Ko Honda, William H. Kazez and Gordana Matic- Contact Structures, Sutured Floer Homology and TQFT

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CONTACT STRUCTURES, SUTURED FLOER HOMOLOGY AND TQFT

KO HONDA, WILLIAM H. KAZEZ, AND GORDANA MATI C

ABSTRACT. We describe the natural gluing map on sutured Floer homology which is induced bythe inclusion of one sutured manifold(M ′, Γ′) into a larger sutured manifold(M, Γ), together with acontact structure onM−M ′. As an application of this gluing map, we produce a(1+1)-dimensionalTQFT by dimensional reduction and study its properties.

1. INTRODUCTION

Since its inception around 2001, Ozsvath and Szabo’s Heegaard Floer homology [OS1, OS2]has been developing at a breakneck pace. In one direction, Ozsvath-Szabo [OS4] and, indepen-dently, Rasmussen [Ra] defined knot invariants, called knotFloer homology, which categorifiedthe Alexander polynomial. Although its initial definition was through Lagrangian Floer homol-ogy, knot Floer homology was recently shown to admit a completely combinatorial description byManolescu-Ozsvath-Sarkar [MOS]. Knot Floer homology is apowerful invariant which detectsthe genus of a knot by the work of Ozsvath-Szabo [OS6], and detects fibered knots by the workof Ghiggini [Gh] and Ni [Ni]. (The latter was formerly calledthe “fibered knot conjecture” ofOzsvath-Szabo).

One of the offshoots of the effort to prove this fibered knot conjecture is the definition of arelative invariant for a3-manifold with boundary. In a pair of important papers [Ju1,Ju2], AndrasJuhasz generalized the hat versions of Ozsvath and Szabo’s Heegaard Floer homology [OS1, OS2]and link Floer homology [OS4] theories, and assigned a Floerhomology groupSFH(M,Γ) to abalanced sutured manifold(M,Γ). (A related theory is being worked out by Lipshitz [Li1, Li2]and Lipshitz-Ozsvath-Thurston [LOT].)

In [HKM3], the present authors defined an invariantEH(M,Γ, ξ) of (M,Γ, ξ), a contact 3-manifold(M, ξ) with convex boundary and dividing setΓ on∂M , as an element inSFH(−M,−Γ).Our invariant generalized the contact class in Heegaard Floer homology in the closed case, as de-fined by Ozsvath and Szabo [OS3] and reformulated by the authors in [HKM2]. The definition ofthe contact invariant was made possible by the work of Giroux[Gi2], which provides special Morsefunctions (calledconvex Morse functions) or, equivalently, open book decompositions which areadapted to contact structures.

Recall that asutured manifold(M,Γ), due to Gabai [Ga], is a compact, oriented, not necessarilyconnected 3-manifoldM with boundary, together with an oriented embedded1-manifoldΓ ⊂ ∂M

Date: This version: July 13, 2008. (The pictures are in color.)1991Mathematics Subject Classification.Primary 57M50; Secondary 53C15.Key words and phrases.tight, contact structure, open book decomposition, Heegaard Floer homology, sutured

manifolds, topological quantum field theories.KH supported by an NSF CAREER Award (DMS-0237386); GM supported by NSF grant DMS-0410066; WHK

supported by NSF grant DMS-0406158.1

http://arXiv.org/abs/0807.2431v1

2 KO HONDA, WILLIAM H. KAZEZ, AND GORDANA MATI C

which bounds a subsurface of∂M .1 More precisely, there is an open subsurfaceR+(Γ) ⊂ ∂M(resp.R−(Γ)) on which the orientation agrees with (resp. is the oppositeof) the orientation on∂Minduced fromM , andΓ = ∂R+(Γ) = ∂R−(Γ) as oriented1-manifolds. A sutured manifold(M,Γ)is balancedif M has no closed components,π0(Γ) → π0(∂M) is surjective, andχ(R+(Γ)) =χ(R−(Γ)) on the boundary of every component ofM . In particular, every boundary component of∂M nontrivially intersects the sutureΓ.

In this paper, we assume that all sutured manifolds are balanced and all contact structuresare cooriented. Although every connected component of a balanced sutured manifold(M,Γ)must have nonempty boundary, our theorems are also applicable to closed, oriented, connected3-manifoldsM . Following Juhasz [Ju1], a closedM can be replaced by a balanced sutured man-ifold as follows: LetB3 be a3-ball insideM , and considerM − B3. On∂(M − B3) = S2, letΓ = S1. SinceSFH(M − int(B3),Γ = S1) is naturally isomorphic toHF (M), we can view aclosedM as(M − int(B3), S1).

The goal of this paper is to understand the effect of cutting/gluing of sutured manifolds. We firstdefine a map which is induced from the inclusion of one balanced sutured manifold(M ′,Γ′) intoanother balanced sutured manifold(M,Γ), in the presence of a “compatible” contact structureξonM − int(M ′). Here we say that(M ′,Γ′) is asutured submanifoldof a sutured manifold(M,Γ)if M ′ is a submanifold with boundary ofM , so thatM ′ ⊂ int(M). If a connected componentNof M − int(M ′) contains no components of∂M we say thatN is isolated.

We will work with Floer homology groups over the ringZ. With Z-coefficients, the contactinvariantEH(M,Γ, ξ) is a subset ofSFH(−M,−Γ) of cardinality1 or 2 of type{±x}, wherex ∈ SFH(−M,−Γ). (The cardinality is1 if and only if x is a2-torsion element.) OverZ/2Z, the±1 ambiguity disappears, andEH(M,Γ, ξ) ∈ SFH(−M,−Γ).

The following theorem is the main technical result of our paper.

Theorem 1.1.Let (M ′,Γ′) be a sutured submanifold of(M,Γ), and letξ be a contact structure onM − int(M ′) with convex boundary and dividing setΓ on ∂M andΓ′ on ∂M ′. If M − int(M ′)hasm isolated components, thenξ induces a natural map:

Φξ : SFH(−M ′,−Γ′) → SFH(−M,−Γ) ⊗ V ⊗m,

which is well-defined only up to an overall± sign. Moreover,

Φξ(EH(M ′,Γ′, ξ′)) = EH(M,Γ, ξ′ ∪ ξ) ⊗ (x⊗ · · · ⊗ x),

wherex is the contact class of the standard tight contact structureonS1×S2 andξ′ is any contactstructure onM ′ with boundary conditionΓ′. HereV = HF (S1 × S2) ≃ Z ⊕ Z is a Z-gradedvector space where the two summands have grading which differ by one, say0 and1.

The choice of a contact structureξ onM −M ′ plays a key role as the “glue” or “field” whichtakes classes inSFH(−M ′,−Γ′) to classes inSFH(−M,−Γ) ⊗ V ⊗m. We emphasize that thegluing mapΦξ is usually not injective. The statement of the theorem, in particular the “naturality”and theV factor, will be explained in more detail in Section 3.

One immediate corollary of Theorem 1.1 is the following result, essentially proved in [HKM3]:

Corollary 1.2. Leti : (M ′,Γ′, ξ′) → (M,Γ, ξ) be an inclusion such thatξ|M ′ = ξ′. If EH(M,Γ, ξ)6= 0, thenEH(M ′,Γ′, ξ′) 6= 0.

1This definition is slightly different from that of Gabai [Ga].

CONTACT STRUCTURES, SUTURED FLOER HOMOLOGY AND TQFT 3

Gluing along convex surfaces. Specifying a sutureΓ on ∂M is equivalent to prescribing atranslation-invariant contact structureζ∂M in a product neighborhood of∂M with dividing setΓ. LetU be a properly embedded surface of(M,Γ) satisfying the following:

• There exists an invariant contact structureζU , defined in a neighborhood ofU , which agreeswith ζ∂M near∂U ;

• U is convex with possibly empty Legendrian boundary and has a dividing set ΓU withrespect toζU .

Let (M ′,Γ′) be the sutured manifold obtained by cutting(M,Γ) alongU and edge-rounding. (See[H1] for a description of edge-rounding.) By slightly shrinking M ′, we obtain the tight contactstructureζ = ζ∂M ∪ ζU onM − int(M ′). The contact structureζ induces the map

Φζ : SFH(−M ′,−Γ′) → SFH(−M,−Γ) ⊗ V ⊗m,

for an appropriatem.Summarizing, we have the following:

Theorem 1.3(Gluing Map). Let (M ′,Γ′) be a sutured manifold and letU+ andU− be disjointsubsurfaces of∂M ′ (with the orientation induced from∂M ′) which satisfy the following:

(1) Each component of∂U± transversely and nontrivially intersectsΓ′.(2) There is an orientation-reversing diffeomorphismφ : U+ → U− which takesΓ′|U+

to Γ′|U−

and takesR±(U+) toR∓(U−).Let (M,Γ) be the sutured manifold obtained by gluingU+ andU− via φ, and smoothing. Thenthere is a natural gluing map

Φ : SFH(−M ′,−Γ′) → SFH(−M,−Γ) ⊗ V ⊗m,

wherem equals the number of components ofU+ that are closed surfaces. Moreover, if(M,Γ, ξ)is obtained from(M ′,Γ′, ξ′) by gluing, then

Φ(EH(M ′,Γ′, ξ′)) = EH(M,Γ, ξ)⊗ (x⊗ · · · ⊗ x),

wherex is the contact class of the standard tight contact structureonS1 × S2.

In particular, whenΓU is ∂-parallel, i.e., each component ofΓU cuts off a half-disk which

intersects no other component ofΓU , then the convex decomposition(M,Γ)(U,ΓU ) (M ′,Γ′) corre-

sponds to a sutured manifold decomposition by [HKM1]. In Section 6 we indicate why our gluingmap

Φ : SFH(−M ′,−Γ′) → SFH(−M,−Γ)

is the same as the direct summand map constructed in [HKM3, Section 6].2

(1 + 1)-dimensional TQFT. We now describe a(1 + 1)-dimensional TQFT, which is obtainedby dimensional reduction of sutured Floer homology and gives an invariant of multicurves onsurfaces. (In this paper we loosely use the terminology “TQFT”. The precise properties satisfiedby our “TQFT” are given in Section 7.)

Let Σ be a compact, oriented surface with nonempty boundary∂Σ, andF be a finite set ofpoints of∂Σ, where the restriction ofF to each component of∂Σ consists of an even number≥ 2

2In [Ju2], Juhasz proves that a sutured manifold gluing induces a direct summand mapSFH(−M ′,−Γ′) →SFH(−M,−Γ). Although it is expected that this map agrees with the natural gluing map, this has not been proven.


of points. Moreover, the connected components of∂Σ − F are alternately labeled+ and−. AlsoletK be a properly embedded, oriented1-dimensional submanifold ofΣ whose boundary isF andwhich dividesΣ intoR+ andR− in a manner compatible with the labeling of∂Σ − F . Let ξK betheS1-invariant contact structure onS1 × Σ which traces the dividing setK on each{pt} × Σ.Let F0 ⊂ ∂Σ be obtained fromF by shifting slightly in the direction of∂Σ. The correspondingcontact invariantEH(ξK) is a subset ofSFH(−(S1 × Σ),−(S1 × F0)) of the form{±x}. TheTQFT assigns to each(Σ, F ) a gradedZ-moduleV (Σ, F ) = SFH(−(S1 ×Σ),−(S1 × F0)) andto eachK the subsetEH(ξK) ⊂ V (Σ, F ).

One application of the TQFT is the following:

Theorem 1.4. The contact invariant in sutured Floer homology does not always admit a single-valued representative withZ-coefficients.

Next, we say thatK is isolating if Σ − K contains a component that does not intersect∂Σ.Using the TQFT properties we will prove:

Theorem 1.5.OverZ/2Z, EH(ξK) 6= 0 if and only ifK is nonisolating.

Theorem 1.5, combined with Corollary 1.2, expands our repertoire of universally tight contactstructures which are not embeddable in Stein fillable contact 3-manifolds.

Corollary 1.6. Let ξK be theS1-invariant contact structure onS1 × Σ corresponding to the di-viding setK ⊂ Σ. ThenξK cannot be embedded in a Stein fillable(or strongly symplecticallyfillable) closed contact3-manifold ifK is isolating.

Finally, we remark thatV (Σ, F ) is the Grothendieck group of a categoryC(Σ, F ), called thecontact category, whose objects are dividing sets on(Σ, F ) and whose morphisms are contactstructures onΣ × [0, 1]. The contact category will be treated in detail in [H3].

