in int a .(i) and (ii) are consequences of taking the limit e a ′ → e a . Given a sequence ofcurves u j that are counted in µ ( y i , y i ′ ) (where i, i ′ may be equal) with e a ′ → e a ,the limit u ∞ consists of a (possibly broken) holomorphic strip v with boundaryon e a and S between y i and y i ′ , together with a gradient trajectory γ to the criticalpoint x a of f . If v is nontrivial, then ind( v ) ≥ by regularity. This implies that ind( u j ) ≥ , a contradiction. If v is trivial, then u ∞ is a single gradient trajectoryand u j are curves of the form (ii).(iii) Referring to Section 3.7 (and using the open book perspective), we can takeneighborhoods N i ⊂ f W of the thin triangles T i and diffeomorphisms ψ ij : N i ∼ → N j taking T i to T j , N i ∩ a to N j ∩ a , N i ∩ a ′ to N j ∩ a ′ , and N i ∩ S to N j ∩ S ;and such that all the orientation data for (i.e., used in defining the sign/orientationof) T i is taken to the orientation data for T j . This guarantees that all the curvescounted in (ii) have the same sign.Therefore, working over R -coefficients, µ ( y, y ) = ± ( P i a i ) x a and there ex-ists an element z ∈ CF ( τ − S ( e a ) , e a ) such that dz = x a . By Lemma 7.3.1, d { z, x , . . . , x κ } = c ( W, β, φ ; τ − S ) ∈ d CF ( W, β, φ ; τ − S ) , which implies the theorem. (cid:3)
9. V
ARIANT OF SYMPLECTIC K HOVANOV HOMOLOGY
Definitions.
Let f W be the -dimensional Milnor fiber of the A κ − singular-ity and let e p : f W → e D ⊂ C z be the standard Lefschetz fibration over the square e D = {− ≤ Re z, Im z ≤ } with regular fiber A = S × [ − , and κ critical values e z = { z , . . . , z κ } ,where Re z i = Re z i + κ , Im z i = − , and Im z i + κ = 1 for i = 1 , . . . , κ . Let p : W := e p − ( D ) → D be its restriction to D = e D ∩ { Im z ≤ } . Let e a = { e a , . . . , e a κ } be the “basis” of κ Lagrangian spheres over the κ disjoint arcs { e γ , . . . , e γ κ } , where e γ i is the straightline segment connecting z i to z κ + i . The restrictions of e z , e a i and e γ i to p are denoted z , a i and γ i .Let h σ ∈ Symp(
W, ∂W ) be the monodromy on W which descends to a braid σ , viewed as an element of Diff + ( D, ∂D, z ) (an orientation-preserving diffeomor-phism of D which is the identity on ∂D and takes z to itself setwise), let b σ be thebraid closure of σ , and let e h σ be the extension of h σ to f W by the identity.Let h e a , e h σ ( e a ) be given by (3.5.1), let { T δ , y } be a complete set of capping surfacesfrom Definition 3.5.1, and let(9.1.1) A = A e a , e h σ ( e a ) = H ([0 , × f W , ( { } × e a ) ∪ ( { } × e h σ ( e a )); Z ) as in (3.5.3).Our variant CKh ♯ ( b σ ) of the symplectic Khovanov (co)chain complex is definedas the higher-dimensional Heegaard Floer (co)chain complex d CF ( f W , e h σ ( e a ) , e a ) with coefficients F [ A ] J ~ , ~ − ] (this means power series in ~ and polynomial in ~ − ). We write Kh ♯ ( b σ ) for the homology d HF ( f W , e h σ ( e a ) , e a ) . Here F [ A ] is thegroup ring over A which we determine to be isomorphic to Z r − in Lemma 9.2.3,where r is the number of connected components of b σ .After discussing coefficients in Section 9.2, we prove Theorem 1.2.1 in Sec-tions 9.3 and 9.4. This entails proving the invariance of d HF ( f W , e h σ ( e a ) , e a ) underhandleslides and Markov stabilizations.9.2. Coefficients.
We explain why we can use coefficients F [ A ] J ~ , ~ − ] insteadof the Novikov ring and also describe F [ A ] .Given κ -tuples y and y ′ of intersection points of e a and e h σ ( e a ) , consider themoduli space M J ( y , y ′ ) , where J is the product J R × [0 , × J f W ∧ such that the pro-jection e p : f W ∧ → C is ( J f W ∧ , i ) -holomorphic. Let J ♦ ∈ J ♦ reg be a perturbationof J . Claim 9.2.1.
For all u : ˙ F → R × [0 , × f W ∧ in M J ( y , y ′ ) , the images underthe projection e p ◦ π f W ∧ ◦ u to C are the same when viewed as weighted domains inthe usual Heegaard Floer sense. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 41
Proof.
Write { y λ (1) , . . . , y κλ ( κ ) } and { y ′ µ (1) , . . . , y ′ κµ ( κ ) } for the projections of y and y ′ to C , where λ , µ are permutations of (1 , . . . , κ ) and y kλ ( k ) ∈ e γ k ∩ σ ( e γ λ ( k ) ) and y ′ kµ ( k ) ∈ e γ k ∩ σ ( e γ µ ( k ) ) . Then e p ◦ π f W ∧ ◦ u | ∂F is theunion of:(1) the unique path in e γ k from y kλ ( k ) to y ′ kµ ( k ) for k = 1 , . . . , κ , and(2) the unique path in σ ( e γ µ ( k ) ) from y ′ kµ ( k ) to y λ − ( µ ( k )) µ ( k ) for k = 1 , . . . , κ ,and e p ◦ π f W ∧ ◦ u | ∂F uniquely determines the weighted domain that it bounds. (cid:3) Lemma 9.2.2.
For fixed y , y ′ , and χ , M χ, ind=1 J ♦ ( y , y ′ ) / R is finite.Proof. Claim 9.2.1 implies area bounds for u ∈ M J ( y , y ′ ) but not necessarilyEuler characteristic bounds. Hence, after perturbing J to J ♦ , the desired curvecount M χ, ind=1 J ♦ ( y , y ′ ) / R is finite. (cid:3) Lemma 9.2.2 justifies the use of the coefficient system F [ A ] J ~ , ~ − ] . Lemma 9.2.3. A e a , e h σ ( e a ) ≃ Z r − , where r is the number of connected componentsof b σ . We also remark that A e a , e h σ ( e a ) = A ′ e a , e h σ ( e a ) since there is no torsion. Proof.
Writing X = [0 , × f W , Y = ( { } × e a ) ∪ ( { } × e h σ ( e a )) , and i : Y → X for the inclusion, the relative homology sequence gives: A e a , e h σ ( e a ) = H ( X, Y ) ≃ H ( X ) /i ∗ H ( Y ) . Now H ( X ) ≃ Z κ − and is generated by a chain of κ − spheres lying over achain of arcs in D ′ and one can compute the quotient H ( X ) /i ∗ H ( Y ) by itera-tively collapsing each pair of points connected by an arc e γ i or σ ( e γ i ) to a point. (cid:3) Proof of invariance under arc slides.
Given { e γ , . . . , e γ κ } , let c ⊂ e D be achord from e γ to e γ which does not intersect any other e γ i . Let e γ ′ be obtainedby arc sliding e γ over e γ along c and let e γ ′ i , i = 2 , . . . , κ , be a pushoff of e γ i which fixes z i , z i + κ ; see Figure 5. Let e a ′ = { e a ′ , . . . , e a ′ κ } be the Lagrangian basiscorresponding to { e γ ′ , . . . , e γ ′ κ } .Let Θ i (resp. Ξ i ), i = 1 , . . . , κ , be the unique intersection point of e a i and e a ′ i thatlies over z i (resp. z i + κ ), and let Θ = { Θ , . . . , Θ κ } and Ξ = { Ξ , . . . , Ξ κ } . Theorem 9.3.1. d CF ( f W , e h σ ( e a ) , e a ) and d CF ( f W , e h σ ( e a ′ ) , e a ′ ) are quasi-isomorphic. This is expected by [SS, Lemma 49], but there is some benefit in carrying outsome holomorphic curve calculations in preparation for the next section. Our proofclosely follows the handleslide invariance proof of the original version of HeegaardFloer homology [OSz1]. Θ Θ Ξ Ξ e γ ′ e γ ′ e γ e γ F IGURE