Organization of the paper. In Section 2 we review the notions of sutured Floer homology andpartial open book decompositions, which appeared in [Ju1, Ju2, HKM3]. Section 3 is devoted toexplaining Theorem 1.1, in particular theV factor and the naturality statement. Theorem 1.1 willbe proved in Sections 4 and 5. The mapΦξ will be defined in Section 4 and the fact thatΦξ is anatural map will be proved in Section 5. We remark that, although the basic idea of the definitionof Φξ is straightforward, the actual definition and the proof of naturality are unfortunately ratherinvolved. Basic properties of the gluing map will be given inSection 6. Section 7 is devoted toanalyzing the(1 + 1)-dimensional TQFT.

2. PRELIMINARIES

We first review some notions which appeared in [Ju1, Ju2] and [HKM3].Let (M,Γ) be a balanced sutured manifold. Then aHeegaard splitting(Σ, α, β) for (M,Γ) con-

sists of a properly embedded oriented surfaceΣ in M with ∂Σ = Γ and two sets of disjoint simpleclosed curvesα = {α1, . . . , αr} andβ = {β1, . . . , βr}. The Heegaard surfaceΣ compresses toR−(Γ) along the collectionα and toR+(Γ) along the collectionβ. The number ofα curves equalsthe number ofβ curves since(M,Γ) is assumed to be balanced.

To define the sutured Floer Homology groups, as introduced byJuhasz, we consider the La-grangian toriTα = α1×· · ·×αr andTβ = β1×· · ·×βr in Symr(Σ). LetCF (Σ, α, β) be the free


Z-module generated by the pointsx = (x1, . . . , xr) in Tα ∩ Tβ . In the definition of the boundarymap for sutured Floer homology, the sutureΓ plays the role of the basepoint. Denote byMx,y

the 0-dimensional (after quotienting by the naturalR-action) moduli space of holomorphic mapsu from the unit diskD2 ⊂ C to Symr(Σ) that (i) send1 7→ x, −1 7→ y, S1 ∩ {Im z ≥ 0} to Tα

andS1 ∩ {Im z ≤ 0} to Tβ, and (ii) avoid∂Σ × Symr−1(Σ) ⊂ Symr(Σ). Then define

∂x =∑

µ(x,y)=1

#(Mx,y) y,

whereµ(x,y) is the relative Maslov index of the pair and#(Mx,y) is a signed count of pointsin Mx,y. The homology ofCF (Σ, α, β) is the sutured Floer homology groupSFH(Σ, α, β) =SFH(M,Γ).

In [HKM3], the present authors defined an invariantEH(M,Γ, ξ) of (M,Γ, ξ), a contact 3-manifold with convex boundary and dividing setΓ on ∂M , as an element inSFH(−M,−Γ).This invariant generalizes the contact class in Heegaard Floer homology in the closed case, asdefined by Ozsvath and Szabo [OS3], and described from a different point of view in [HKM2].For the sake of completeness, we sketch the definition of the invariantEH(M,Γ, ξ).

First consider the case whenξ is a contact structure on a closed manifoldM . In [HKM2] we usedan open book decomposition compatible withξ to construct a convenient Heegaard decomposition(Σ, α, β) forM in which the contact class was a distinguished element inHF (Σ, α, β). Recall thatan open book decomposition forM is a pair(S, h) consisting of a surfaceS with boundary and ahomeomorphismh : S

∼→ S with h|∂S = id, so thatM ≃ S× [0, 1]/ ∼h, where(x, 1)∼h(h(x), 0)

for x ∈ S and(x, t)∼h(x, t′) for x ∈ ∂S, t, t′ ∈ [0, 1]. A Heegaard decomposition(Σ, β, α) for

−M (recall that the contact class lives in the Heegaard Floer homology of−M) is obtained fromthe two handlebodiesH1 = S × [0, 1

2]/ ∼h andH2 = S × [1

2, 1]/ ∼h, which are glued along the

common boundaryΣ = (S × {12}) ∪ −(S × {0}) by id ∪ h. Take a family of properly embedded

disjoint arcsai that cuts the surfaceS into a disk, and small push-offsbi of ai (in the direction ofthe boundary) such thatbi intersectsai in exactly one point. The compressing disks forH1 andH2, respectively, areDai

= ai × [0, 12] andDbi

= bi × [12, 1]; setαi = ∂Dai

andβi = ∂Dbi. We

call the family of arcsai a basisfor S, and show in [HKM2] that the element of Heegaard Floerhomology that corresponds to the generatorx = (x1, . . . , xn), wherexi is the unique intersectionpoint ofai ×{1

2} andbi ×{1

2}, is independent of the choice of basis forS and the compatible open

book decomposition. Moreover, it is the contact class defined by Ozsvath and Szabo.To define the contact classEH(M,Γ, ξ) in the case of a balanced sutured manifold, we general-

ize the notions of an “open book” and a “basis”, involved in the definition of the contact invariantabove. Let(A,B) be a pair consisting of a surfaceA with nonempty boundary and a subsurfaceB ⊂ A. A collection{a1, . . . , ak} of properly embedded disjoint arcs inA is called abasis for(A,B) if eachai is disjoint fromB andA − ∪k

i=1ai deformation retracts toB. A partial openbook(S,R+(Γ), h) consists of the following data: a compact, oriented surfaceS with nonemptyboundary, a subsurfaceR+(Γ) ⊂ S, and a “partial” monodromy maph : P → S, whereP ⊂ S

is the closure ofS − R+(Γ) andh(x) = x for all x ∈ (∂S) ∩ P . We say that(S,R+(Γ), h) is apartial open book decomposition for(M,Γ) if M ≃ S× [0, 1]/ ∼h, where the equivalence relationis (x, 1)∼h(h(x), 0) for x ∈ P and(x, t)∼h(x, t

′) for x ∈ ∂S, t, t′ ∈ [0, 1]. Since the monodromyh is defined only onP , the space obtained after gluing has boundary consisting ofR+ × {1} and


R− × {0}, whereR− = S − h(P ). The sutureΓ is the common boundary ofR+ × {1} andR− × {0}.

To see a handlebody decomposition ofM from this point of view, letH1 = S× [0, 12]/ ∼, where

(x, t) ∼ (x, t′) if x ∈ ∂S andt, t′ ∈ [0, 12] and letH2 = P × [1

2, 1]/ ∼, where(x, t) ∼ (x, t′) if x ∈

∂P andt, t′ ∈ [12, 1]. It is clear that we can think ofM ≃ S × [0, 1]/ ∼h asM ≃ H1 ∪H2/gluing,

where the handlebodies are glued along portions of their boundary as follows:(x, 12) ∈ H1 is

identified to(x, 12) ∈ H2 and(x, 1) ∈ H2 is identified with(h(x), 0) ∈ H1 for x ∈ P . This leaves

R+ × {12} andR− × {0} as the boundary of the identification space. Now let{a1, . . . , ak} be

a basisfor (S,R+(Γ)) in the sense defined above. Letbi, i = 1, . . . , k, be pushoffs ofai in thedirection of∂S so thatai andbi intersect exactly once. Then it is not hard to see that if we setΣ = (S×{0})∪ (P ×{1

2}), αi = ∂(ai× [0, 1

2]) andβi = (bi×{1

2})∪ (h(bi)×{0}), then(Σ, β, α)

is a Heegaard diagram for(−M,−Γ).The two handlebodiesH1 andH2 defined above by the open book decomposition(S,R+(Γ), h)

carry unique product disk decomposable contact structures. After gluing, they determine a contactstructureξ(S,R+(Γ),h) on (M,Γ). We say that a partial open book decomposition(S,R+(Γ), h) anda contact structureξ arecompatibleif ξ = ξ(S,R+(Γ),h). On the other hand, as in the closed manifoldcase, every contact structureξ with convex boundary on a sutured manifold(M,Γ) gives rise to acompatible partial open book decomposition(S,R+(Γ), h).

3. EXPLANATION OF THEOREM 1.1

3.1. Naturality. We now explain what we mean by a “natural map”Φξ. Recall the followingtheorem from Ozsvath-Szabo [OS7]:

Theorem 3.1 (Ozsvath-Szabo). Given two Heegaard decompositions(Σ, α, β), (Σ, α, β) of aclosed 3-manifoldM , the isomorphism

Ψ : HF (Σ, α, β)∼→ HF (Σ, α, β),

given as the composition of stabilization/destabilization, handleslide, and isotopy maps, is well-defined up to an overall factor of±1 and does not depend on the particular sequence chosen from(Σ, α, β) to (Σ, α, β).

This lack of monodromy allows us to “naturally” identify theisomorphic Heegaard Floer homol-ogy groupsHF (Σ, α, β) andHF (Σ, α, β), up to an overall sign. Sutured Floer homology enjoysthe same naturality property, that is, the isomorphism

Ψ : SFH(Σ, α, β)∼→ SFH(Σ, α, β)

is also well-defined up to an overall factor of±1 and is independent of the same type of choices if(Σ, α, β), (Σ, α, β) are two Heegaard decompositions for(M,Γ).

Next, suppose(Σ′, β ′, α′), (Σ′, β

′, α′) are Heegaard splittings for(−M ′,−Γ′) and (Σ, β, α),

(Σ, β, α) are their extensions to(−M,−Γ). We will restrict ourselves to working with a certainsubclass of Heegaard splittings of(−M ′,−Γ′), namely those that are contact-compatible on aneighborhood of∂M ′, with respect to an invariant contact structureζ which induces the dividingsetΓ′ on ∂M ′. The Heegaard splittings for(−M,−Γ) we will use extend those of(−M ′,−Γ′)and are contact-compatible with respect toξ onM −M ′. AssumeM − int(M ′) has no isolated


components. Then, in the statement of Theorem 1.1, we take the commutativity of the followingdiagram to be the definition of thenaturalityof Φξ:

(3.1.1)

SFH(Σ′, β ′, α′)(Φξ)1

- SFH(Σ, β, α)

SFH(Σ′, β

′, α′)

Ψ1

?

(Φξ)2- SFH(Σ, β, α)

Ψ2

?

Here the vertical mapsΨ1,Ψ2 are the natural isomorphisms of Theorem 3.1 and(Φξ)1, (Φξ)2 arethe maps induced byξ, to be defined in Section 4.

3.2. Explanation of the V factor. Consider(M,Γ, ξ) and a compatible partial open book de-composition(S,R+(Γ), h). Let {a1, . . . , ak} be a basis for(S,R+(Γ)). Consider a larger col-lection {a1, . . . , ak, ak+1, . . . , ak+l} of properly embedded disjoint arcs inS which satisfyai ⊂P , so thatS − ∪k+l

i=1ai is a disjoint union of disksDj , j = 1, . . . , l, and a surface that de-formation retracts toR+(Γ). For eachj, pick zj ∈ Dj and consider a small neighborhoodN(zj) ⊂ Dj. Then{a1, . . . , ak+l} becomes a basis for(S,R+(Γ) ∪ (∪l

j=1N(zj))). The Hee-gaard surface for(S,R+(Γ)) is Σ = (P × {1}) ∪ (S × {0}), whereas the Heegaard surface for(S,R+(Γ)∪(∪l

j=1N(zj))) isΣ′ = Σ−∪lj=1N(zj). As in Section 2, theai determine arcsbi as well

as closed curvesαi, βi. We refer to the procedure of adding extra arcs to a basis for(S,R+(Γ))and extraN(zj)’s toR+(Γ) as “placing extra dots” or “placing extrazj ’s”.

Claim. The effect of placing an extra dot on(S,R+(Γ)) on sutured Floer homology is that oftaking the tensor product withHF (S1 × S2) ≃ V .

Proof. Consider the following situation: Suppose{a1, . . . , ak} is a basis for(S,R+(Γ)). Then addan extra properly embedded arcak+1 ⊂ P of S which is disjoint froma1, . . . , ak and such that onecomponentD1 of S − ak+1 is a half-disk which is contained inP . Also add an extra dotz1 in thecomponentD1. Theαk+1 andβk+1 corresponding toak+1 intersect in exactly two points, and donot interact with the otherαi andβi. By the placement of the extra dot,

SFH(Σ′, {β1, . . . , βk+1}, {α1, . . . , αk+1})

≃ SFH(Σ, {β1, . . . , βk}, {α1, . . . , αk}) ⊗ HF (S1 × S2).

Next, after a sequence of arc slides as in Section 3.1 of [HKM2] (or just handleslides), we canpass between any two bases of(S,R+(Γ) ∪ N(z1)), whereN(z1) is a small disk inS − R+(Γ)aboutz1. SinceSFH is invariant under any sequence of handleslides, the claim follows. �

Now we explain theV factors that appear in Theorem 1.1. Let us consider a partialopen bookdecomposition(S ′, h′) for any contact structureξ′ which is compatible with(M ′,Γ′), and let(S, h)be a partial open book decomposition forξ∪ ξ′ which extends(S ′, h′). If no connected componentof M − int(M ′) is isolated, then a basis{a′1, . . . , a

′k} for (S ′, h′) easily extends to a basis for

(S, h). If there arem isolated components ofM − int(M ′), thenS − ∪ki=1a

′i hasm connected

components which do not intersectR+(Γ) (and hence can never be completed to a basis for(S, h)).Instead, by addingm extra dotsz1, . . . , zm, we can extend{a′1, . . . , a

′k} to a basis for(S,R+(Γ) ∪

(∪mi=1N(zi))).


4. DEFINITION OF THE MAP Φξ

In this section we define the chain map:

Φξ : CF (−M ′,−Γ′) → CF (−M,−Γ),

which induces the map, also calledΦξ by slight abuse of notation, on the level of homology. Let usassume thatM−int(M ′) has no isolated components. The general case follows without additionaleffort, by putting extra dots.

Sketch of the construction.We start by giving a quick overview of the construction ofΦξ. Theactual definition needed to prove naturality is considerably more complicated and occupies theremainder of the section.

Let us first decomposeM = M ′ ∪ M ′′, whereM ′′ = M − int(M ′). Let Σ′ be a Heegaardsurface for the sutured manifold(M ′,Γ′). By definition,Σ′∩∂M ′ = Γ′. Next choose compressingdisksα′, β ′ onΣ′. Also letΣ′′ be a Heegaard surface for the sutured manifold(M ′′,Γ′′ ∪−Γ′).

Although it might appear natural to take the union ofΣ′ andΣ′′ along their common boundaryΓ′ to create a Heegaard surface forM , we are presented with a problem. If we glueM ′ andM ′′ toobtainM , then, on the common boundary ofM ′ andM ′′, R±(Γ′) from M ′ is glued toR∓(−Γ′)from M ′′. As a result, theα-curves forΣ′ andβ-curves forΣ′′ will be paired, and theβ-curvesfor Σ′ andα-curves forΣ′′ will be paired, and we will be mixing homology and cohomology.A way around this problem is to insert the layerN = T × [0, 1], whereT = ∂M ′, so thatM = M ′ ∪ N ∪M ′′, M ′ ∩ N = T × {0}, andM ′′ ∩ N = T × {1}. Let ΣN be a Heegaardsurface for(N, (−Γ′ × {0})∪ (Γ′ × {1})). ThenΣ = Σ′ ∪ΣN ∪Σ′′ is a Heegaard surface forM .

A second issue which arises is that the union of compressing disks forM ′, N , andM ′′ is notsufficient to give a full set of compressing disks forM . Our remedy is to use the contact invari-ant: First we take(Σ′, β ′, α′) to becontact-compatiblenear∂M ′. Roughly speaking, this meansthat(Σ′, β ′, α′), near∂M ′, looks like a Heegaard decomposition arising from a partialopen bookdecomposition of a contact structureζ which is defined near∂M ′ and has dividing setΓ′ on∂M ′.Let Σ′′ be a Heegaard surface which is compatible withξ|M ′′ and letΣN be a Heegaard surfacewhich is compatible with the[0, 1]-invariant contact structureξ|N . We then extendα′ andβ ′ byaddingα′′ andβ ′′ which are compatible withξ ∪ ζ , and then define

Φξ : CF (Σ′, β ′, α′) → CF (Σ, β ′ ∪ β ′′, α′ ∪ α′′),

y 7→ (y,x′′),

wherex′′ is the contact classEH(ξ ∪ ζ), consisting of a point from eachβ ′′i ∩ α′′

i . �

We now give precise definitions. LetT = ∂M ′ and letT × [−1, 1] be a neighborhood ofT = T × {0} with a [−1, 1]-invariant contact structureζ which satisfies the following:

• Tt = T × {t}, t ∈ [−1, 1], are convex surfaces with dividing setΓ′ × {t};• T × [−1, 0] ⊂M ′ andT × [0, 1] ⊂M − int(M ′);• ξ|T×[0,1] = ζ |T×[0,1].

In order to defineΦξ, we need to construct a suitable Heegaard splitting(Σ′, β ′, α′) for thesutured manifold(−M ′,−Γ′) and a contact-compatible extension to(Σ, β, α) for (−M,−Γ). Thiswill be done in several steps.


Step 1: Construction of (Σ′, β ′, α′). In this step we construct(Σ′, β ′, α′) which is contact-compatible with respect toζ near∂M ′. (Although a little unwieldy, we take the construction belowas thedefinitionof a contact-compatible(Σ′, β ′, α′) with respect toζ near∂M ′.) The technique issimilar to the proof of Theorem 1.1 of [HKM3].

Let 0 < ε′ < 1. Start by choosing a cellular decomposition ofT−ε′ so that the following hold:• The1-skeletonK ′

0 is Legendrian;• Each edge of the cellular decomposition lies on the boundaryof two distinct2-cells∆, ∆′;• The boundary of each 2-cell∆ intersects the dividing setΓT

−ε′exactly twice.

Here we use the Legendrian realization principle and isotopT−ε′ , if necessary. LetK ′1 be a finite

collection of Legendrian segments{p} × [−ε′, 0], so that every endpoint(p,−ε′) in T−ε′ lies inK ′

0 ∩ (Γ′ × {−ε′}) and for each connected componentγ of Γ′ there are at least twop’s in γ. Nowlet K ′

2 be a graph attached toK ′0 so thatK ′

0 ∪ K ′2 is a1-skeleton of a cellular decomposition of

M ′−(T×(−ε′, 0]) andint(K ′2) ⊂M ′−(T×[−ε′, 0]). The graphK ′

2 is obtained without referenceto any contact structure. If we setK ′ = K ′

0 ∪K′1 ∪K

′2, then∂N(K ′) is the union of the tubular

portionU and small disksD1, . . . , Ds ⊂ ∂M ′. HereN(G) denotes the tubular neighborhood of agraphG. See Figure 1.

FIGURE 1.

Define the Heegaard surfaceΣ′ to be (a surface isotopic to) the union(R−(Γ′)−∪iDi)∪U . AlsomodifyR+(Γ′) slightly so thatR+(Γ′)−∪iDi is the newR+(Γ′). Theβ ′-curves are meridians (=boundaries of compressing disks) ofN(K ′), and theα′-curves are meridians of the complement,as chosen below.

After a contact isotopy, we may take the standard contact neighborhoodN(K ′0) to beT ×

[−3ε′

2,−ε′

2] with standard neighborhoods of Legendrian arcs of type{q} × [−3ε′

2,−ε′

2], q ∈ Γ′,

removed. Now define the following decomposition ofT × [−3ε′

2, 0] into two handlebodies:

H ′1 = (T × [−3ε′/2, 0]) −N(K ′

0 ∪K′1),

H ′2 = N(K ′

0 ∪K′1),

whereN(K ′0 ∪ K ′

1) denotes the standard contact neighborhood. BothH ′1 andH ′

2 are productdisk decomposable. (The product disk decomposability ofH ′

2 is clear. As forH ′1, observe that


(T × [−ε′

2, 0])−N(K ′

1) is product disk decomposable.) Hence we may writeH ′1 = S ′× [0, 1]/ ∼,

(x, t) ∼ (x, t′) if x ∈ ∂S ′ andt, t′ ∈ [0, 1]. Here∂S ′ × [0, 1]/ ∼ is the dividing set of∂H ′1, and

R+(Γ′) ⊂ S ′ × {1}. Let P ′ = S ′ − R+(Γ′). Similarly we can writeH ′2 = S ′

(2) × [0, 1]/ ∼. If

−Γ′−3ε′/2 is the dividing set ofT−3ε′/2, with the outward orientation induced fromT × [−3ε′

2, 0],

then letP ′(2) = S ′

(2) − R−(−Γ′−3ε′/2). Observe thatP ′ × {1} is identified withP ′

(2) × {0}; let

ψ : P ′ ∼→ P ′

(2) be the corresponding identification map. Also leth′ : Q′ → S ′ be the monodromymap for the partially defined open book, where the domain of definition Q′ is a subset ofP ′ andcontains arcs that correspond to compressing disks ofN(K ′

1).Next let {a′1, . . . , a

′k} be a maximal set of properly embedded arcs onS ′ such that the corre-

spondingα′i = ∂(a′i × [0, 1]), i = 1, . . . , k, on ∂H ′

1 form a maximal collection of curves whichcan be extended to a fullα′ set. (We will abuse notation and call such a maximal collection ofarcs abasisfor (S ′, R+(Γ′)). Let b′i be the usual pushoff ofa′i, and defineβ ′

i = ∂(ψ(b′i) × [0, 1]),i = 1, . . . , k, on∂H ′

2.

Lemma 4.1. {α′1, . . . , α

′k} and {β ′

1, . . . , β′k} can be completed to fullα′ and β ′ sets which are

weakly admissible.

Proof. The decompositionT × [−3ε′

2, 0] = H ′

1 ∪ H ′2 can be extended to a decomposition ofM ′

into two handlebodies. To accomplish this, letK ′2 be the graph defined above, and chooseK ′

3

to be the graph such thatN(K ′2) ∪ N(K ′

3) is a decomposition ofM ′ − (T × [−3ε′

2, 0]) into two

handlebodies. ThenM ′ is the union of the two handlebodiesH ′1 ∪ N(K ′

3) = M ′ − N(K ′) andH ′

2 ∪ N(K ′2) = N(K ′). The collection{α′

1, . . . , α′k} can be completed to a fullα′ set by adding

α′k+1, . . . , α

′k+l which are meridians ofN(K ′

3). On the other hand,{β ′1, . . . , β

′k} can be completed

by adding meridians ofN(K ′2), in addition to∂(c′i×[0, 1]) ⊂ ∂H ′

2, wherec′i are properly embeddedarcs ofR−(−Γ′

−3ε′/2). (Add enough compressing disks ofN(K ′2) so thatH ′

2 ∪N(K ′2) compresses

toH ′2. Then add enough arcsci so thatS ′

(2) −∪ic′i −∪iψ(b′i) deformation retracts to the “ends” of

S ′(2), namely the arcs of intersection withT0.)We now prove that the above extension can be made weakly admissible, without modifyingα′

i

andβ ′i, i = 1, . . . , k. If a periodic domain uses anyα′

i or β ′i with 1 ≤ i ≤ k, then the position

of R+(Γ′) and the relative positions ofa′i andb′i imply that the periodic domain has both positiveand negative signs. Hence assume that we are not usingα′

i or β ′i with 1 ≤ i ≤ k. It is easy to

find disjoint closed curvesγ′k+1, . . . , γ′k+l which are duals ofα′

k+1, . . . , α′k+l, i.e.,γ′i andα′

j havegeometric intersection numberδij , and which do not enter∂H ′

1. (Hence theγ′i do not intersectα′

j with j = 1, . . . , k.) If we wind theα′i, i = k + 1, . . . , k + l, about the curvesγ′i as in [OS1,

Section 5], then the result will be weakly admissible. �

Remark. An alternate way of thinking of the contact compatibility with respect toζ near∂M ′ is asfollows: Start with any Heegaard decomposition(Σ′, β ′, α′) for (−M ′,−Γ′). TakeT × [0, 1] withthe invariant contact structureζ , and form a partial open book decomposition for(T × [0, 1], ζ)by choosing a Legendrian skeleton consisting of sufficiently many arcs of type{p} × [0, 1], wherep ∈ Γ′. Let Σ′

ζ be the corresponding contact-compatible Heegaard surface. Then the Heegaardsurface for(−M ′,−Γ′) which is contact-compatible with respect toζ near∂M ′ is obtained fromΣ′

by attaching two Heegaard surfaces of typeΣ′ζ , one forT×[−3ε′

2,−ε′

2] and another forT×[−ε′

2, 0].


(Note that the choice of arcs of type{p}× [0, 1] for the twoΣ′ζ ’s may be different.) In other words,

we are gluing two copies of(T × [0, 1], ζ) to (−M ′,−Γ′).

Remark. Another approach is to restrict attention to the class of contact-compatible Heegaardsplittings for an arbitrarily chosen, tight or overtwisted, contact(M ′,Γ′, ξ′) compatible with thedividing set. Suppose we show that the definition ofΦξ depends only on the partial open book(S ′, h′) for (M ′,Γ′, ξ′), up to positive and negative stabilizations. By the result of [GG], twoopen books become isotopic after a sequence of positive and negative stabilizations, provided theycorrespond to homologous contact structures. This would show thatΦξ is only dependent on thehomology class ofξ′. However, we would still need to remove the dependence on thehomologyclass.

Step 2: Extension of the Heegaard splitting to(−M,−Γ). We extend the Heegaard splitting(Σ′, β ′, α′) constructed in Step 1 to a Heegaard splitting(Σ, β, α) for (−M,−Γ) which iscontact-compatible with respect toξ ∪ ζ . (Again, we take the construction below as the definition ofcontact-compatibility.)