5. Arc sliding e γ over e γ . Proof.
Omitting f W from the notation, it suffices to prove that d CF ( e h σ ( e a ) , e a ) and d CF ( e h σ ( e a ) , e a ′ ) are quasi-isomorphic since the quasi-isomorphism of d CF ( e h σ ( e a ) , e a ′ ) and d CF ( e h σ ( e a ′ ) , e a ′ ) is proved analogously.We claim that Θ is a cycle in d CF ( e a ′ , e a ) and Ξ is a cycle in d CF ( e a , e a ′ ) . Forsimplicity let κ = 2 . The Fredholm index differences are:(9.3.1) ind( Θ , Ξ ) = 4 , ind( Θ , { Θ , Ξ } ) = ind( Θ , { Θ , Ξ } ) = 2 . We will treat the first case; the other cases are easier. A holomorphic map u :˙ F → R × [0 , × W that limits to Θ and Ξ at the positive and negative endssatisfies ind( Θ , Ξ ) = µ ( u ) by Lemma 3.4.1. Let A and B be the outer and innercomponents of e D that are bounded by e γ , e γ , e γ ′ , e γ ′ . Then the projection of u to e D has degree over A and degree over B and regardless of the number of branchpoints of this projection we have µ ( u ) = 4 . Equation (9.3.1) implies that Θ iscycle; the situation for Ξ is similar.In view of the claim we can define the cochain map Φ : d CF ( e h σ ( e a ) , e a ) → d CF ( e h σ ( e a ) , e a ′ ) , (9.3.2) y µ ( Ξ ⊗ y ) , where µ is the product map µ : d CF ( e a , e a ′ ) ⊗ d CF ( e h σ ( e a ) , e a ) → d CF ( e h σ ( e a ) , e a ′ ) . Similarly we have the cochain map
Ψ : d CF ( e h σ ( e a ) , e a ′ ) → d CF ( e h σ ( e a ) , e a ) , y µ ′ ( Θ ⊗ y ) , where µ ′ is the product map µ ′ : d CF ( e a ′ , e a ) ⊗ d CF ( e h σ ( e a ) , e a ′ ) → d CF ( e h σ ( e a ) , e a ) . Composing the two maps corresponds to degenerating the base D in one way.We now degenerate the base in the “other way” as in Figure 6.Again assuming that κ = 2 we consider the moduli space M J ( Ξ , Θ ) , where the symplectic fibration and Lagrangian boundary conditions are: π : R × [0 , × f W ∧ → R × [0 , , L = R × { } × e a ′ , L = R × { } × e a , IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 43 e h σ ( e a ) e a e a e a ′ ΘΞ F IGURE J = J R × [0 , × J f W ∧ . (To avoid cumbersome notation, we will not puta hat over e p .) We will use the following modifiers: • χ = j , which means the domain ˙ F of the holomorphic map u has Eulercharacteristic j ; • w = { w , w } , which means that u passes through the points (0 , , w ) and (0 , , w ) , where (0 , ∈ R × [0 , and w i ∈ e a i .Since the projection of u to C is a holomorphic map which is a degree mapto region A and a degree map to region B , where A and B are as before and ind( Ξ , Θ ) = µ ( u ) = 4 , the only type of degeneration as in Figure 6 is if theleft-hand side had ind = 4 and the right-hand side had ind = 0 .By Theorem 9.3.7 below, the count of χ = 0 curves from Ξ to Θ (i.e., annuli)passing through a generic w = { w , w } (by this we mean passing through a pairof points (0 , , w ) and (0 , , w ) with w , w generic) is ± . Although it wouldbe interesting to do, we do not calculate the contributions from χ < . Thus Ψ ◦ Φ is chain homotopic to ~ · p ( ~ ) times the identity, where p ( ~ ) is a power series in ~ whose constant term is ± . (Note that the fact that Ψ ◦ Φ is chain homotopic to ~ · p ( ~ ) id forces us to work over F J ~ , ~ − ] .) The situation for Φ ◦ Ψ is analogous.This completes the proof of Theorem 9.3.1, modulo a discussion of coefficientsand the rather nontrivial calculations involved in the proof of Theorem 9.3.7. (cid:3) Coefficients.
We relate h e a , e h σ ( e a ) and A e a , e h σ ( e a ) for the pair ( e h σ ( e a ) , e a ) withthose for ( e h σ ( e a ) , e a ′ ) .First consider the map Ξ ∗ : h e a , e h σ ( e a ) → h e a ′ , e h σ ( e a ) , [ δ ] [ Ξ δ ] , where Ξ δ is obtained from Ξ , viewed as an κ -tuple of chords on [ , × f W , and δ , viewed as an κ -tuple of chords on [0 , ] × f W , by concatenating with an κ -tupleof connecting arcs on { } × e a . Since each component of e a is simply-connected, [ Ξ δ ] does not depend on the choice of connecting arcs. The map Ξ ∗ is a bijectionsince there is an analogously defined inverse Θ ∗ : h e a ′ , e h σ ( e a ) → h e a , e h σ ( e a ) , [ δ ′ ] [ Θ δ ′ ] . Next we choose complete sets of capping surfaces { T δ , y } for the pair ( e h σ ( e a ) , e a ) , { T Ξ ∗ δ , y ′ } for the pair ( e h σ ( e a ) , e a ′ ) , and { T δ , Ξ , Ξ ∗ δ } for the triple ( e h σ ( e a ) , e a , e a ′ ) . The map Φ given by Equation (9.3.2) can then be defined with coefficients F [ A e a , e h σ ( e a ) ] ≃ F [ A e a ′ , e h σ ( e a ) ] in view of the following lemma: Lemma 9.3.2. A e a , e h σ ( e a ) ≃ A e a ′ , e h σ ( e a ) .Proof. Let us write X = [0 , × f W , Y = ( { }× e a ) ∪ ( { }× e h σ ( e a )) , Y ′ = ( { }× e a ′ ) ∪ ( { }× e h σ ( e a )) , and i : Y → X , i ′ : Y ′ → X for the inclusions. We have i ∗ H ( Y ) = i ′∗ H ( Y ′ ) since e γ ′ is obtained from e γ by arc sliding over e γ . Hence A e a , e h σ ( e a ) = H ( X, Y ) ≃ H ( X ) /i ∗ H ( Y ) ≃ H ( X ) /i ′∗ H ( Y ′ ) ≃ H ( X, Y ′ ) = A e a ′ , e h σ ( e a ) . (cid:3) Holomorphic curves after Lagrangian surgery.
As preparation for Theo-rem 9.3.7, we state a useful fact which is a restatement of [FOOO2, Theorem55.11]. Let b , . . . , b m be transversely intersecting Lagrangians in c W that are ei-ther compact or with cylindrical Legendrian ends and project to arcs under the map b p : c W → C .Let M ( y , . . . , y m ) be a moduli space of holomorphic maps v : D m = D − { p , . . . , p m } → ( c W , J c W ) , such that:(1) ∂ i D m , i = 0 , . . . , m , is mapped to b i ;(2) p i , i = 0 , . . . , m , corresponds to the intersection point y i ∈ b i − ∩ b i ,where i is viewed modulo m + 1 ;(3) the intersection point y lies above a singular value z of b p ;(4) the p i may be allowed to vary or they may be fixed, and the maps v may ormay not have point constraints;(5) M ( y , . . . , y m ) is compact; and(6) all the curves of M ( y , . . . , y m ) project to thin sectors between b m and b near p as shaded in Figure 7.Here we are using the notation from the beginning of Section 4.Now there are two types of Lagrangian surgery operations at p ; the resultsare called b + and b − , corresponding to the right and middle pictures of Figure 7.The modified moduli spaces M + ( y , . . . , y m ) and M − ( y , . . . , y m ) are the anal-ogously defined moduli spaces of maps v ± : D − { p , . . . , p m } → ( c W , J c W ) such that (setting b ± = b = b m ):(1) the counterclockwise arc component of ∂ ( D − { p , . . . , p m } ) connecting p i and p i +1 is mapped to b i for i = 1 , . . . , m ;(2) p i , i = 1 , . . . , m , corresponds to the intersection point y i ∈ b i − ∩ b i ; IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 45 (3) if the points p , . . . p m were fixed for M ( y , . . . , y m ) , then p , . . . , p m arealso for M ± ( y , . . . , y m ) ; if there were point constraints for M ( y , . . . , y m ) ,then the same point constraints are used for M ± ( y , . . . , y m ); and(4) all the curves of M ± ( y , . . . , y m ) project to the shaded regions in Figure 7.F IGURE
7. The result of performing the two different kinds ofLagrangian surgery. The picture on the left shows the intersectionof b m and b , and the pictures in the middle and on the right are b − , b + . Theorem 9.3.3 (Fukaya-Oh-Ohta-Ono) . Suppose J c W has a specific form on asmall neighborhood of y and M ( y , . . . , y m ) is regular. Then M ± ( y , . . . , y m ) are regular and there are maps θ ± : M ± ( y , . . . , y m ) → M ( y , . . . , y m ) , where θ − is a C -diffeomorphism and θ + is a C -double covering map.Remark . In general, if dim W = 2 n , then θ − is still a C -diffeomorphismbut θ + is a fiber bundle with fiber S n − . Remark . Theorem 9.3.3 is essentially a consequence of the following fact:Consider the Lefschetz fibration(9.3.3) b p : C z ,z → C ζ , ( z , z ) z z . Then the Clifford torus(9.3.4) T = {| z | = 1 } × {| z | = 1 } ⊂ C is a Lagrangian that “lies over” {| ζ | = 1 } ⊂ C . There exist two holomorphic disks u in C with boundary on T that pass through a generic point ( a , a ) of T andsuch that b p ◦ u is a degree map to {| ζ | < } . They are of the form z ( z, a ) and z ( a , z ) .9.3.3. Warm-up problem.