Let ε′′ > 0. Then we writeM = M ′ ∪ N ∪M ′′, whereN = Nε′′ = T × [0, ε′′] andM ′′ =M ′′

ε′′ = M − int(M ′ ∪N).The contact manifold(M ′′, ξ|M ′′) admits a Legendrian graphK ′′ with endpoints onΓ∂M ′′ and a

decomposition intoN(K ′′) andM ′′ − N(K ′′), according to [HKM3, Theorem 1.1]. Assume thatevery connected component ofK ′′ intersectsΓ at least twice. (This will be useful in Lemma 5.1.)Similarly,(N, ξ|N) admits a Legendrian graphK ′′′ consisting of Legendrian segments{q}×[0, ε′′],where there is at least oneq for each component of the dividing set ofT0. We also assume that theendpoints ofK ′,K ′′, andK ′′′ do not intersect.

We then decomposeM into H1 = (M ′ − N(K ′)) ∪ N(K ′′′) ∪ (M ′′ − N(K ′′)) andH2 =N(K ′) ∪ (N −N(K ′′′)) ∪N(K ′′), respectively. SinceN −N(K ′′′) andN(K ′′) are product diskdecomposable with respect toξ, their union is also product disk decomposable. Hence, we have:

• H2 is a neighborhood of a graphK,• the restriction ofK toM ′′ ∪ (T × [−ε′, ε′′]) is Legendrian, and• restricted toM ′′ ∪ (T × [−3ε′

2, ε′′]),H2 is a standard contact neighborhood ofK ∩ (M ′′ ∪

(T × [−ε′, ε′′])).

Similarly, ((T×[−3ε′

2, 0])−N(K ′

0∪K′1))∪N(K ′′′)∪(M ′′−N(K ′′)) is product disk decomposable

with respect toξ ∪ ζ . Therefore,H1 extendsH ′1 ∪ N(K ′

3) = (S ′ × [0, 1]/ ∼) ∪ N(K ′3) so that

H1 = (S × [0, 1]/ ∼) ∪ N(K ′3), where(x, t) ∼ (x, t′) if x ∈ ∂S and t, t′ ∈ [0, 1]. Here,

R+(Γ) ⊂ S × {1} andS ′ is a subsurface ofS.Therefore, we may extend(Σ′, β ′, α′) to (Σ, β, α) as follows: Consider a collection of arcs

a′′1, . . . , a′′m which form a basis for(S − P ′, R+(Γ)). Then letα′′

i = ∂(a′′i × [0, 1]), andβ ′′i be the

corresponding closed curves derived from the pushoffsb′′i of a′′i . The monodromyh for b′′i can becomputed from the partial open book decomposition onM ′′∪(T×[−3ε′

2, ε′′]). Thenα = α′∪α′′ and

β = β ′ ∪ β ′′, whereα′′ (resp.β ′′) is the collection of theα′′i (resp.β ′′

i ). The contact-compatibilityonΣ − Σ′ immediately implies that the extension is weakly admissible.

We are now in a position to define the chain mapΦξ. Let (Σ′, β ′, α′) be a Heegaard splittingfor (−M ′,−Γ′) which is contact-compatible near∂M ′, and let(Σ, β, α) be a contact-compatibleextension of(Σ′, β ′, α′) to (−M,−Γ). Now letx′′i be the preferred intersection point (i.e., the only


one onS × {1}) betweenα′′i andβ ′′

i , and denote their collection byx′′. Giveny ∈ CF (Σ′, β ′, α′),we define the map:

Φξ : CF (Σ′, β ′, α′) → CF (Σ, β ′ ∪ β ′′, α′ ∪ α′′),

y 7→ (y,x′′).

The fact thatΦξ is a chain map follows from observing that every nonconstantholomorphic mapwhich emanates fromx′′i must nontrivially intersectR+(Γ). Hencex′′ will be used up, and theonly holomorphic maps from(y,x′′) to (y′,x′′) are holomorphic maps fromy to y

′ within Σ′. Thetuplex

′′ will be calledtheEH class onS−P ′. It is immediate from the definition ofΦξ that when(M ′,Γ′, ξ′) is contact,Φξ(EH(M ′,Γ′, ξ′)) = EH(M,Γ, ξ′ ∪ ξ).

Remark. Observe that the set{a′′1, . . . , a′′m} contains arcs ofR+(Γ′) ⊂ S ′. This is one of the

reasonsΣ′ must be contact-compatible near∂M ′.

5. NATURALITY OF Φξ

In this section we prove thatΦξ does not depend on the choices made in Section 4. The proofsare similar to the proofs of well-definition of theEH class in [HKM2, HKM3], and we will onlyhighlight the differences. The proof of naturality under isotopy is identical to the proof of [HKM2,Lemma 3.3], and will be omitted.

5.1. Handlesliding. Consider the Heegaard surfaceΣ and two sets of compressing disks(β, α),(β, α) which are contact-compatible with respect toξ ∪ ζ . In particular,(β ′′, α′′) and (β

′′, α′′)

correspond to bases{a′′1, . . . , a′′m} and{a′′1, . . . , a

′′m} for (S − P ′, R+(Γ)).

There are two types of operations to consider:(A) Arc slides in theS − P ′ region, while fixingα′ andβ ′.(B) Handleslides withinΣ′, while preserving the contact-compatibleα′′ andβ ′′.

Lemma 5.1. Suppose the closure of each component ofS−P ′−R+(Γ) intersectsΓ along at leasttwo arcs. Then one can take(β, α) to (β, α) through a sequence of moves of type(A) or (B).

The required connectivity ofS − P ′ − R+(Γ) was already incorporated in the definition inSection 4.

Proof. According to [HKM2, Lemma 3.3], any basis{a′′i }mi=1 for (S − P ′, R+(Γ)) can be taken to

any other basis{a′′i }mi=1 for (S−P ′, R+(Γ)) through a sequence of arc slides withinS−P ′−R+(Γ),

assuming sufficient connectivity ofS − P ′ − R+(Γ). We must, however, not forgetΣ′. If Σ′ istaken into account, the situation given in Figure 2 must be dealt with: Locally P ′ is attached toS − P ′ along an arcc′ (in the diagram, we have pushedc′ into P ′), and we would like to arc slidea′′i overc′ to obtaina′′i . However, thisc′ may not be ana′i. If this is the case, we must perform asequence of handleslides onα′ andβ ′ first (while fixingα′′ andβ ′′), so thatα′

1 = ∂(c′ × [0, 1]) andβ ′

1 = (c′ ×{1})∪ (h(c′)×{0}). This is possible since we required at least two arcs{p}× [−ε′, 0]in the definition ofK ′

1. Then we may arc slidea′′i overc′. �

We now discuss naturality under the moves (A) and (B).(A). Recall that an arc slide corresponds to a sequence of twohandleslides by [HKM2]. For each

handleslide of an arc slide in theS−P ′ region, the “tensoring withΘ” mapΨ sends theEH classx′′ onS − P ′ − R+(Γ) to theEH class onS − P ′ − R+(Γ), also calledx′′ by abuse of notation.


S − P ′ P ′c′

a′′

i

a′′

i

FIGURE 2.

Since theα′′ andβ ′′ are used up, the restriction ofΨ to the remainingr-tupley ∈ CF (Σ′, β ′, α′)

is the natural “tensoring withΘ′” mapΨ′ fromCF (Σ′, β ′, α′) toCF (Σ′, β ′, α′). Therefore,

Ψ(y,x′′) = (Ψ′(y),x′′).

The proof is identical to the proof of [HKM2, Lemma 5.2].(B). The “tensoring withΘ” operation for a handleslide in theΣ′ region clearly sendsx′′ to x

′′

as well. Therefore we have:Ψ(y,x′′) = (Ψ′(y),x′′).

5.2. Stabilization. In this subsection we prove naturality under stabilization. For this, we needto prove two things: (A) naturality under stabilizations (contact or otherwise) insideM ′, and (B)naturality under positive (contact) stabilizations insideM −M ′.

Let A be a surface with nonempty boundary andB ⊂ A be a subsurface. Letc be a properlyembedded arc inA; after isotopy rel boundary, we assumec intersects∂B transversely and effi-ciently. Then we define thecomplexity ofc with respect to(A,B) as the number of subarcs ofcwhich are contained inB and have both endpoints on the common boundary ofA− B andB.

Given two Heegaard splittings(Σ′, β ′, α′) and (Σ′, β

′, α′) for (−M ′,−Γ′) which are contact

compatible with respect toζ near∂M ′ (i.e., of the type constructed in Step 1 of Section 4) andtheir extensions(Σ, β, α) and(Σ, β, α) to (−M,−Γ) which are contact compatible with respectto ξ ∪ ζ (i.e., of the type constructed in Step 2 of Section 4), we firstfind a common stabilization(Σ, β, α), which is also contact compatible with respect toξ ∪ ζ . If we place a line (resp. tilde)over a symbol, then it stands for the corresponding object for (Σ, β, α) (resp.(Σ, β, α)), e.g.,K

′is

K ′ for (Σ, β, α). (The exception isR+(Γ′), which refers to the closure ofR+(Γ′).)

(A) We will first discuss the subdivision on(−M ′,−Γ′). Takeε′ > 0 so thatε′ ≪ ε′, ε′. Given theLegendrian portionL′

0 = K ′0 ∪K

′1 of the1-skeletonK ′, we successively attach Legendrian arcsc′i

toL′i to obtainL′

i+1 in the following order:(α) First attach arcs to construct the Legendrian1-skeleton of a sufficiently fine Legendrian

cell decomposition ofT−eε′ , after possibly applying Legendrian realization.(β) Then attach Legendrian arcs of the type{p} × [−ε′, 0] with p ∈ Γ′.


The arcs are attached so that in the end we obtain a Legendriangraph containingL′0 = K ′

0 ∪ K′1,

whereK ′0 is a Legendrian skeleton ofT−eε′ andK ′

1 is the union of arcs of type{p}× [−ε′, 0], and sothat the restrictions ofL′

0 andL′

0 to T × [−ε′, 0] are subsets ofL′0. If we start withL

′

0 = K′

0 ∪K′

1

instead, then there is a sequenceL′

i which eventually yields a Legendrian graph containingL′0. The

stabilizations of contact type will be treated in (A1). Next extendL′0 to a common refinementK ′

of K ′ andK′by subdividing onM ′ − (T × [−ε′, 0]). These stabilizations will be treated in (A2).

(A1) The above attachments of Legendrian arcs are done in the same way as in [HKM3, Theo-rem 1.2] and in particular [HKM3, Figures 1, 3, and 4].

The attachment of arcsc′i ⊂ T−eε′ of type (α) can be decomposed into three stages (α1), (α2) and(α3). Figure 3 depicts an arc of type (α1). An arcc′i of type (α1) connects between{p} × [−ε′, 0]and{q} × [−ε′, 0], wherep, q ∈ Γ′. After attaching all the arcs of type (α1), we attach the arcsof type (α2), depicted in Figure 4. Here, the arcc′i connects between two arcs of type (α1) anddoes not cross the dividing set ofT−eε′. Finally, an arc of type (α3) is an arc that intersects thedividing set ofT−eε′ exactly once, and in its interior. Arcs of type (α1), (α2), and (α3) are sufficientto construct the Legendrian skeleton ofT−eε′. Figure 5 depicts an arc attachment of type (β).

T0 = ∂M ′

Γ′

d′i

c′i

FIGURE 3. Arc of type (α1). The surface in the back isT0 = ∂M ′, whose orienta-tion as the boundary ofM ′ points into the page. The cylinders on the left and rightare thickenings of arcs{p} × [−ε′, 0] and{q} × [−ε′, 0] of K ′

1, and the horizontalcylinder is a thickening ofc′i. The blue arc isc′i and the green arc is its isotopic copyd′i.

In particular, we observe that the following holds:

• Each endpoint ofc′i lies onΓ∂(M ′−N(L′

i)), andint(c′i) ⊂ int(M ′ −N(L′

i)).• N(L′

i+1) = N(c′i)∪N(L′i), andL′

i+1 is a Legendrian graph so thatN(L′i+1) is its standard

neighborhood.• There is a Legendrian arcd′i on∂(M ′−N(L′

i)) with the same endpoints asc′i, after possibleapplication of the Legendrian realization principle. The arc d′i intersectsΓ∂(M ′−N(L′

i))only

at its endpoints.• The Legendrian knotγ′i = c′i ∪ d

′i bounds a disk inM ′ −N(L′

i) and hastb(γ′i) = −1 withrespect to this disk. This implies thatc′i andd′i are isotopic relative to their endpoints insidethe closure ofM ′ −N(L′

i).


c′i

d′i

FIGURE 4. Arc of type (α2).

c′id′i

FIGURE 5. Arc of type (β).