We will first treat a slightly easier warm-up problem.Consider Figure 8, which is half of Figure 5 and corresponds to the Lefschetzfibration p : W → D . Strictly speaking, we are working in the cylindrical endsituation of b p : c W → C and γ i , γ ′ i , a i , a ′ i have been completed by attachingcylindrical ends. In general, the completion of ∗ will be denoted by b ∗ . Let c ij bethe shortest counterclockwise chord in ∂D that connects γ i to γ ′ j . Correspondingto each chord c ij there exists an S -Morse-Bott family of chords in ∂W ; in the perturbed version we consider we consider chords ˇ c ij and ˆ c ij which are the longerand shorter Reeb chords. We write c = { ˇ c , ˇ c , ˇ c , . . . , ˇ c κκ } . Θ Θ γ ′ γ ′ γ γ c c F IGURE κ = 2 , we consider the moduli space M J ♦ ( c , Θ ) , where the symplectic fibration and Lagrangian boundary conditions are: π : R × [0 , × c W → R × [0 , , L = R × { } × b a ′ , L = R × { } × b a , and we take J to be the product J R × [0 , × J c W such that the projection b p : c W → C is ( J c W , i ) -holomorphic. Theorem 9.3.6. M χ =1 , w J ♦ ( c , Θ ) = 1 mod for generic w .Proof. We write u for an element of M χ =1 , w J ♦ ( c , Θ ) and v for its projection π c W ◦ u :˙ F → c W , where F is a unit disk in C .When χ = 1 , there are two types of maps v : • Type int (for “interior”) which has a branch point b that maps to the inte-rior of B ; • Type ∂ (for “boundary”) which does not have a branch point that maps tothe interior of B .We claim that Type ∂ falls into two subcases: ∂ and ∂ , where for v of Type ∂ (resp. ∂ ) there are two “switch points” b , b ∈ ∂ ˙ F (they may be the same point)that both map to b γ ′ (resp. both map to b γ ) and where b p ◦ v | ∂ ˙ F switches directionsalong b γ ′ (resp. b γ ). This a consequence of the fact that there is only one component ζ of ∂ ˙ F that maps to b γ ′ (resp. b γ ) and v ( ζ ) must connect the terminal point of c to b p (Θ ) (resp. b p (Θ ) to the initial point of c ).If b = b , let b be the point that is closer to the puncture corresponding to c (resp. c ) along the arc of ∂ ˙ F that maps to b γ ′ (resp. b γ ).The calculation of M χ =1 , w J ♦ ( c , Θ ) does not depend on the choice of generic w since c and Θ are closed. We therefore choose w as follows, subject to genericity: This appears to be the opposite of the previous usage. Our rule is to put a check on thelower-degree generator in the appropriate Floer cohomology group, which in the current situationis d CF ( b a , b a ′ ) . IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 47 (R1) w ∈ b a so that its projection b p ( w ) to b γ is close to z ;(R2) w ∈ b a so that its projection b p ( w ) to b γ satisfies | b p ( w ) | ≫ .We also assume that:(R3) the region in C between b γ ′ and b γ corresponding to B is an arbitrarily thinstrip whose width is w . Order of choice of w , w , w . This will become important when taking limits usingGromov compactness:(S1) First choose w satisfying (R2) and an interval [0 , so that | b p ( w ) − z | ∈ [0 , . (S2) Then choose w = w ( w ) for (R3).(S3) Finally choose w satisfying (R1).We also remark that the proof of the regularity of any explicit holomorphiccurves with constraints that we use is straightforward and is omitted. Step 1.
We first apply Theorem 9.3.3 to push the holomorphic curve v off of thecritical value z ; see Figure 9. The mod curve counts remain the same as longas no point constraints are prescribed at the clean S -intersection on the right-handside of the diagram. F IGURE Step 2.
Fix w with b p ( w ) large and w = w ( w ) small. We write ι = Im ◦ b p , where Im refers to the imaginary part. We claim that ι ◦ v ( b ) ≫ ι ( w ) or ι ◦ v ( b ) ≫ ι ( w ) for | b p ( w ) − z | small. (Keep in mind the order (S1)–(S3).) We will treat the b case; the b case is similar. We use the notation q (Θ ) to denote the point on ∂F that “maps to” Θ , etc. See Figure 10. Θ w ˇ c ˇ c w Θ F IGURE
10. The location of the points q (Θ ) , etc. on ∂F , assum-ing ι ◦ v ( b ) , ι ◦ v ( b ) ≫ ι ( w ) does not hold. Here q is omittedfrom the notation. Arguing by contradiction, if the claim does not hold, then there exists a sequence { v i : ˙ F i → c W } ∞ i =1 of v ’s with respect to a sequence w ( i )1 of w ’s approaching Θ such that all the ι ◦ v i ( b ( i ) ) are bounded above, where b ( i ) is b for v i . By Gromovcompactness, there exists a subsequence that converges to v ∞ : ˙ F ∞ → c W with abranch point b ∞ that is a subsequential limit of b ( i ) . There also exist δ > and anormalization of the points on ∂F ∞ such that(i) q (ˇ c ) = 1 , q (ˇ c ) = − , q (Θ ) = − i ;(ii) | q (Θ ) − q (ˇ c ) | , | q (Θ ) − q (ˇ c ) | > δ ;(iii) | q ( w ) − q (Θ ) | > δ .On the other hand, (i)–(iii) contradicts the requirement that there be an involu-tion of F i for i ≫ taking(9.3.5) q (Θ ) q (Θ ) , q (ˇ c ) q (ˇ c ) , q ( w ) q ( w ) . This is because | q ( w ( i )1 ) − q (Θ ) | → as i → ∞ . (Recall that such an involutionexists if and only if there is a fractional linear transformation of the disk F i whichfixes q (ˇ c ) and q (ˇ c ) and simultaneously moves q (Θ ) and q (Θ ) to antipodalpoints on ∂F i and q ( w ) and q ( w ) to antipodal points on ∂F i .) Step 3.
We claim that M χ =1 , w ,♯J ♦ ( c , Θ ) = 0 mod , where ♯ means we arecounting curves that satisfy ι ◦ v ( b ) ≫ ι ( w ) or ι ◦ v ( b ) ≥ ι ( w ) − C, for C > a fixed constant. Recall our convention is that ι ◦ v ( b ) ≥ ι ◦ v ( b ) .Also ι ◦ v ( b ) ≫ ι ( w ) by Step 2.We apply SFT-type stretching with ι ( w ) → ∞ and count -level curves asgiven on the left-hand side of Figure 11. The components are labeled , , andany v ∈ M χ =1 , w ,♯J ♦ ( c , Θ ) is close to breaking into v (1) ∪ v (2) ∪ v (3) . (The casedrawn is for Type int , but v (3) may also have two switch points b and b .) Type ∂ Type ∂ b b b b b ˇ c ˇ c ˇ c ˇ c ˇ c ˇ c ˇ c ˇ c ˇ c w w w w w w F IGURE
IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 49
The chords at the negative ends of v (3) must match the chords at the positive endsof v (1) and v (2) . By Theorem 9.3.3, the algebraic count of curves v (1) satisfyingthe point constraint w is mod ; in particular the set of chords at the positive endof v (1) is finite. Since ι ◦ v ( b ) , ι ◦ v ( b ) ≫ ι ( w ) , there is a portion of v (3) passingthrough w that is close to a curve corresponding to a gradient trajectory through w ; this effectively constrains the positive end of v (2) to a single chord.We can therefore convert the count of v (3) to the count of ν ∈ M χ =1 J ♦ ( c , c ′ ) thatsatisfy the following: The symplectic fibration is π : R × [0 , × ( C × T ∗ S ) → R × [0 , , and the Lagrangian boundary conditions are L = R × { } × a ′ ⋆ , L = R × { } × a ⋆ , where a ⋆ = { x = 1 } × S , a ⋆ = { x = 0 } × S ,a ′ ⋆ = { x = − } × S , a ′ ⋆ = { x = − ǫ } × S , are in C × T ∗ S , ǫ > is small, and S is the zero section of T ∗ S . Also thepositive and negative ends of ν are c = { ˇ c , ˇ c } , c ′ = { ˇ c , ˇ c } , where ˇ c ij , ˆ c ij are the longer and shorter Reeb chords in the perturbed version ofthe S -Morse-Bott family of Reeb chords from a ⋆i to a ′ ⋆j . a ⋆ a ⋆ a ′ ⋆ a ′ ⋆ ˇ c ˇ c ˇ c ˇ c F IGURE
12. The fiber T ∗ S where the sides are identified.On the other hand, generically M χ =1 J ♦ ( c , c ′ ) = ∅ . We give two proofs: (P1)Applying Hamiltonian perturbations to S (on T ∗ S ) we can push off the a ⋆i and a ′ ⋆i as in Figure 12. We then note that there is no domain in the picture with allpositive weights which could be the projection of ν with positive corners ˇ c , ˇ c and negative corners ˇ c , ˇ c . (P2) Alternatively, we can verify that the Fredholmindex ind( ν ) = 0 ; this is consistent with ind( v (1) ) = 2 and ind( v (2) ) = 2 whenwe view v (2) as having ˇ c at the positive end.The claim then follows. Step 4.
We make one model calculation.
Any notation introduced here is limited tothis step.