For simplicity, consider the situation of attaching a single arcc′0 toL′0 to obtainL′

0. Consider the(very) partial open book decomposition onT × [−3ε′

2, 0], corresponding to the decomposition into

H ′2 = N(K ′

0∪K′1) andH ′

1 = (T×[−3ε′

2, 0])−N(K ′

0∪K′1) = S ′×[0, 1]/ ∼. The monodromy map

ish′ : Q′ → S ′ as before. The arcc′0 can be viewed as a Legendrian arc onS ′×{12} with endpoints

on ∂Q′ × {12}. Hence, removing a neighborhood ofc′0 from H ′

1 and adding it toH ′2 is equivalent

to the following positive (contact) stabilization: Lete′0 be the Legendrian arc onS ′ = S ′ × {1}which is Legendrian isotopic toc′0 rel endpoints, via an isotopy insideH ′

1. Add a1-handle toS ′

along the endpoints ofe′0 to obtainS ′, and completee′0 to a closed curveγ0 on S ′ by attaching the


core of the1-handle. Then the stabilization is the data(S ′, R+(Γ′), Q′, h′ = Rγ0◦ h′), whereRγ0

is a positive Dehn twist aboutγ0 andQ′ is the domain ofh′. Let (S, h) (resp.(S, h)) be the partialopen book which extends(S ′, h′) (resp.(S ′, h′)).

Lemma 5.2. The arcc′0 can be chosen so that the correspondinge′0 ⊂ S ′ has complexity0 withrespect to(S ′, P ′) and complexity at most1 with respect to(S ′, R+(Γ′)).

Proof. We treat the (α1) case, and leave the other cases to the reader. Refer to Figure 3; in thefigure replacec′i, d

′i by c′0, d

′0. If d′0 intersectsR+(Γ′), thend′0, viewed onS ′ × {1}, is the desired

isotopic copye′0 of c′0. It is clear thate′0 has complexity0 with respect to(S ′, P ′) and complexity1with respect to(S ′, R+(Γ′)). On the other hand, ifd′0 intersectsR−(Γ′), then we need to isotopc′0towardsT−ε′/2 instead, in order to obtaine′0. The procedure is still the same — in Figure 3 assumethat the surface in the back isT−ε′/2 (instead ofT0) so that∂(M ′ − (T × (−ε′

2, 0])) points out of

the page. The resultinge′0 has complexity0 with respect to both(S ′, P ′) and(S ′, R+(Γ′)). �

In view of Lemma 5.2, there exists a basis{a′1, . . . , a′k} of (S ′, R+(Γ′)) and an extension to a

basis{a′1, . . . , a′k, a

′′1, . . . , a

′′m} of (S,R+(Γ)), so thate′0 does not intersect any basis element. Let

a′0 be the cocore of the1-handle of the stabilization alonge′0, and letb′0 be the pushoff ofa′0. Thenlet α′

0 = ∂(a′0 × [0, 1]) andβ ′0 = (b′0 × {1}) ∪ (Rγ0

(b′0) × {0}), where both are viewed on∂H ′1.

Observe thatβ ′0 does not intersect any of{α′

1, . . . , α′k, α

′′1, . . . , α

′′m}, whereα′

i = ∂(a′i × [0, 1]) andα′′

i = ∂(a′′i × [0, 1]). Sinceβ ′0 also does not intersect the remainingα′-curvesα′

k+1, . . . , α′k+l, the

only intersection betweenβ ′0 and someα-curve is the sole intersection withα′

0.Let Ψ : CF (Σ, β, α) → CF (Σ, β, α) be the composition of (Heegaard decomposition) stabi-

lization and handleslide maps corresponding to the stabilization alonge′0. We have the following:

Lemma 5.3. The mapΨ : CF (Σ, β, α) → CF (Σ, β, α)

is given by:(y,x′′) 7→ (Ψ′(y), x′′),

wherey ∈ CF (Σ′, β ′, α′), x′′ (resp.x′′) is theEH class in theS−P ′ region(resp.S− P ′ region)for (Σ, β, α) (resp.(Σ, β, α)), andΨ′ is the natural map from(Σ′, β ′, α′) to (Σ′, β ′, α′).

Proof. This follows from the technique in [HKM3, Lemma 3.5]. We use the fact that the onlyintersection betweenβ ′

0 and anα-curve is the unique intersection withα′0. We decompose the

positive stabilization alonge′0 into a trivial stabilization, followed by a sequence of handleslides.(By a trivial stabilizationwe mean the addition of a1-handle toΣ′, together with curvesα′

0 andβ ′0

that intersect each other once, say atx′0, and no otherα′i, β

′i, i = 1, . . . , s, and where the regions

of Σ′ − ∪si=0α

′i − ∪s

i=0β′i adjacent tox′0 are path-connected toR+(Γ′).) This is done exactly as

described in [HKM3, Lemma 3.5]: wheneverβ ′i (could beβ ′′

i ) intersectse′0 × {0}, andβ′

i is theresult of applying a positive Dehn twist aboutγ0×{0}, thenβ

′

i can be obtained fromβ ′i by applying

a trivial stabilization, followed by handleslides overβ ′0 as in [HKM3, Figure 7]. Here, the triple

diagrams are weakly admissible for the same reasons as [HKM3, Lemma 3.5].The slight complication that we need to keep in mind is that the arcsh(b′′i ), whereb′′i is the

usual pushoff ofa′′i in S − P ′, may enter the regionS ′ and intersecte′0. If h(b′′i ) intersectse′0, thenthe “tensoring withΘ” map corresponding to handleslidingβ ′′

i overβ ′0 sendsEH to EH in the


S − P ′ region and restricts to the natural “tensoring withΘ” map in theΣ′ region. (The proof isthe same as that of [HKM3, Lemma 3.5]. Also refer to [HKM3, Figure 8].) On the other hand, ifβ ′

i intersectse′0 × {0}, theS −P ′ region is unaffected (henceEH is mapped toEH in theS −P ′

region), and we are doing a standard handleslide map in theΣ′ region. �

(A2) Next we discuss the effect of a stabilization, in the handlebody sense, in the portion ofM ′

which is not contact-compatible, i.e., away fromT × [−ε′, 0]. Assume all the contact stabilizationshave already taken place onT × [−ε′, 0]. By abuse of notation, we resetε′ = ε′ and use the samenotationS ′, P ′, Q′, h′, Σ′, K ′, K ′

0, K′1, K

′2, K

′3, H

′1, H

′2, used in Step 1 of Section 4, for the new

(finer) Heegaard decomposition which is contact-compatible onT × [−ε′, 0] = T × [−ε′, 0].

Claim. The stabilization can be decomposed into a trivial stabilization, followed by a sequence ofhandleslides which avoidsR−(Γ′).

Proof. Observe that the arc of stabilizationc′0 is contained inN(K ′3). The meridian of the tubular

neighborhood ofc′0 will be calledβ ′0 and it is not difficult to see that there exists a curveα′

0 whichintersectsβ ′

0 once and lies on∂N(K ′3) − ∂H ′

1. (Note thatβ ′0 only intersectsα′

0.) After a sequenceof handleslides that takes place away from∂H ′

1 (in fact the change takes place in a neighborhoodof α′

0 ∪ β′0), we may assume thatα′

0 andβ ′0 satisfy the conditions of a trivial stabilization. �

The claim implies that the handleslides and stabilization do not interact withh(a′′i ) (or equiva-lently with β ′′

i ). Hence theEH class is mapped to theEH class in theS − P ′ region, and we aredoing a standard sequence of handleslide maps plus one stabilization in theΣ′ region.

(B) Next we discuss the subdivision onM − M ′. Take ε′′ > 0 so thatε′′ ≪ ε′′, ε′′. ConsiderNeε′′ = T × [0, ε′′]. OnNeε′′ attach the following arcs in the given order toK ′′′ to obtain a commonrefinement of the restriction ofK ′′′ andK

′′′toNeε′′:

(1) First attach the Legendrian skeleton of a sufficiently fine Legendrian cell decomposition ofTeε′′ , after possibly applying Legendrian realization.

(2) Then attach Legendrian arcs of type{q} × [0, ε′′] with q ∈ Γ′.

Each of the above Legendrian arc attachments leads to a stabilization — however, since the arcsare contained in the complement ofS× [0, 1]/ ∼, the stabilization is aprecompositionh 7→ h◦Rγ .More precisely, letc be the Legendrian attaching arc. There is an isotopy ofc rel endpoints, insidethe complement ofS × [0, 1]/ ∼, to an arce ⊂ S − P ′, viewed onS × {1}, and also toh(e),viewed onS×{0}. Observe thath(e) may enter theR−(Γ′) region. Add a1-handle toS along theendpoints ofe to obtainS, and completee to a closed curveγ by attaching the core of the handle.

In the following lemma, we identifyS = S×{0} and determine the complexity of the restrictionof h(e) to S ′ and toP ′. Observe thath(e) ∩ S ′ = h(e) ∩ R−(Γ′).

Lemma 5.4.

(1) h(e) has complexity at most one with respect to(S, S ′).(2) h(e) has complexity at most one with respect to(S, P ′).

Proof. We isotopc rel endpoints in two stages: first through the product structure given by thecomplement ofS × [0, 1]/ ∼, and then through the product structure given byS × [0, 1]/ ∼.


(1) follows from examining the proof of [HKM3, Theorem 1.2] as in Lemma 5.2. The threetypes of arc attachments are (α1), (α2), and (α3). Consider an arc of type (α1), given in Figure 3.In the current case, the surface in the back is stillT0 = ∂M ′, but the orientation is pointing out ofthe page; alsoc′i andd′i should be changed toc andd. If the arcd intersectsR−(Γ′) (where theorientation on∂M ′ is the orientation induced fromM ′), thend is an arc onS × {0}, which meansthatd = h(e). Henceh(e) has complexity1 with respect to(S,R−(Γ′)), and also complexity atmost1 with respect to(S, S ′). On the other hand, ifd intersectsR+(Γ′), thend = e. Henceh(e)is contained in(S − S ′) × {0} and has complexity zero with respect to(S, S ′). It follows thath(e) also has complexity zero with respect to(S, P ′). The arcs of type (α2) and (α3) are treatedsimilarly.

(2) follows from considerations similar to [HKM3, Section 5, Example 2]. Supposed = h(e),i.e.,d intersectsR−(Γ′). (The situation ofd = e is easier, and is left to the reader.) Then Figure 6depicts what happens when we pushh(e), viewed as an arc onS×{0}, toS×{1}. The surface inthe front isT0 and the surface in the back isT−ε′/2. The blue arc(h(e)|S′)0 is the isotopic copy ofh(e)|S′ onT0 or S ′×{0}, and the green arc(h(e)|S′)−ε′/2 is the copy onS ′ ×{1} which intersectsT−ε′/2. We easily see thath(e) has complexity1 with respect to(S, P ′). The arc corresponding to

(h(e)|S′)0

−+

(h(e)|S′ )−ε′/2

T0

T−ε′/2

FIGURE 6.

[HKM3, Figure 4] is simpler, and does not enterP ′. �

The following lemma follows from Lemma 5.4.

Lemma 5.5. There exists a basis{a′1, . . . , a′k} for (S ′, R+(Γ′)) and basis{a′′1, . . . , a

′′m} for (S −

P ′, R+(Γ)) such that the following hold:

(1) a′′i does not intersecte for all i;(2) all but one of thea′i or a′′i are disjoint fromh(e);(3) one ofa′1 or a′′1 intersectsh(e).

The basis{a′1, . . . , a′k} can be used to constructα′, β ′ for Σ′, and the basis{a′′1, . . . , a

′′m} gives

an extension toα, β onΣ.


Proof. Consider the (α1) case whered intersectsR−(Γ′). By (2) of Lemma 5.4, there exists a basis{a′1, . . . , a

′k} for (S ′, R+(Γ′)) so thata′1 intersectsh(e) once, and the remaininga′i, i = 2, . . . , k,

do not intersecth(e). Next observe thate ⊂ S − P ′ and does not intersect theR+(Γ) region. It ispossible to choose a basis{a′′1, . . . , a

′′m} for (S −P ′, R+(Γ)) which does not intersecte, as well as

h(e). The other cases are similar. �

Let us consider the case wherea′1 intersectsh(e). (The other case is similar.) When we stabilizeS alonge, we add the cocorea′′0 of the 1-handle and obtain the correspondingα′′

0 andβ ′′0 . The

only intersection point ofα′′0 with anyβ arc is withβ ′′

0 , which we callx′′0. Hence we expect thefollowing diagram to commute:

SFH(β ′, α′)Φξ

- SFH(β, α)

SFH(β ′, α′)

Ψ

? Φξ- SFH(β ∪ {β ′′

0}, α ∪ {α′′0})

Ψ

?