Consider the Lefschetz fibration b p given by Equation (9.3.3). Let T be the Clif-ford torus {| z | = 1 } × {| z | = 1 } over | ζ | = 1 and let L be the Lagrangianthimble in C emanating from ζ = 0 along [ − ,
0] = {− ≤ Re( ζ ) ≤ , Im( ζ ) = 0 } . We note that for any ζ = re iθ ∈ b p ( T ∪ L ) , b p − ( ζ ) ∩ ( T ∪ L ) = {√ r ( e iφ , e i ( − φ + θ ) ) , φ ∈ [0 , π ] } . Let µ be the S -family of intersections T ∩ L .Let M w J be the moduli space of holomorphic maps u = ( u , u ) : R σ × [0 , τ → C with respect to the standard complex structure J satisfying the following:(B1) u ( R × { } ) ⊂ T and u ( R × { } ) ⊂ L ;(B2) u limits to points of µ as σ → ±∞ ;(B3) b p ◦ u has degree over {| ζ | ≤ }− b p ( T ∪ L ) and degree outside {| ζ | ≤ } ;(B4) u (0 ,
0) = w = ( w , w ) .In our current situation we take w = (1 , . Its boundary ∂ M w J consists ofpairs v ∪ c , where c is a trivial strip that maps to a point in µ and v : D → C is a disk bubble that passes through (1 , and such that b p ◦ v has degree over {| ζ | ≤ } . Recall that ∂ M w J = 2 by Remark 9.3.5 and that the two maps v areof the form z ( z, and z (1 , z ) . See Figure 13, which gives a schematicdescription of M w J . The left-hand side of each row represents the image of b p ◦ u or b p ◦ v in C and the right-hand side of each row represents the image of u or v in C . The top and bottom maps are v where v ∪ c ∈ ∂ M w J and the middle map isof the form ( g ( z ) , g ( z )) where g is a conformal map to the half-disk.Next pick a point e i ( π + ǫ ) with ǫ > small and consider the evaluation map(9.3.6) ev J : M w J → S | z | =1 , u w , where w = ( w , w ) is the unique point of intersection between u ( R × { } ) and b p − ( e i ( π + ǫ ) ) . While J is not necessarily regular, there exists a small perturbation J ♦ of J with an analogously defined moduli space M w J ♦ and an evaluation map ev J ♦ : M w J ♦ → S , such that M w J ♦ is transversely cut out and ∂ M w J = ∂ M w J ♦ .The condition ∂ M w J = ∂ M w J ♦ is possible because the maps v are transversely cutout by automatic transversality.We claim that, if z ∈ C = { e iθ | π + ǫ < θ < π } , then ev − J ♦ ( z ) = 1 modulo . The evaluation map over C is a degree map since: • The analogously defined ev − J ♦ values for v ∈ ∂ M w J ♦ are the endpoints e i ( π + ǫ ) and e πi of C . • For i = 1 , , consider the restriction u i ( s, of u i to R s × { } , notingthat u i ( s, ∈ S | z i | =1 . Since u ( s, , u ( s, , and u ( s, u ( s, allmonotonically rotate in the counterclockwise manner as s ∈ R increasesand u i (0 ,
0) = 0 , ev J ( M w J ) ⊂ C . By Gromov compactness, ev J ♦ ( M w J ♦ ) is contained in a small neighborhood of C . IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 51 w w w w w w w w w w w w w w w w z z z z F IGURE
13. A schematic description of M w J . The top and bot-tom rows represent curves v where v ∪ c ∈ ∂ M w J and the middlerow represents a curve u in the “middle of” M w J . All the circlesare unit circles. A solid circle indicates the boundary of a holo-morphic disk and a dotted circle with a red dot on it indicates aconstant map to . Step 5.
We claim that M χ =1 , w ,♭J ♦ ( c , Θ ) = 1 mod , where we are counting curvesof Type ∂ that satisfy(9.3.7) ι ◦ v ( b ) ≫ ι ( w ) but ι ◦ v ( b ) ≤ ι ( w ) − C, for C > fixed. We can apply SFT-type stretching and count -level curves asgiven on the middle and right-hand side of Figure 11. In view of (9.3.7), we maytake the long neck to be above ι ( w ) .We will treat the Type ∂ case; the Type ∂ case is complementary and analo-gous. Component corresponds to a gradient trajectory and there is a unique curvefrom ˇ c to Θ passing through w . Component 3 has ˇ c at the positive end and ˆ d and ˇ c at the negative end, where ˆ d is the shorter Reeb chord from a ′ to a ′ .There is a single such curve (modulo ) by a calculation analogous to that of Fig-ure 12: we perturb the T ∗ S -projections of the ends of a , a ′ , a ′ as in Figure 12and count a single holomorphic triangle with vertices ˇ c , ˆ d , ˇ c . Next for Com-ponent 1 we view ˆ d as a point constraint which corresponds to w in Step 4 ( w in Step 4 directly corresponds to w in our case). By Step 4, there is a single curve(modulo ) corresponding to Component which passes through w and “half” ofthe values of ˆ d (which in turn come from half of the values of w ). The reasonthe Type ∂ case is complementary is that the “other half” of the values of w aretaken care of by Type ∂ .Finally, still assuming we are in the Type ∂ case, we glue the components andimpose the involution condition (9.3.5) on F ; also refer to Figure 10. As we take ι ◦ v ( b ) → ∞ , q (Θ ) approaches q (ˇ c ) but | q ( w ) − q (Θ ) | ≪ | q ( w ) − q (ˇ c ) | .Hence, assuming w is sufficiently close to Θ , there is a single value of ι ◦ v ( b ) for which there exists an involution of F satisfying (9.3.5). This completes theproof of Theorem 9.3.7. (cid:3) Curve count.
Theorem 9.3.7. M χ =0 , w J ♦ ( Ξ , Θ ) = 1 mod for generic w .Proof. We write u : ˙ F → R × [0 , × f W ∧ for an element of M χ =0 , w J ♦ ( Ξ , Θ ) and v = v ( u ) for its projection to f W ∧ .We stretch the base D ′ in the Im z -direction, as given in Figure 14. This meansthat we are keeping Re z i = Re z i + κ fixed and taking Im z i = − K , Im z i + κ =2 K , for i = 1 , . . . , κ and K ≫ . The region R bounded by e γ ′ and e γ is dividedinto three regions R , R , and R , which are intersections of R with { Im z ≤− K } , {− K ≤ Im z ≤ K } , and { K ≤ Im z } . Θ Θ Ξ Ξ w w e γ ′ e γ e γ e γ ′ F IGURE
14. The horizontal direction is the Im z -direction. Theregions R , R , R are from left to right.Since the calculations do not depend on the choice of generic w , we choose w to satisfy the following, subject to genericity:(R1 ′ ) w ∈ e a so that its projection e p ( w ) to e γ is close to z ;(R2 ′ ) w ∈ e a so that its projection e p ( w ) to e γ is close to Im z = 0 .We also assume that:(R3 ′ ) the region in C between e γ ′ and e γ is an arbitrarily thin strip whose widthis w .Additional conditions will be imposed later.We classify the types of maps v by their branch points and switch points. Thetypes are , L , R , LL , LR , RR , where the number is the number of interiorbranch points and each occurrence of L (resp. R ) indicates two switch points thatmap to e γ ′ (resp. e γ ). If they exist, the interior branch points are enumerated by b , b ′ ∈ int( ˙ F ) and the switch points by b , . . . , b ∈ ∂ ˙ F . The switch points comein adjacent pairs ( b , b ) and ( b , b ) that map to the same e γ ′ i and we assume that ι ( b j +1 ) ≥ ι ( b j ) for j = 1 , ; and if there is more than one branch point we assumethat ι ( b ′ ) ≥ ι ( b ) . Here we are writing ι ( x ) = Im( e p ◦ v ( x )) for x ∈ ˙ F . Step 1.
Suppose K ≫ . We describe the limit of v ( i ) : F ( i ) → f W ∧ as w i → ,where w i is the width of the thin strip and u ( i ) ∈ M χ =0 , w J ♦ ( Ξ , Θ ) and v ( i ) = v ( u ( i ) ) are with respect to w i .After possibly passing to a subsequence the limit of v ( i ) is v ∞ ∪ δ + ∪ δ − , where IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 53 • v ∞ : F ∞ → f W ∧ is a holomorphic annulus, • δ + is a gradient trajectory on e a from Ξ to a point p + on Im v ∞ , and • δ − is a gradient trajectory on e a from a point p − on Im v ∞ to Θ .Here v ∞ could be degenerate in the sense that F ∞ is a disk with an interior point F ∞ that “behaves like” a boundary component of F ∞ and maps to e a . We also usethe following conventions: • If w lies on the gradient trajectory δ ± , then q ( w ) = q ( p ± ) ∈ ∂F ∞ . • q (Θ ) = q ( p − ) and q (Ξ ) = q ( p + ) on ∂F ∞ .In particular, in the limit it is possible for q ( w ) = q (Θ ) or q (Ξ ) . (In the “closeto breaking” picture for v ( i ) with i ≫ , q ( w ) is close to q (Θ ) or q (Ξ ) .)The limit of Figure 14 is given by Figure 15. Θ Θ Ξ Ξ w w F IGURE
15. The limit of Figure 14 as we take w i → . In thepicture, the red dot w is placed just below the slit, indicating that v ∞ passes through w along the lower branch of the slit. Thearrow indicates the possible locations of e p ◦ v ∞ ( q ( w )) .Let b ( i ) , b ′ ( i ) , b ( i ) j be b , b ′ , b j for w i small and let b ∞ , b ′ ∞ , b ∞ j be their limitsafter passing to a subsequence. We will often omit the superscripts ( i ) and ∞ .Step 2. We claim that, for K ≫ and w i small, we can restrict to the case where: max { ι ( b ′ ) , ι ( b ) , ι ( b ) } ≥ K and(9.3.8) min { ι ( b ) , ι ( b ) , ι ( b ) } ≤ − K, (9.3.9)and the left-hand sides of (9.3.8) and (9.3.9) correspond to the endpoints of the slitin the limit w i → .We argue as in Step 3 of Theorem 9.3.6. If (9.3.9) does not hold, then therestriction of v ∞ to R (for K ≫ ) can be viewed as a curve of the type describedin Remark 9.3.5. Since such curves come in pairs, the mod count is unaffectedby restricting to curves satisfying (9.3.9). The argument for (9.3.8) is similar. Step 3.
We show that, for K ≫ and w i small, M χ =0 , w , ∗ J ♦ ( Ξ , Θ ) = 0 mod , where ∗ = 2 , L, LL , or RR . By Step 2 we may restrict to the case where(9.3.8) and (9.3.9) hold. Hence in all the cases, v ∞ has a long slit along e γ ′ = e γ atleast from Im z = − K to Im z = K . Type . Note that v ∞ maps q (Θ ) and q (Ξ ) to the endpoints of the slit. Hence q (Θ ) cannot be close to q ( w ) , a contradiction, and v cannot be of Type . Type L or R . There are three cases. First assume that ι ( b ) ≤ ι ( b ) ≤ ι ( b ) .In this case ι ( b ) and ι ( b ) are the endpoints of the slit, q (Θ ) maps to ι ( b ) , and q (Ξ ) maps to ι ( b ) . If ι ( b ) ≥ , then the images of q (Θ ) and q ( w ) are too far, a contradiction. If ι ( b ) ≤ , then the images of q (Θ ) and q (Ξ ) are too close,which is also a contradiction.Next assume that ι ( b ) ≤ ι ( b ) ≤ ι ( b ) . In this case ι ( b ) and ι ( b ) are theendpoints of the slit and q (Θ ) and q (Ξ ) map to ι ( b ) . Hence the images of q (Θ ) and q (Ξ ) are too close, a contradiction.Finally assume that ι ( b ) ≤ ι ( b ) ≤ ι ( b ) , where q (Θ ) maps to ι ( b ) and q (Ξ ) maps to ι ( b ) . If ι ( b ) ≤ − , then the images of q (Θ ) and q ( w ) are too far, acontradiction. If ι ( b ) ≥ − , then the images of q (Θ ) and q (Ξ ) are too close,which is also a contradiction. Type LL or RR . We treat Type LL ; Type RR is similar. We may assumethat ι ( b ) ≤ ι ( b ) ≤ ι ( b ) ≤ ι ( b ) . This is because ι ( b ) < ι ( b ) is incompatible with v ∞ having degree over R and degree over the thin strip. Then ι ( b ) and ι ( b ) are the endpoints of the slit, q (Θ ) maps to ι ( b ) , and q (Ξ ) maps to ι ( b ) . If ι ( b ) ≤ − , then the images of q (Θ ) and q ( w ) are too far, a contradiction. If ι ( b ) ≥ − , then the images of q (Θ ) and q (Ξ ) are too close, which is also a contradiction.It remains to consider Type LR . Step 4.
Let K ≫ . We describe the limit v ∞ ∪ δ + ∪ δ − of v ( i ) as w i → (fromStep 1), where u ( i ) ∈ M χ =0 , w , LRJ ♦ ( Ξ , Θ ) .Suppose for v ( i ) the switch points b ( i )1 , b ( i )2 map to e γ ′ and the switch points b ( i )3 , b ( i )4 map to e γ . As before we are assuming ι ( b ( i )2 ) ≥ ι ( b ( i )1 ) , ι ( b ( i )4 ) ≥ ι ( b ( i )3 ) . By Step 2, we have:
Claim 9.3.8. v ∞ has a long slit along e γ ′ = e γ at least from Im z = − K to Im z = K with endpoints max( ι ( b ∞ ) , ι ( b ∞ )) , min( ι ( b ∞ ) , ι ( b ∞ )) . See Figure 16 for an example. The restriction C + of v ( i ) over the shaded regionconverges to the trajectory δ + ; there is an analogous curve C − converging to thetrajectory δ − . Θ Ξ b ( i )1 b ( i )2 b ( i )3 b ( i )4 b ∞ b ∞ b ∞ b ∞ F IGURE
16. A schematic picture of the switch points and the limitslit as i → ∞ . In the figure b ( i )1 is shorthand for e p ◦ v ( i ) ( b ( i )1 ) , forexample. IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 55
We also have:
Claim 9.3.9. q ( p + ) = (cid:26) b ∞ if ι ( b ∞ ) ≥ ι ( b ∞ ) , b ∞ if ι ( b ∞ ) ≥ ι ( b ∞ ) .q ( p − ) = (cid:26) b ∞ if ι ( b ∞ ) ≥ ι ( b ∞ ) , b ∞ if ι ( b ∞ ) ≥ ι ( b ∞ ) . The various possibilities for b ∞ j around the slit are given in Figure 17 and aredenoted by (A)–(D). b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ b ∞ (A)(B)(C)(D)F IGURE
17. The various possibilities for b ∞ j around the slit. Step 5.
We make one model calculation, which is a slightly more involved variantof Step 4 of Theorem 9.3.6. Let b p , T , L , M w J be as in Step 4 of Theorem 9.3.6.We will define two moduli spaces M θ and M of holomorphic maps R × [0 , → C , define the evaluation maps ev , ev : M θ , M → S × S = R / π Z × R / π Z , and determine their images. The fiber product of ev and ev , i.e., the set of pairs ( v ′ , v ′′ ) ∈ M θ × M such that ev ( v ′ ) = ev ( v ′′ ) with θ = − π + ˜ ǫ (shown to begiven by the intersection of the blue line and the pink region in Figure 18) will bethe model for the gluing of v ′∞ and v ′′∞ (cf. (9.3.12)) from Step 6. Step 5A.
For θ ∈ [0 , π ) , let M θ = M w J , where w = ( e iθ , e − iθ ) . Let w ± = e i ( π ± ǫ ) ∈ C ζ , where ǫ > is small, and let ev ± ( v ′ ) be the z -coordinate of ( b p ◦ v ′ ) − ( w ± ) , where v ′ ∈ M θ . We then define the map ev : M θ → S × S ,v ′ ( ev ( v ′ ) , ev − ( v ′ )) . The calculation of Step 4 of Theorem 9.3.6 (i.e., examining the -parameter fam-ily of maps in Figure 13 that depicts M θ =01 ) implies that ev ( M ) is homotopicrel endpoints to { (1 − t )( π + ǫ, π − ǫ ) + t (2 π, | t ∈ [0 , } . (9.3.10)Note that there is a diffeomorphism M ∼ → M θ , v ′ v ′ θ , where v ′ θ is obtained from v ′ by rotating the first component by θ and the secondcomponent by − θ . It follows that ev ( M θ ) = ev ( M ) + ( θ, θ ) . (9.3.11)We will take θ = − π + ˜ ǫ where ˜ ǫ > is small. See Figure 18. π π π π F IGURE
18. The torus S × S with coordinates ( θ , θ ) . Thesides are identified and the top and the bottom are identified. Thedotted line is the diagonal. The blue line represents ev ( M θ ) ,where θ = − π + ˜ ǫ , and the pink region represents ev ( M ) , aswe take ǫ → . Step 5B.
The moduli space M is the set of maps v ′′ : R × [0 , → C thatsatisfy (B1)–(B3) and(B4 ′ ) v ′′ nontrivially intersects ℓ := { ( − r, r ) | r ∈ [0 , } ⊂ L . (Note that ℓ isobtained by parallel transporting a point on L .)We will denote a point of intersection between v ′′ and ℓ by w = w ( v ′′ ) . Let ev ± ( v ′′ ) be the z -coordinate of ( b p ◦ v ′′ ) − ( w ± ) and define the map ev : M → S × S ,v ′′ ( ev − ( v ′′ ) , ev ( v ′′ )) . Note that we have switched the + and − compared to ev ; this is because we wantto identify w ± for v ′ with w ∓ for v ′′ .We claim that, as we take ǫ → , ev ( M ) limits to the two pink trianglesin Figure 18: T with vertices (0 , π ) , ( π, π ) , ( π, π ) and T with vertices ( π, , ( π, π ) , (2 π, π ) . Let G (resp. G ) be the -dimensional family of maps v ′′ satisfying(B1)–(B3) and (B4 ′ ) with image T (resp. T ). Figure 19 describes maps v ′′ =( v ′′ , v ′′ ) that are on or close to ∂ G ; the left-hand side depicts the image of b p ◦ v ′′ in C and the right-hand side depicts the images of the components v ′′ and v ′′ . Thered (resp. blue) dots on the left-hand side are w ± (resp. w ) and the red dots on theright-hand side with labels w ± are the preimages of w ± (resp. w ). The maps aredenoted φ , . . . , φ from top to bottom. The first map φ is z ( z, . The thirdmap φ is close to z ( g ( z ) , g ( z )) , where g is a conformal map from D to thehalf-disk. Maps φ , φ , φ are of the form ( φ ) θ , where the subscript θ indicates φ IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 57 has been modified by rotating the first component by θ and the second componentby − θ . There is a bijection G ∼ → G , v ′′ = ( v ′′ , v ′′ ) ( − v ′′ , − v ′′ ) . Let s ( v ′′ ) be the length of the slit ⊂ [ − , of the image b p ◦ v ′′ . We partition M = ⊔ s ∈ [0 , M s by the slit length s and note that ev ( M s ) , in the limit ǫ → , consists of two linesegments of slope : a segment from θ = π to θ = π in T and a segment from θ = 0 to θ = 2 π in T . Here the set of ( φ ) θ is half of M s corresponding to T ,for s close to . Since ev ( φ ) , ev ( φ ) , ev ( φ ) are close to the vertices of T and s ( φ ) ≈ and s ( φ i ) ≈ for i = 2 , , , the claim follows. φ φ φ φ z z z z w + w − w w w w w + w − w + w − w − w − w − w + w + w + w + w − w − w − w + w + F IGURE
Step 6.