However, our stabilization is not a trivial stabilization,asα′1 intersectsβ ′′

0 in one point. Thereforewe need to decompose the stabilization into a trivial stabilization, followed by a handleslide. Thiswill be done in a manner similar to [HKM3, Lemma 3.5]. Letγ′′i be pushoffs ofα′′

i for all i, γ′j bepushoffs ofα′

j for all j 6= 1, andγ′1 be obtained by pushingα′1 overα′′

0, as depicted in Figure 7. InFigure 7, we place black dots in regions that are path-connected toΓ; in other words, holomorphiccurves are not allowed to enter such regions.

α′′

0

β′′

0

α′

1

γ′′

0

γ′

1

FIGURE 7.


Now consider the following diagram:

SFH(β ′, γ′)Φξ

- SFH(β, γ)

SFH(β ∪ {β ′′0}, γ ∪ {γ′′0})

Ψ2

?

SFH(β ′, α′)

Ψ1

? Φξ- SFH(β ∪ {β ′′

0}, α ∪ {α′′0})

Ψ3

?

For the termSFH(β ′, γ′) in the upper left-hand corner,γ′ is the set consisting of all theγ′i; forSFH(β, γ) in the upper right-hand corner,γ is the set consisting of all theγ′′i andγ′i, with theexception ofγ′′0 . This means that(β, γ) is obtained from the middle diagram of Figure 7 by adestabilization; henceγ effectively consists of pushoffs ofα. The mapΨ2 is the map whichcorresponds to the trivial stabilization, andΨ3 is the handleslide map which is the “tensoring withΘ” map, whereΘ is the top generator ofCF (γ ∪ {γ′′0}, α ∪ {α′′

0}). The slightly tricky feature ofthis diagram is that atSFH(β ∪ {β ′′

0}, γ ∪ {γ′′0}) we leave the category of diagrams which nicelydecompose into theM ′ part and theM −M ′ part. The mapΨ1 is the “tensoring withΘ′” map,whereΘ′ is the top generator ofCF (γ′, α′). The mapsΦξ are the “tensoring with theEH class”maps. By the placement of the dots in the right-hand diagram of Figure 7, it is not difficult to seethe following:

Lemma 5.6. TheEH class on(β ′′ ∪ {β ′′0}, γ

′′ ∪ {γ′′0}) is mapped to theEH class on(β ′′ ∪{β ′′

0}, α′′ ∪ {α′′

0}) via Ψ3.

Proof. The Heegaard triple diagram is weakly admissible for the same reason as Lemma 3.5 of[HKM3], and the details are left to the reader. In the right-hand diagram of Figure 7, consider thelargest closed connected componentR which is bounded by theα ∪ {α′′

0}, β ∪ {β ′′0}, γ ∪ {γ′′0}

curves, does not contain a dot (i.e., does not intersectΓ), and contains the unique intersection pointof β ′′

0 andγ′′0 . The setR is an annulus which is bounded byα′′0 andγ′′0 on one side, and byα′

1

andγ′1 on the other. There are two points ofΘ in R, but only one intersection point ofβ ∪ {β ′′0}

andγ ∪ {γ′′0}. Hence one of theΘ points cannot be used towardsR, namely the intersection pointbetweenα′

1 andγ′1. This allows us to “erase” the boundary component ofR consisting ofα′1 and

γ′1, and conclude thatα′′0 ∩ γ′′0 is mapped toα′′

0 ∩ β ′′0 . The rest of the tuples of theEH class are

straightforward. �

Once theEH portion is used up by Lemma 5.6,Ψ3 restricts toΨ1 on the rest of the tuples, i.e.,those that lie onΣ′. The commutativity of the above diagram follows.

Now, insideM ′′eε′′ = M − int(M ′ ∪ Neε′′), we attach Legendrian arcs to the Legendrian graph

which plays the role ofK ′′ so that we have a common refinement ofK ′′ andK′′. An arc attachment

in this region corresponds to a straightforward stabilization alongc which lies inS − S ′. The map


on Floer homology induced by such a stabilization clearly sendsEH to EH and has a naturalrestriction to theΣ′ region.

6. PROPERTIES OF THE GLUING MAP

In this section we collect some standard properties of the gluing map.

Theorem 6.1 (Identity). Let (M,Γ) be a sutured manifold andξ be a [0, 1]-invariant contactstructure on∂M × [0, 1] with dividing setΓ × {t} on∂M × {t}. The gluing map

Φξ : SFH(−M,−Γ) → SFH(−M,−Γ),

obtained by attaching(∂M × [0, 1], ξ) onto(M,Γ) along∂M ×{0}, is the identity map(up to anoverall± sign if overZ).

The proof of Theorem 6.1 will be given in Subsection 6.1, after some preliminaries.

Proposition 6.2 (Composition). Consider the inclusions(M1,Γ1) ⊂ (M2,Γ2) ⊂ (M3,Γ3) ofsutured manifolds, and letξ12 be a contact structure onM2 − int(M1) which has convex boundaryand dividing setsΓi on∂Mi, i = 1, 2. Similarly defineξ23. If

Φ12 : SFH(−M1,−Γ1) → SFH(−M2,−Γ2),

Φ23 : SFH(−M2,−Γ2) → SFH(−M3,−Γ3),

Φ13 : SFH(−M1,−Γ1) → SFH(−M3,−Γ3),

are natural maps induced byξ12, ξ23, andξ12 ∪ ξ23, respectively, thenΦ23 ◦ Φ12 = Φ13, (up to anoverall± sign if overZ).

Proof. This is immediate, once we unwind the definitions. Let(S1, R+(Γ1), h1) be a partial openbook decomposition for(M1,Γ1, ξ1). Hereξ1 is arbitrary and may be tight or overtwisted. Let(Σ1, β1, α1) be the corresponding contact-compatible Heegaard splitting. We assume that the par-tial open book forξ1 is sufficiently fine and the Heegaard splitting is of the type given in Step 1of Section 4. Extend(S1, R+(Γ1), h1) to (S2, R+(Γ2), h2) via ξ12 (of the type given in Step 2 ofSection 4), and letx12 be theEH class for the arcs which complete a basis for(S1, R+(Γ1), h1) toa basis for(S2, R+(Γ2), h2). Similarly definex23. Then the chain mapΦ12 maps:

y 7→ (y,x12),

andΦ23 maps:(y,x12) 7→ (y,x12,x23).

This is the same asΦ13(y), since(x12,x23) is theEH class for the arcs which complete a basisfor (S1, R+(Γ1), h1) to a basis for(S3, R+(Γ3), h3). Moreover the extension is of the type given inStep 2 of Section 4. �

Proposition 6.3(Associativity). Let(M1,Γ1), (M2,Γ2), and(M3,Γ3) be pairwise disjoint suturedsubmanifolds of(M,Γ). Let ξ be a contact structure defined onM − int(M1 ∪M2 ∪M3) whichhas convex boundary and dividing setsΓ on ∂M and Γi on ∂Mi. Let (M12,Γ12) be a suturedsubmanifold of(M,Γ) which is disjoint fromM3, containsM1 andM2, and has dividing setΓ12

on∂M12 with respect toξ. Similarly define(M23,Γ23). Then the maps

(6.0.1) SFH(−M1,−Γ1) ⊗ SFH(−M2,−Γ2) ⊗ SFH(−M3,−Γ3)


Φξ|M12−M1−M2⊗ id

- SFH(−M12,−Γ12) ⊗ SFH(−M3,−Γ3)Φξ|M−M12−M3- SFH(−M,−Γ)

and

(6.0.2) SFH(−M1,−Γ1) ⊗ SFH(−M2,−Γ2) ⊗ SFH(−M3,−Γ3)

id⊗ Φξ|M23−M2−M3- SFH(−M1,−Γ1) ⊗ SFH(−M23,−Γ23)Φξ|M−M1−M23- SFH(−M,−Γ)

are identical(up to an overall± sign if overZ).

Proof. Let (Si, R+(Γi), hi) be a partial open book decomposition for(Mi,Γi, ξi), i = 1, 2, 3, whereξi is arbitrary. Let(Σi, βi, αi) be the corresponding contact-compatible Heegaard splitting. Todefine the chain mapΦ12 = Φξ|M12−M1−M2

, we extend(Si, R+(Γi), hi) to a partial open bookdecomposition(S12, R+(Γ12), h12) for (M12,Γ12, ξ|M12−M1−M2

∪ ξ1 ∪ ξ2). Then

Φ12 : (y1,y2) 7→ (y1,y2,x12),

wherex12 is theEH class for the arcs which complete a basis for

(S1, R+(Γ1), h1) ∪ (S2, R+(Γ2), h2)

to a basis for(S12, R+(Γ12), h12). Next we complete a basis for

(S12, R+(Γ12), h12) ∪ (S3, R+(Γ3), h3)

to a basis for the open book(S123, R+(Γ123), h123) corresponding to(M, ξ ∪ ξ1 ∪ ξ2 ∪ ξ3), and letx(12)3 be theEH class for the completing arcs. Hence,Φ(12)3 = Φξ|M−M12−M3

maps:

(y1,y2,x12) ⊗ y3 7→ (y1,y2,y3,x12,x(12)3).

Similarly,Φ1(23) ◦ (id⊗ Φ23) sends

(y1,y2,y3) 7→ (y1,y2,y3,x23,x1(23)).

By applying a sequence of positive stabilization and basis change moves in theM − int(M1 ∪M2 ∪M3) region, as proven in Section 5, we see that(x12,x(12)3) is taken to(x23,x1(23)). �

Proposition 6.4. Let (M ′,Γ′) be obtained from(M,Γ) by decomposing along a properly embed-ded surfaceT with ∂-parallel dividing setΓT . The inclusion/direct summand map

SFH(−M ′,−Γ′) → SFH(−M,−Γ)

given in[HKM3, Section 6]is the same as the gluing map of Theorem 1.3.

Proposition 6.4 can be proved using techniques as that are similar to the proof of Theorem 6.1below, and is left to the reader.

6.1. Proof of Theorem 6.1. In this subsection we prove Theorem 6.1.


6.1.1. Attaching a trivial bypass.Let (S ′, R+(Γ′), h′) be a partial open book decomposition forthe triple(M ′,Γ′, ξ′), whereξ′ is any contact structure. We determine the effect of attaching atrivial bypass on the partial open book(S ′, R+(Γ′), h′). Let (M,Γ) be the result of attaching abypassD to (M ′,Γ′) along a trivial arc of attachmentc, and thickening. (Of course(M,Γ) and(M ′,Γ′) are isotopic, but we keep the distinction.) The boundary∂D decomposes into two arcswhich intersect at their common endpoints: the arc of attachmentc ⊂ ∂M ′ and the bypass arcd.As described in [HKM3, Section 5, Example 5], attaching a neighborhoodN(D) ofD is equivalentto attaching a tubular neighborhood ofd (a 1-handle), followed by a neighborhoodD0 × [0, 1] ofa diskD0 ⊂ D which is a slight retraction ofD (a 2-handle). Now, letK ′ be the Legendriangraph in(M ′,Γ′) with endpoints onΓ′, which gives rise to the partial open book decomposition(S ′, R+(Γ′), h′). Then the Legendrian graphK for (S,R+(Γ), h) is obtained fromK ′ by takingthe union with a Legendrian arc{pt} × [0, 1] ⊂ D0 × [0, 1], which is the cocore of the2-handle.The complement ofN(K) in M is product disk decomposable. This decomposition gives rise toan extension(S,R+(Γ), h) of (S ′, R+(Γ′), h′) to (M,Γ), obtained by attaching a1-handle toS ′.Let a0 be the cocore of the1-handle. The monodromyh′ on theS ′-portion remains unchanged.

We now apply the calculations done in [HKM3, Section 5, Example 5] to obtain a description of(S,R+(Γ), h). There are two cases of trivial bypasses:c cuts off a half-diskD1 of ∂M ′−Γ′ whichis either inR+(Γ′) or inR−(Γ′). (If c cuts off two half-disksD1,D2 and∂D1, ∂D2 intersect alongan arc ofΓ′, then we takeD1 to be the “smaller” half-disk, i.e.,∂D1 ∩ Γ′ ⊂ ∂D2 ∩ Γ′.) The twocases will be called theR+ andR− cases, respectively. See Figure 8 for the determination of themonodromy corresponding to the portion that is attached.

+

−

+

−

+

−

FIGURE 8. The top row is theR+ case and the bottom row is theR− case. Thediagrams on the right-hand side depict the1-handle attached toS ′ to obtainS. Theblue arca0 completes a basis for(S ′, R+(Γ′), h′) to a basis(S,R+(Γ), h), and thegreen arc is its imageh(a0).