We finally show that M χ =0 , w ,♭, LRJ ♦ ( Ξ , Θ ) = 1 mod , where w satisfies(R1 ′ ) and (R2 ′ ) and w i is close to . In view of Claim 9.3.8, the holomorphic map v ∞ from Step 4 can viewed as thegluing of(9.3.12) ( v ′∞ : F ′∞ → f W ∧ ) ∈ M θ and ( v ′′∞ : F ′′∞ → f W ∧ ) ∈ M , where v ′∞ where corresponds to the left-hand side R and v ′′∞ corresponds to theright-hand side R ∪ R . We abuse notation and will not distinguish between Θ Θ Ξ Ξ w w c c c ∗ c ∗ F IGURE v ′∞ , v ′′∞ and the restrictions of v ∞ to R and R ∪ R . Under this identification,(i) the chords c and c (see Figure 20) correspond to w − and w + for M θ and the chords c ∗ and c ∗ correspond to w + and w − for M ;(ii) v ∞ passing through w corresponds to the constraint (B4) with w =( e iθ , e − iθ ) for v ′∞ and v ∞ ∪ δ + ∪ δ − passing through w corresponds tothe constraint (B4 ′ ) for v ′′∞ .We further assume w has been chosen so that the constraint of passing through w is transformed to (B4) and (B4 ′ ) with θ = − π + ˜ ǫ .We now consider the matching condition ev ( v ′∞ ) = ev ( v ′′∞ ) for the gluingof v ′∞ and v ′′∞ . Let I be the intersection of the blue line and the pink region inFigure 18. In view of the location of I , we have the following:(X) The slits for v ′∞ and v ′′∞ must be long, i.e., end near Θ and Ξ .(Y) The location of b ∞ is as depicted in Cases (A) and (B).We will explain (Y): Since q ( w ) must be close to q (Θ ) and F ( i ) must have aninvolution taking q ( w ) to q ( w ) and q (Θ ) to q (Θ ) , it follows that q ( w ) is closeto q (Θ ) and in the limit q ( w ) = q ( p − ) = q (Θ ) . The intersection I indicatesthat v ′′∞ intersects w along the upper branch of the slit. Let M I be the set ofcurves v ∞ that are obtained by gluing ( v ′∞ , v ′′∞ ) .At this point the possible gluings of v ′∞ and v ′′∞ are parametrized by I , but ifwe fix b ∞ , corresponding to fixing a circle { θ − θ = π + δ } ⊂ S × S forsufficiently small δ > , then the endpoints of the slit are uniquely determined.Once b ∞ (i.e., where q (Θ ) is mapped) and the slit are determined, the position of q (Ξ ) = q ( p + ) (i.e., b ∞ in Case (A) or b ∞ in Case (B), as appropriate) is uniquelydetermined using Claim 9.3.9 and the involution of F ∞ . In summary, fixing b ∞ uniquely determines all the other b ∞ j .Finally we consider v ( i ) for i ≫ . Taking w to be sufficiently close to Θ forces w to be on C − in Figure 16. The key observation is the following:(*) the distance between q ( w ) and q (Θ ) decreases monotonically as b ( i )2 moves from left to right.The unique solution v ( i ) with an involution(9.3.13) q ( w ) q ( w ) , q (Θ ) q (Θ ) , q (Ξ ) q (Ξ ) IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 59 is obtained by taking an open set of holomorphic curves v ( i ) that are close to v ∞ ∪ δ + ∪ δ − , where v ∞ ∈ M I , and picking the unique one satisfying (9.3.13) fromthis set using (*).This completes the proof of Theorem 9.3.7. (cid:3) Proof of invariance under Markov stabilizations.
A Markov stabilizationis given as follows: Let σ be an κ -strand braid which intersects D along z = { z , . . . , z κ } . We view σ as an element of Diff + ( D, ∂D, z ) which additionallyrestricts to the identity on a neighborhood N ( γ ) ⊂ D of a short arc γ from apoint z to ∂D . Given an arc c from z to some z i , i > , which is disjoint fromthe other z j , let σ c be the positive half twist along c . Then a positive (resp. negative ) Markov stabilization is the ( κ + 1) -strand braid given by σ ◦ σ c (resp. σ ◦ σ − c ).Let γ := { γ , . . . , γ κ } be a basis of half arcs, where γ i connects z i to ∂D . Bythe handleslide invariance, we may assume that c connects from z to z and doesnot intersect any γ i , i > , in its interior. Then let γ ′ := { γ , . . . , γ κ } . Let p : W → D − N ( γ ) and p ′ : W ′ → D be the standard Lefschetz fibrations with critical values z = { z , . . . , z κ } and z ′ = { z , . . . , z κ } and regular fiber A = S × [ − , , and such that p ′ | W = p . Let a = { a , . . . , a κ } be the Lagrangian thimbles over γ and let a ′ = { a , . . . , a κ } be the Lagrangian thimbles over γ ′ . If h σ ∈ Symp(
W, ∂W ) descends to σ , thenlet h ′ σ ∈ Symp( W ′ , ∂W ′ ) be its extension to W ′ by the identity. Finally let τ c ∈ Symp( W ′ , ∂W ′ ) be the Dehn twist along the Lagrangian sphere over c .9.4.1. Model calculation. Any notation introduced here is limited to this subsec-tion.
Consider the product fibration b p : C × T ∗ S → C and Lagrangians a i = { x = i } × S , i = 1 , , and b j = { y = j } × S , j = 1 , , where S is the zero sectionof T ∗ S . We write a = { a , a } and b = { b , b } . Let x ij = ( i, j ) ∈ C and let ˇ x ij , ˆ x ij ∈ CF ( b j , a i ) be the bottom and top generators of the clean intersection x ij × S . We can alternatively take a ′ = { a ′ , a ′ } and b ′ = { b ′ , b ′ } , where a ′ i is { x = i } (resp. b ′ j is { y = j } ) times a Hamiltonian perturbation of S , where allthe perturbations intersect transversely; see the right-hand side of Figure 21. a ′ a ′ b ′ b ′ x x x x a ′ a ′ b ′ b ′ F IGURE
21. The base C on the left and the fiber T ∗ S on the right(the sides are identified). Lemma 9.4.1. d CF ( b ′ , a ′ ) is generated by generators { x † , x † } and { x † , x † } ,where † may be a hat or check, and the differentials are given by ~ times the arrowsgiven in Figure 22. { x † , x † } { x † , x † } F IGURE
22. Description of the differentials of d CF ( b ′ , a ′ ) . Thegenerators in the top row have checks, those in the middle rowhave check, and those in the bottom row have no checks. Proof.
Let u : ˙ F → R × [0 , × ( C × T ∗ S ) be a holomorphic map with { ˇ x , ˇ x } at the positive end and { ˆ x , ˇ x } at the negative end. Its projection to C is a degree map over [1 , × [1 , ; this determines the complex structure of the -punctureddisk ˙ F = ˙ F .Next we consider holomorphic maps w : ˙ F → T ∗ S which could be projectionsof u to T ∗ S . The two possible domains A, B that w can map to are shaded inFigure 23. The domain A on the left (resp. B on the right) has a π corner at ˆ x (resp. ˇ x ) and there may be slits to the right or pointing up (resp. to the right orpointing up). We will refer to the four types of slits by AR , AU , BR , BU . Asthe slit AR goes all the way to the right (i.e., until it hits a ′ ), the punctures q (ˆ x ) and q (ˇ x ) approach one another; when AU goes all the way up, we can view itas continuing to BR going all the way to the right; finally, as BU goes all the wayup, q (ˇ x ) and q (ˇ x ) approach one another. Hence the algebraic count of w withthe given domain ˙ F = ˙ F is one. ˇ x ˇ x ˇ x ˆ x A Ba ′ a ′ b ′ b ′ F IGURE
23. The two domains A and B representing the possibleclosures of w ( ˙ F ) .The determination of the other arrows of Figure 22 is analogous. (cid:3) Proof of stabilization invariance.
In this subsection we prove the invarianceunder Markov stabilization. We will use the open book interpretation of the Hee-gaard Floer groups.
IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 61
Theorem 9.4.2. d CF ( W, h σ ( a ) , a )) and d CF ( W ′ , h ′ σ ◦ τ c ( a ′ ) , a ′ ) are isomorphiccochain complexes for specific choices of almost complex structures and h σ ( a ) and h ′ σ ◦ τ c ( a ′ ) after a Hamiltonian isotopy.Proof. We will treat the positive stabilization case; the negative stabilization caseis analogous.The generators of d CF ( W, h σ ( a ) , a )) come in two types of κ -tuples: { x } ∪ y ′ and y , where { x , . . . , x κ } is the contact class and y does not contain x .The generators of d CF ( W ′ , h ′ σ ◦ τ c ( a ′ ) , a ′ ) are in bijection with the generatorsof d CF ( W, h σ ( a ) , a )) and have the form { x , x } ∪ y ′ and { Θ } ∪ y , where { x , . . . , x κ } is the contact class and Θ is the unique intersection point between a and h ′ σ ◦ τ c ( a ) = τ c ( a ) . Refer to Figure 24.Let γ = { Re z } × [ − , ⊂ D . Choose ǫ > small. We normalize h σ ( a ) and h ′ σ ◦ τ c ( a ) within their Hamiltonian isotopy classes (rel boundary) such thattheir projections σ ( γ ) and σ ◦ σ c ( γ ) can be written as concatenations ζ ∪ ζ ∪ ζ and ζ ′ ∪ ζ ∪ ζ of arcs, where: • ζ starts at p ′ ( x ) and ends at (Re z − ǫ, − / ; • Re z − ǫ < Re(int ζ ) < Re z ; • ζ ′ starts at p ′ ( x ) and ends at (Re z − ǫ, − / ; and • ζ = { Re z − ǫ } × [ − / , − / .We claim that the linear isomorphism Φ s : d CF ( W, h σ ( a ) , a )) → d CF ( W ′ , h ′ σ ◦ τ c ( a ′ ) , a ′ ) , (9.4.1) { x } ∪ y ′
7→ { x , x } ∪ y ′ , y
7→ { Θ } ∪ y , commutes with the differentials for ǫ > small. Here we are assuming that D hasthe standard complex structure.We will use the notation u ′ : ˙ F ′ → R × [0 , × c W ′ for a holomorphic map thatis counted in the differential of d CF ( W ′ , h ′ σ ◦ τ c ( a ′ ) , a ′ ) and u for a holomorphicmap ˙ F → R × [0 , × c W that is counted in the differential of d CF ( W, h σ ( a ) , a )) .If a curve u ′ goes from { x , x } ∪ y ′ to { x , x } ∪ y ′ , then, apart from thetrivial strip from x to itself, it projects to the region D − N ( γ ) and hence is inbijection with a curve u that goes from { x } ∪ y ′ to { x } ∪ y ′ . Similarly, a curve u ′ from { Θ } ∪ y to { Θ } ∪ y is in bijection with a curve u from y to y .There are no curves from { x , x } ∪ y ′ to { Θ } ∪ y and likewise no curves from { x } ∪ y ′ to y .It remains to identify curves u ′ from { Θ } ∪ y to { x , x } ∪ y ′ with curves u from y to { x } ∪ y ′ . By our normalization of σ ( γ ) and σ ◦ σ c ( γ ) with ǫ > small, if ind( u ) = 1 , then there exists u ′ with ind = 1 which is obtained by gluing u (where the negative end x is now viewed as the thin neck between h ′ σ ◦ τ c ( a ) and a that projects to [Re z − ǫ, Re z ] × [ − / , − / ) and a holomorphic -punctured disk u ′′ : ˙ F ′′ → R × c W ′ that projects to the shaded region in Figure 24,i.e, a curve of the type calculated in Lemma 9.4.1. The algebraic count of such acurve u ′′ is . Conversely a curve u ′ with ind( u ′ ) = 1 is obtained by gluing u and u ′′ . (cid:3) γ γ γ x x Θ c F IGURE
24. The region cut off by the dotted line and containing γ , . . . γ κ is D − N ( γ ) . The red half-arcs are σ ◦ σ c ( γ ) and σ ◦ σ c ( γ ) .9.5. Contact class.
In this section we prove Theorem 1.2.2.
Proof of Theorem 1.2.2.
We first prove the invariance under handleslides. Con-sider the cochain map Φ given by Equation (9.3.2). We take e h σ ( e a i ) to be the e γ e γ e γ ′ e γ ′ e δ e δ F IGURE e δ i in Figure 25. The key is to position e δ i as shown soit is “locally to the right” of e γ i and e γ ′ i as viewed from z κ + i : by this we mean thaton a small neighborhood N ( z κ + i ) of z κ + i ,(1) the arcs e δ i , e γ ′ i , e γ i restrict to rays emanating from z κ + i ;(2) they are contained in a thin circular sector S κ + i with center z κ + i ;(3) inside S κ + i the arcs are e δ i , e γ ′ i , e γ i in counterclockwise order.Strictly speaking, we then further apply a C ∞ -small perturbation to e h σ ( e a i ) so thatit is slightly pushed off of the intersection point Ξ i between e a i and e a ′ i ; this can bedone in the local model b p : C z ,z → C ζ , ( z , z ) z z , from Equation (9.3.3).The contact class x = { x , . . . , x κ } ∈ d CF ( e h σ ( e a ) , e a ) is the κ -tuple of pointswhere x i is over z κ + i . Let x ′ = { x ′ , . . . , x ′ κ } ∈ d CF ( e h σ ( e a ) , e a ′ ) be the pushoff ofthe κ -tuple of points over z κ +1 , . . . , z κ .We claim that Φ( x ) = x ′ : Using the local model one can verify that every curve u that is counted in µ ( Ξ , x ) must have a component which is a small triangle Ξ i , x i , x ′ i for i = 2 . Once e a i , e a ′ i , and e h σ ( e a i ) are used up for i = 2 , the onlypossible holomorphic triangle involving Ξ and x is the small triangle Ξ , x , x ′ .Hence Φ( x ) = x ′ .The analogous map Φ ′ : d CF ( e h σ ( e a ) , e a ′ ) → d CF ( e h σ ( e a ′ ) , e a ′ ) IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 63 similarly takes x ′ to the contact class. This proves the invariance under han-dleslides.The invariance under positive Markov stabilizations is immediate from the defi-nition of the cochain map Φ s given by Equation (9.4.1). (cid:3) Relationship with symplectic Khovanov homology.
We will work over thering F [ A ] J ~ , ~ − ] J U − K . Consider the following filtration for CKh ♯ ( b σ ) ⊗ J U − K : F := ( · · · ⊂ F − ⊂ F − ⊂ F ) , F i = U i · CKh ♯ ( b σ ) ⊗ J U − K , where the tensor product is over F .Given u : ˙ F → R × [0 , × f W in M ind=1 ,A,χJ ♦ ( y , y ′ ) and its projections v to R × [0 , and v to the base e D of the Lefschetz fibration e p : f W → e D ⊂ C , weconsider the map ( v , v ) : ˙ F → R × [0 , × e D. Note that the target is -dimensional. Although we will not go into detail here, itis possible to choose a regular almost complex structure J ♦ on R × [0 , × f W , aregular almost complex structure J R × [0 , × e D on R × [0 , × e D for curves of theform ( v , v ) , and the standard complex structure J R × [0 , on R × [0 , such thatthe projections ( R × [0 , × f W , J ♦ ) → ( R × [0 , × e D, J R × [0 , × e D ) → ( R × [0 , , J R × [0 , ) are holomorphic and the fibers of the projections are holomorphic.Let W ( u ) be the total weight of the singularities of ( v , v ) , including self-intersections. By the positivity of intersections for pseudoholomorphic curves indimension , this is a nonnegative number which is equal to the difference betweenthe Heegaard Floer index of [CGH1] and the Fredholm index in the usual Hee-gaard Floer homology. Writing d U for the differential of CKh ♯ ( b σ ) ⊗ J U − K , thecontribution of u to d U ( U a y ) is ~ κ − χ e A U a − W ( u ) y ′ (modulo ) and ( CKh ♯ ( b σ ) ⊗ J U − K , d U , F ) is a filtered cochain complex. Reinterpreting Manolescu [Ma], the symplectic Kho-vanov condition is that the holomorphic map ( v , v ) be embedded, i.e., that weonly count curves with W ( u ) = 0 .The following lemma is immediate from the above discussion. Lemma 9.6.1.
With F [ A ] J ~ , ~ − ] J U − K -coefficients, the E term of the spectralsequence of ( CKh ♯ ( b σ ) ⊗ J U − K , d U , F ) is the symplectic Khovanov homology Kh symp ( b σ ) tensored with J U − K . The main issue that we face is the convergence of the spectral sequence. Thefollowing is the key question:
Question . Is there a bound on W ( u ) for u ∈ M ind=1 ,A,χJ ♦ ( y , y ′ ) ? This isclosely related to whether there is a bound on χ .What we can say for the moment is the following: Theorem 9.6.3.
There is a filtered cochain map d CF ( f W , e h σ ( e a ) , e a ) ⊗ J U − K → d CF ( f W , e h σ ( e a ′ ) , e a ′ ) ⊗ J U − K for a handleslide (cf. Theorem 9.3.1) which is a quasi-isomorphism of cochaincomplexes. There is also a filtered cochain map for a Markov stabilization (cf.Theorem 9.4.2) which is a quasi-isomorphism of cochain complexes. Hence the E k term of the spectral sequence of ( CKh ♯ ( b σ ) ⊗ J U − K , d U , F ) is a link invariant.Proof. The cochain map is an enhancement of d CF ( f W , e h σ ( e a ) , e a ) → d CF ( f W , e h σ ( e a ′ ) , e a ′ ) from the proof of Theorem 9.3.1 which keeps track of the powers of U − . Sincethe total singularity weight W ( u ) of the appropriate projection of any curve u counted in the maps Φ and Ψ is always nonnegative, the cochain map is filtration-preserving. The quasi-isomorphism as cochain complexes follows from observingthat in the proof of Theorem 9.3.7 the only nonzero (mod ) curve count is that of M χ =0 , w ,♭, LRJ ♦ ( Ξ , Θ ) in Step 6, whose elements satisfy W = 0 .The case of Markov stabilizations is straightforward and is left to the reader. (cid:3)
10. T HE B ALDWIN -P LAMENEVSKAYA EXAMPLES
The goal of this section is to prove Theorem 1.2.4.
Proof.