6.1.2. Effect of a trivial bypass attachment on sutured Floer homology. Let (Σ′, β ′, α′) be thecontact-compatible Heegaard splitting for a basis{a′1, . . . , a

′k} for (S ′, R+(Γ′), h′) and (Σ, β =

β ′∪{β0}, α = α′∪{α0}) be its extension with respect to{a′1, . . . , a′k, a0} for (S,R+(Γ), h). Here

α0 = ∂(a0 × [0, 1]) andβ0 = (b0 × {1}) ∪ (h(b0) × {1}), whereb0 is the usual pushoff ofa0. Letx0 be theEH class corresponding toa0.

Let c be the trivial arc of attachment along∂M ′ and letD1 ⊂ ∂M ′ be the half-disk coboundedby a subarc ofc and an arc ofΓ′, as described above. Assume, without loss of generality, that noendpoint ofK ′ lies on∂D1. If D1 ⊂ R+(Γ′), then the only intersection ofα0 with anyβi is x0.On the other hand, ifD1 ⊂ R−(Γ′), then the only intersection ofβ0 with anyαi is x0. Therefore,for bothR+ andR−, the inclusion map

CF (Σ′, β ′, α′) → CF (Σ, β, α),

y 7→ (y, x0)

is an isomorphism of chain complexes. Therefore, tensoringwith x0 gives an isomorphism

Φ : SFH(−M ′,−Γ′)∼→ SFH(−M,−Γ).

However, in order to show that the map is an identity morphism, we need to decompose thestabilization (i.e., attaching a handle toΣ′ and addingα0, β0 to α′, β ′) into a trivial stabilizationand a sequence of handleslides. Let us consider theR− case. (TheR+ case is left to the reader.)In this case,β0 only intersectsα0, but α0 can intersectβ ′

i. If there are no other intersections,then we are done, since we have a trivial stabilization. Otherwise, consider the pushoffa0 of a0,obtained by isotoping the endpoints ofa0 along∂S ′, against the orientation ofΓ′. If we stabilize(S ′, R+(Γ′), h′) alonga0, then all the intersections betweenα0 andβ ′

i will be eliminated, sincethe composition with the positive Dehn twist forces the arcsto go around the core of the attached1-handle. In its place, ifa′k+1 is the cocore of the1-handle, then its image under the monodromywill intersecta0 exactly once. Let us rename open books and assume(S ′, R+(Γ′), h′) already hasthis property, namely we may assume that there is only one intersection betweenα0 and∪iβ

′i. The

rest of the argument is identical to that of Lemma 5.6, and will be omitted.

6.1.3. Reduction to a sequence of trivial bypasses.Suppose now that(M ′,Γ′) is a sutured sub-manifold of(M,Γ), M −M ′ = ∂M ′ × [0, 1], ∂M ′ = ∂M ′ × {0}, and the contact structureξ on∂M ′ × [0, 1] with convex boundary conditionΓ ∪ Γ′ is [0, 1]-invariant. We now prove that there isan extension of(S ′, R+(Γ′), h′) for (M ′,Γ′) to (S,R+(Γ), h) for (M,Γ), of the type constructedin Step 2 of Section 4, which can be decomposed into a sequenceof trivial bypass attachments.The nature of this extension is such that it is obtained by adding “horizontal” Legendrian arcs oftypeδ × {t} ⊂ ∂M ′ × [0, 1] and “vertical” Legendrian arcs of type{p} × [0, t]. We will see howthe extension can be thought of as a sequence of trivial bypass additions.

Observe that, when we attach a neighborhoodN(d) of a trivial bypass arcd, then the resultcan be viewed more symmetrically as in Figure 9. (This we leave as an exercise for the reader.)This means thatd can be viewed as the concatenation of three Legendrian arcs:two “vertical” arcs{p1, p2} × [0, t] and a “horizontal” arcδ × {t}, whereδ connectsp1 andp2. In this subsectionwe make the assumption that allδ’s, possibly with subscripts, do not intersectΓ′ in the interior ofδ, and allpi’s are inΓ′. Let c′ be the component ofc − Γ′ which is not part of∂D1. Slide theendpoints ofc′ in the direction ofΓ′ if c′ ⊂ R−(Γ′) and in the direction of−Γ′ if c′ ⊂ R+(Γ′). Wewill call the resulting Legendrian arca0; this notation agrees with the notation for the stabilizing


N(d)

D0

a0

FIGURE 9. Attaching a trivial bypass arcd. We stabilize alonga0 before attachingthe bypass.

arc, used in Section 6.1.2. If we take a Legendrian-isotopiccopy ofa0 insideM ′ via an isotopywhich fixes the endpoints, then we perform a stabilization asin Section 6.1.2 along the copy ofa0

before attaching the bypass.Now, if we have a Legendrian graph consisting of{p1, p2, p3} × [0, t], together withδi × {t},

i = 1, 2, with ∂δi = {pi, pi+1}, then attaching its standard Legendrian neighborhood is equivalentto attaching two bypass arcs as given in Figure 10; this is readily seen by sliding an endpoint ofδ2 × {t} along the dividing set on the boundary of the union ofM ′ and the neighborhood of theLegendrian arc({p1} × [0, t]) ∪ (δ1 × {t}) ∪ ({p2} × [0, t]).

p1 p3p2

=

a0

FIGURE 10. Sliding the bypass arc.

Finally, let γ1 be a Legendrian arc given by the concatenation of{p1, p2} × [0, t] andδ1 × {t}with ∂δ1 = {p1, p2}, and we attach a Legendrian arcγ2 consisting of{p3} × [0, t] andδ2 × {t}with ∂δ2 = {p3, q}, whereq is an interior point ofδ1. By sliding the endpoint ofδ2, we see thatattachingγ1 andγ2 is equivalent to attaching the two Legendrian arcs given in Figure 11. Whenattaching the first arcγ1, we first stabilize alonga0; for the second arcγ2, the attachment of thefirst arc has the same effect as a stabilization.

Therefore, using the above trivial bypass arcs, we can construct a Legendrian graphL in ∂M ′ ×[0, 1 − ε], which is the union of arcs of type{p} × [0, 1 − ε] and the1-skeletonL1−ε of a cell


p1 p2

p3

a0

FIGURE 11.

decomposition of∂M ′ × {1 − ε}, each of whose cells have boundary withtb = −1. (Here(p, 1 − ε) must lie inL1−ε.) If we take the standard Legendrian neighborhoodN(L), then itscomplement(∂M ′× [0, 1−ε])−N(L) is also a standard neighborhood of a Legendrian graphK ′′.The Legendrian neighborhoodN(L1−ε) can be enlarged via a contact isotopy so thatN(L1−ε) is∂M ′×[1−2ε, 1], with neighborhoods of Legendrian arcs of type{q}×[1−2ε, 1], q ∈ Γ′, removed.On the other hand,N({p} × [0, 1 − ε]) is viewed as a sufficiently thin/small neighborhood of theLegendrian arc{p}× [0, 1− ε]. The above description clearly indicates that the extension ofK ′ toK ′ ∪K ′′ is an extension of the partial open book decomposition(S ′, R+(Γ′), h′) to (S,R+(Γ), h)of the type described in Step 2 of Section 4. This completes the proof of Theorem 6.1.

7. A (1 + 1)-DIMENSIONAL TQFT

In this section we describe a(1 + 1)-dimensional TQFT, obtained by dimensional reduction.(Strictly speaking, the theory does not quite satisfy the TQFT axioms but has similar compositionrules.)

7.1. Invariants of multicurves on surfaces. In this subsection we describe a TQFT-type invariantof a multicurve on a surface. LetΣ be a compact, oriented surface with nonempty boundary∂Σ,andF be a finite set of points of∂Σ, where the restriction ofF to each component of∂Σ consistsof an even number≥ 2 of points. Part of the structure of a pair(Σ, F ) is a labeling of eachcomponent of∂Σ − F by + or − so that crossing a point ofF while moving along∂Σ reversessigns. Let#F = 2n be the cardinality ofF . Also letK be a properly embedded, oriented1-dimensional submanifold ofΣ whose boundary isF and which dividesΣ into R+ andR− in amanner compatible with the labeling of∂Σ− F . As on∂Σ− F , the sign changes every timeK iscrossed. Such aK will be called adividing set for(Σ, F ).

We now list the properties satisfied by the TQFT.


TQFT Properties.

I. It assigns to each(Σ, F ) a gradedZ-moduleV (Σ, F ). If Σ is connected, then

V (Σ, F ) = Z2 ⊗ · · · ⊗ Z

2,

where the number of copies ofZ2 is r = n − χ(Σ), andZ

2 = Z ⊕ Z is a gradedZ-module whose first summand has grading1 and the second summand has grading−1. Wewill refer to this grading as theSpinc-grading. Moreover, if(Σ, F ) is the disjoint union(Σ1, F1) ⊔ (Σ2, F2), then

V (Σ1 ⊔ Σ2, F1 ⊔ F2) ≃ V (Σ1, F1) ⊗ V (Σ2, F2).

II. To eachK it assigns a subsetc(K) ⊂ V (Σ, F ) of cardinality1 or 2 of type{±x}, wherex ∈ V (Σ, F ). If K has a homotopically trivial closed component, thenc(K) = {0}.

III. Given (Σ, F ), let γ, γ′ ⊂ ∂Σ be mutually disjoint1-dimensional submanifolds of∂Σ, sothat their endpoints do not lie inF . Suppose there is a diffeomorphismτ : γ

∼→ γ′ which

sendsγ ∩ F∼→ γ′ ∩ F and preserves the orientations onγ ∩ ∂Σ andγ′ ∩ ∂Σ′. If we glue

(Σ, F ) by identifyingγ andγ′ via τ , then the result will be denoted by(Σ′, F ′). Then thereexists a map

Φτ : V (Σ, F ) → V (Σ′, F ′),

which satisfiesc(K) 7→ c(K),

whereK is obtained fromK by gluingK|γ andK|γ′. HereΦτ is well-defined up to anoverall±1 multiplication.

See Figure 12 for an illustration of the gluing in Property III, when (Σ, F ) = (Σ′′, F ′′) ⊔(Σ′′′, F ′′′),K = K ′′ ⊔K ′′′, andγ, γ′ are arcs. In this case, the gluing map is:

Φτ : V (Σ′′, F ′′) ⊗ V (Σ′′′, F ′′′) → V (Σ′, F ′).

Σ′′ Σ′′′

K ′′

K ′′′

FIGURE 12. Gluing(Σ′′, K ′′) and(Σ′′′, K ′′′). The red dots areF ′′ andF ′′′.

We will use the subscripts(i) to denote the Spinc-grading:V (Σ, F )(i) is the graded piece withgradingi andZn

(i) is theZn-summand representing theith graded piece.


Theorem 7.1.There exists a nontrivial TQFT satisfying Properties I-IIIabove.

Proof. This TQFT arises by dimensional reduction of sutured Floer homology.

I. Given (Σ, F ), letF0 ⊂ ∂Σ be obtained fromF by shifting slightly in the direction of∂Σ. (Wemay think of points ofF0 as being situated halfway between successive points ofF on ∂Σ.) WeconsiderS1-invariant balanced sutured manifolds(S1 × Σ, S1 × F0), and let

V (Σ, F ) = SFH(−(S1 × Σ),−(S1 × F0)).

The reason for usingF0 instead ofF in the definition is explained below in II when the role ofcontact structures is explained. The Spinc-grading forV (Σ, F ) corresponds to the different relativeSpinc-structures on(S1 × Σ, S1 × F0).

The next lemma determinesV (Σ, F ), up to isomorphism.

Lemma 7.2. If Σ is connected, then

SFH(−(S1 × Σ),−(S1 × F0)) ≃ (Z(−1) ⊕ Z(1))⊗r,

wherer = n− χ(Σ).

Proof. This follows from Juhasz’ tensor product formula [Ju2, Proposition 8.10] for splitting su-tured manifolds along product annuli, together with the observation that whenn = 2 andΣ = D2,we haveSFH(−(S1 × D2),−(S1 × F0)) ≃ Z(−1) ⊕ Z(1), split according to the relative Spinc-structure. (See [HKM3, Section 5, Example 2].) �

Finally, the property

V (Σ1 ⊔ Σ2, F1 ⊔ F2) ≃ V (Σ1, F1) ⊗ V (Σ2, F2)

is immediate from the definition of the sutured Floer homology groups.