In view of Lemma 7.3.2 it suffices to show that ψ ♯ ( b σ BP, ) = 0 . We claimthat d ( { ˆ y , ˆ y , x } + { ˇ y , z , x } ) = ~ { x , x , x } mod in Figure 26. Referring to Figure 26, there is a single intersection point of a ∩ h BP, ( a ) over each z i , denoted by ˇ z i or simply z i ; two intersection points (herewe are doing Morse-Bott theory) over each of r i , s i , t i , y i , given by ˇ r i of lowerdegree, ˆ r i of higher degree, etc.; and one intersection point over x i , denoted by ˆ x i or simply x i . The point ˆ x i is a component of the contact class.Since the only holomorphic curve that has x i at the positive end is a trivialstrip, we may erase the arcs γ and σ BP, ( γ ) and omit x from the notation; seeFigure 27. Excluding trivial strips, all the curves u : ˙ F → R × [0 , × c W that weend up counting have a domain ˙ F which is a disk with boundary punctures andcontributes the coefficient ~ . Remark . In fact, the proof of Theorem 1.2.4 shows that ψ ♯ ( b σ ) = 0 for any κ -braid σ for which there exist arcs γ and γ such that σ ( γ ) and σ ( γ ) are asgiven in Figure 27 and there are no other critical points in regions bounded bysubarcs of γ , γ , σ ( γ ) , σ ( γ ) . First Calculation.
The next several pages are devoted to showing that d { ˆ y , ˆ y } = ~ ( { x , x } + { ˇ r , x } + { ˇ t , x } ) mod . All the loops in D that are concatenations of an arc each of γ , γ , σ BP, ( γ ) , σ BP, ( γ ) IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 65 x x x y y r s r s t t z z F IGURE
26. The braid σ BP, acting on γ , γ , γ on D .F IGURE y , y are given by: y → x , t , t → y → r , r , x → y , where x , t , t means any one of them can be used. The only loops that canpossibly bound domains with nonnegative weights (denoted D , D ′ , D , D , D )are: (1) y → x → y → r → y ,(1’) y → t → y → x → y ,(2) y → x → y → x → y ,(3) y → x → y → r → y ,(4) y → t → y → x → y . Cases (1) and (1’). D is a quadrilateral and an easy Maslov index calculationgives:(i) ind( u ) = 0 if u has { ˆ y , ˆ y } at the positive end and { ˆ r , x } at the negativeend;(ii) ind( u ) = − if u has { ˆ y , ˆ y } at the positive end and { ˇ r , x } at thenegative end.Hence curves u that project to D do not have the right indices and are not counted.The D ′ case is analogous. x x y y t r y x y r r F IGURE
28. The domains D (left) and D (right). x y t y F IGURE
29. The domain D . Case (2).
Denoting the closure of D by D , we choose D as in Figure 28 suchthat the intersections D ∩ γ and D ∩ γ are close and that D ∩ σ BP, ( γ ) and D ∩ σ BP, ( γ ) are close. Claim 10.0.2.
The count of u : ˙ F → R × [0 , × c W from { ˆ y , ˆ y } to { x , x } modulo R -translation is mod if its projection to C is D . Note that such u satisfies ind( u ) = 1 and that the index difference between { y † , y † } and { x , x } is plus the number of † =ˇ . The intersection points ˆ y , ˆ y should be viewed as point constraints and x , x as imposing no point constraints(and can be disregarded). Proof of Claim 10.0.2.
There are two types of slits in D that start at y : Type D that goes straight down along γ and Type L that initially goes to the left along σ BP, ( γ ) . As the slit D goes all the way down so that it hits r , the punctures q ( y ) , q ( x ) , q ( y ) can be viewed as approaching one another on the domain ˙ F ;as the slit L goes all the way to γ , the punctures q ( y ) , q ( x ) , q ( y ) approach IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 67 one another on ˙ F . Moreover, for Type D , q ( y ) comes right after q ( y ) , travelingcounterclockwise, whereas q ( y ) comes right before q ( y ) for Type L . These twodegenerations can be viewed as the ends of a -dimensional family of maps b p ◦ v :˙ F → C .Recall the Lefschetz fibration b p : C z ,z → C ζ , ( z , z ) z z , and the Clifford torus T = {| z | = 1 } × {| z | = 1 } over | ζ | = 1 . Let ǫ > ˆ y ˆ y ˆ y ˆ y ′ F IGURE
30. The base of the Lefschetz fibration b p . The circles are b p ( T ) = {| ζ | = 1 } . The path b p ◦ η goes counterclockwise from e iǫ to e − iǫ , given by red dots.be small. Referring to Figure 30, we view ˆ y , ˆ y as points on T sitting over e iǫ , and take a path η on T which starts at ˆ y , is obtained by parallel transport around ∂D in the counterclockwise direction via a symplectic connection, and ends at ˆ y ′ sitting over e − iǫ . The desired curve count is equivalent to the curve count inLemma 10.0.3 below and implies Claim 10.0.2. (cid:3) Lemma 10.0.3.
The count of disks w : F → C modulo reparametrizations satis-fying (i) and (ii) below is modulo . (i) w ( ∂F ) ⊂ T and w ( F ) = D with degree . (ii) w intersects ˆ y and η .Proof of Lemma 10.0.3. Refer to Figure 31, which depicts the Lagrangian torus T ,where the vertical direction represents the fibers of b p : T → {| ζ | = 1 } . Theblue curves are boundaries of the curves w and w that pass through ˆ y byRemark 9.3.5, and the curve η has slope and connects from the fiber T e iǫ over e iǫ to the fiber T e − iǫ . As long as ǫ is small and ˆ y does not go to ˆ y as ǫ → ,the intersection number between η and w ∪ w is modulo , which implies thelemma. (cid:3) We now claim that the count of u from { ˆ y , ˆ y } to { x , x } mod does notchange under a (generic) isotopy of D : As we isotop the shape of D , the -dimensional family of curves from { ˆ y , ˆ y } to { x , x } can degenerate to an ind =0 curve from { ˆ y , ˆ y } to { ˆ r , x } and an ind = 1 curve from { ˆ r } to { x } (or an ind = 0 curve from { ˆ y , ˆ y } to { ˆ t , x } and an ind = 1 curve from { ˆ t } to { x } ).Since the ind = 1 curve types come in pairs by Remark 9.3.5, the mod curvecount remains invariant as we pass through this bifurcation. ˆ y ˆ y ˆ y ˆ y ′ ηw w F IGURE
31. The sides of the torus T are identified and the topand the bottom are also identified. Case (3).
We choose D as in Figure 28 such that the two critical values z and z are far apart and that D has a long neck in the middle. A Maslov index calculationimplies that, in Case (3), ind( u ) = 1 if and only if u is from { ˆ y , ˆ y } to { ˇ r , x } .We claim that the count of u : ˙ F → R × [0 , × c W from { ˆ y , ˆ y } to { ˇ r , x } is mod . Observe that ˆ y , ˆ y , ˇ r should be viewed as point constraints and x as imposing no point constraints. There are slits going to the left or down at y (labeled yL or yD ) and slits going to the right or down at r (labeled rR or rD ).The slits yL and rR cannot be too long: if rR is long, then we are effectivelygluing a disk u about z with no constraint and a disk u about z with threeconstraints, which is a contradiction; the situation for yL long is similar.On the other hand, if yL and rR are not too long, then we are effectively gluinga disk u about z with constraints and a disk u about z with constraint.In the next paragraph we sketch a model calculation for u similar to Step 4 ofTheorem 9.3.6, which implies that the count of u is mod . The gluing of u to u imposes another constraint on u and the count of u with constraints is mod by a similar calculation. The claim then follows. Model calculation.
We use the notation from Step 4 of Theorem 9.3.6 and consider M w J , where w = ( w , w ) = ( e i ( π − ǫ ) , with ǫ > small. Figure 32 givesa schematic description of M w J analogous to Figure 13. The evaluation map ev J from Equation (9.3.6) therefore maps M w J almost completely once around S | z | =1 .As long as w and w are not too close on S | z | =1 , the degree calculation impliesthat ev − J ♦ ( w ) = 1 modulo .We also note that w and w not being close translates to the points corre-sponding to ˆ y and ˇ r not being close. This implies that we cannot obtain a disk u that passes through ˆ y and ˇ r by adjusting the slit length of rR and that we mustuse rD . Our model calculation corresponds to using the slit rD . Case (4).
Consider D as in Figure 29. In Case (4), ind( u ) = 1 if and only if u isfrom { ˆ y , ˆ y } to { ˇ t , x } .We claim that the count of u : ˙ F → R × [0 , × c W from { ˆ y , ˆ y } to { ˇ t , x } is mod . Refer to Figure 29 for D . The Fredholm index calculations areanalogous to those of Case (3). Consider the slits BR, P R of D that start at t , y , respectively, and initially go to the right. If both slits BR, P R are too short,
IGHER-DIMENSIONAL HEEGAARD FLOER HOMOLOGY 69 w w w w w = w w w w w w w w w w w z z z z F IGURE
32. A schematic description of M w J analogous to Fig-ure 13 but with a different w .then we are effectively gluing a disk u about z with no constraint and a disk u about z with three constraints, which is a contradiction. If P R is too long, then u about z will have no constraints, which is also a contradiction. Hence P R isshort and BR is long. Also note that BR must be long but cannot be so long thatits other endpoint gets too close to z , in which case u about z will also have noconstraints. This means that we are effectively gluing u about z with constraintcorresponding to ˆ y and u about z with constraints corresponding to ˆ y and ˇ t .Calculations similar to those of Case (3) imply the claim. Second Calculation.
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