II. Next, there is a1−1 correspondence between dividing setsK of (Σ, F ) without homotopicallytrivial closed curves and tight contact structures with boundary condition(S1 × Σ, S1 × F0), upto isotopy rel boundary. For the correspondence to hold we require that∂Σ 6= ∅. The map fromdividing sets to contact structures is easy: simply consider theS1-invariant contact structureξK onS1 × Σ so that the dividing set on each{pt} × Σ is {pt} ×K. It was shown in [Gi3, H2] that themap, when restricted to the subset of dividing setsK without homotopically trivial curves, gives abijection with the set of isotopy classes of tight contact structures on(S1 × Σ, S1 × F0). Now, toeachK we assignEH(ξK) ⊂ SFH(−(S1 × Σ),−(S1 × F0)). If K has a homotopically trivialcurve, thenξK is overtwisted, andEH(ξK) = {0}.

Finally we explain why we useF0 instead ofF in (S1 × Σ, S1 × F0). The dividing setS1 × F0

of ∂(S1 ×Σ) does not intersect the dividing set of{pt}×Σ, since the two surfaces are transverse.This means thatF0 must lie between the endpointsF of K.

III. This is a corollary of Theorem 1.1: In order to apply Theorem 1.1, slightly shrinkΣ toΣ0 insidethe glued-up surfaceΣ′. See Figure 13. If we writeΣ − Σ0 = ∂Σ × [0, 1] with ∂Σ × {0} = ∂Σ0

and∂Σ×{1} = ∂Σ, then the dividing setK0 onΣ′−Σ0 is obtained fromF × [0, 1] by identifyingF |γ × {1} with F |γ′ × {1} via φ. Let ξK0

be theS1-invariant contact structure onS1 × (Σ′ −Σ0)corresponding to the dividing setK0. The contact structureξK0

induces the mapΦξK0= Φτ from

V (Σ, F ) to V (Σ′, F ′). This completes the proof of Theorem 7.1. �


Σ0 Σ0

FIGURE 13. The dividing setK0 onΣ′ − Σ0 is given in red.

Remark 7.3. There is another grading forV (Σ, F ), a relative grading called theMaslov grading,which is largely invisible for the time being since all the generators have the same Maslov grading.

7.2. Analysis ofΣ = D2. SupposeΣ = D2 andF consists of2n points on∂D2. In this case, theset of dividing setsK without closed components corresponds to the set of crossingless matchingsof F . A crossingless matching ofF is a collection ofn properly embedded arcs inD2 withendpoints onF so that each endpoint is used once and no two arcs intersect inD2. The orientationcondition is trivially satisfied for a crossingless matching. If K has a closed component, thenthe component must be homotopically trivial. Thus the corresponding contact structureξK isovertwisted, andc(K) = {0}.

n=1. Whenn = 1, V (Σ, F ) = Z(0), which is generated by the uniqueK which connects the twopoints. (By this we meanZ is generated by either element ofc(K).)

n=2. Whenn = 2, V (Σ, F ) = Z(1) ⊕ Z(−1). We claim thatV (Σ, F ) is generated byc(K+) andc(K−), given as in Figure 14. HereK+ andK− are the two dividing sets, both∂-parallel. The

−

K+ K−

+

+

+− −

FIGURE 14.

grading forc(K) can be calculated by takingχ(R+)−χ(R−), whereR+ (resp.R−) is the positive(resp. negative) region ofΣ − K. Hence the degrees are1 and−1 for K+ andK−, respectively.As calculated in [HKM3, Section 5, Example 3], there is a Heegaard diagram for which theEHclass forξK+

is the unique tuple representing its Spinc-structure (and similarly forK−). Hencec(K+) generates the first summand andc(K−) generates the second summand, with respect to anycoefficient system.

n=3. When n = 3, V (Σ, F ) decomposes intoZ(2) ⊕ Z2(0) ⊕ Z(−2). The first and last sum-

mands are generated byc(K) for ∂-parallelK. The middleZ2(0) must support three configurations

K1, K2, K3. See Figure 15.


+

−

+

−

FIGURE 15. K1, K2, K3, from left to right.

We have the following:

Lemma 7.4. The setsc(K1), c(K2), c(K3) are nonzero and distinct. Moreover, their elements areprimitive.

OverZ/2Z, the lemma implies thatc(K1) + c(K2) = c(K3), i.e., c(K3) is a superposition ofthe other two statesc(K1) andc(K2).

Proof. Consider an arcγ ⊂ ∂Σ with #(F ∩ γ) = 2. Take a diskΣ′ = D2 with #F ′ = 2, and pickan arcγ′ ⊂ ∂Σ′ with #(F ′ ∩ γ′) = 2. Then attachΣ′ ontoΣ so thatγ andγ′ are identified andF ′′ = F ∪ F ′ − γ satisfies#F ′′ = 4. Observe that theZ-moduleV (Σ′, F ′) ≃ Z is generated bya unique elementK ′, which is a∂-parallel arc. Label the points ofF in clockwise order from1 to6, so that1 is 2pm,2 is 4pm, etc., and letΦj , j = 1, 2, 3, be the gluing map

V (Σ, F ) → V (Σ ∪ Σ′, F ′′),

obtained by attaching the∂-parallel arcK ′ from j to j + 1. It sendsc(Ki) 7→ c(Ki ∪ K ′). SeeFigure 16. Restricted toZ2

(0), the image ofΦj is one of the two summandsZ(1) or Z(−1). Hencewe viewΦj as a mapZ2

(0) → Z(±1).

+

−

+

−

FIGURE 16. The diagram representsΦ1(c(K1)). K1 is the crossingless matchinginside the circle, and the∂-parallel arc, representing a generator ofV (Σ′, F ′) = Z

with #F ′ = 2, is attached from1 to 2.

SupposeKi ∪K′ does not have a closed component; there is always someΦj for which this is

true. Then we have reduced to the casen = 2, where we already know that each representative ofc(Ki ∪K

′) is nonzero and primitive. SinceΦj : Z2 → Z mapsc(Ki) 7→ c(Ki ∪K′) and the latter

is primitive, it follows thatc(Ki) must also be primitive.Next,c(Ki ∪K

′) = EH(ξKi∪K ′) = {0} if Ki ∪K′ has a closed (and necessarily homotopically

trivial) component. Hence, by attachingΣ′ at the appropriate locations (i.e., checking whichΦ1,Φ2 or Φ3 annihilatesc(Ki)), we can determine the locations of all the∂-parallel (or outermost)


arcs ofKi ⊂ Σ. Since the location of the∂-parallel arcs determinesKi, it follows that thec(Ki)must be distinct. �

By inductively applying the above procedure, we obtain the following:

Proposition 7.5. All crossingless matchingsK of (Σ = D2, F ) with #F = 2n are distinguishedby c(K) ⊂ V (Σ, F ) and are primitive. Equivalently, all the tight contact structures onS1 × D2,#F = 2n, are distinguished by their contact invariant inSFH(−(S1 ×D2),−(S1 × F0)).

The proof is left to the reader. Lemma 7.4 and its generalization Proposition 7.5 are rathersurprising, since the dimension ofV (D2, F ) with #F = 2n is 2n−1, whereas the number of

crossingless matchings on(D2, F ) is theCatalan numberCn =1

n + 1

(2nn

), which is greater

than or equal to2n−1, and grows roughly twice as fast as a function ofn. This means that all thec(K)’s are “tightly packed” insideV (D2, F ), especially when the coefficient ring isZ/2Z.

Also recall that the dimensions of ourV (D2, F ) with #F = 2n are the same as that of(1 + 1)-dimensional, levelk = 2, sl(2,C) TQFT. It would be interesting to compare the two TQFT’s.

7.3. The±1 ambiguity over Z-coefficients. In this subsection we prove Theorem 7.6 and deducefrom it that the±1 ambiguity of the contact invariantEH(M,Γ, ξ) in SFH(−M,−Γ) over Z

cannot be removedand that the gluing mapΦξ of Theorem 1.1 is well-definedonly up toan overall± sign overZ. This proves Theorem 1.4, stated in the Introduction.

Theorem 7.6.There is no single-valued lift ofc(K) ⊂ V (Σ, F ) for all K,Σ, F , withZ-coefficients.

Proof. Assume the invariants of dividing curves are single-valued. Consider the exampleΣ = D2

and#F = 6. RecallK1, K2, K3 from Figure 15. By Lemma 7.4, each element ofc(Ki), i =1, 2, 3, is primitive inZ2

(0). We also use the same mapsΦj : Z2(0) → Z2, j = 1, 2, 3.

We compute the following:

Φ1 : c(K1) 7→ c(K+), c(K2) 7→ c(K+), c(K3) 7→ 0,

Φ2 : c(K1) 7→ 0, c(K2) 7→ c(K−), c(K3) 7→ c(K−),

Φ3 : c(K1) 7→ c(K+), c(K2) 7→ 0, c(K3) 7→ c(K+).

Here c(K+) and c(K−) are generators ofV (D2, F ′) with #F ′ = 4. Since the image of eachΦj(Z

2) is Z, generated by eitherc(K+) or c(K−), we viewΦj as a mapZ2 → Z.Let us analyzeΦ1 in more detail. Writec(K1) as(1, 0) ∈ Z2, since it is primitive. ThenΦ1 :

Z2 → Z maps(1, 0) 7→ 1. We can then decomposeZ

2 into Z ⊕ Z so that(0, 1) generatesker Φ1,possibly after an appropriate isomorphism ofZ2. Without loss of generality,c(K3) = (0, 1). SinceΦ1 : c(K2) 7→ 1, it follows thatc(K2) = (1, a), a ∈ Z.

Next considerΦ2. Since(1, 0) 7→ 0, (0, 1) 7→ 1, and(1, a) 7→ 1, it follows thata = 1.Finally, Φ3 maps(1, 0) 7→ 1, (0, 1) 7→ 1, and should map(1, 1) 7→ 2, but instead sends it to0, a

contradiction. �

Theorem 7.7.±1 monodromy exists in sutured Floer homology. That is, there is a sequence ofstabilization, destabilization, handleslide, and isotopy maps which begins and ends at the sameconfiguration, so that their composition is−id.


Proof. When working overZ, EH(M,Γ, ξ) andΦξ are defined up to a factor of±1. The onlyreason for the introduction of this factor was thepossibilityof the existence of±1 monodromy.Since single-valued lifts do not always exist by Theorem 7.6, it follows that±1 monodromy mustexist. �

This proof is unsatisfying in the sense that it does not explain the root cause of the existence ofmonodromy, nor does it give a specific sequence of maps which exhibits nontrivial monodromy.

Question 7.8. Is there monodromy in Heegaard Floer homology, i.e., when the 3-manifold isclosed? In particular, is there an explicit sequence of stabilization, destabilization, handleslide,and isotopy maps which begins and ends at the same configuration, so that their composition is−id for HF (S1 × S2) ≃ Z ⊕ Z?

7.4. A useful gluing isomorphism. In this subsection we give a useful gluing map and exploresome consequences.

Let γ be a properly embedded arc onΣ which is transverse toK and intersectsK exactly once.Suppose we cut(Σ, F ) andK alongγ to obtain(Σ′, F ′) andK ′. This is the reverse procedure ofgluing (Σ′, F ′) andK ′ along disjoint subarcsγ′, γ′′ ⊂ ∂Σ′, where each arc intersectsF ′ exactlyonce. We then have:

Lemma 7.9. The gluing mapΦ : V (Σ′, F ′) → V (Σ, F ) is an isomorphism.

If γ decomposesΣ into two components(Σ′′, F ′′) and(Σ′′′, F ′′′), then the gluing map is:

Φ : V (Σ′′, F ′′) ⊗ V (Σ′′′, F ′′′)∼→ V (Σ, F ).

Proof. We interpret the gluing mapΦ as a gluing mapΦ0 : V (Σ′, F ′) → V (Σ, F ), where thegluing occurs along a∂-parallel convex annulusA as given in Figure 17.

+

+

+

A

FIGURE 17. The top and bottom of the annulus are identified.

First we prove thatΦ = Φ0. Let (M,Γ) = (S1 × Σ, S1 × F0), whereF0 is the pushoff ofF inthe direction of∂Σ. Also let (M ′,Γ′) be the sutured manifold obtained from(S1 × Σ′, S1 × F ′

0)by slightly retractingΣ′ to Σ′

0; hereF ′0 is the pushoff ofF ′. Let ξ0 be the contact structure on

M − int(M ′), given as the union of the invariant contact structures on a neighborhood of∂M ′

with dividing setS1 × F ′0 and on a neighborhood ofA with ∂-parallel dividing set. Since the

dividing set on∂(M − int(M ′)) is of the typeS1 × {finite set}, ξ0 is anS1-invariant contactstructure by [Gi3, H2], and is encoded by the “minimal” dividing setK0 onΣ − Σ′

0.


We now briefly sketch whyξK ′ ∪ ξ0 is isotopic toξK . Let γ1, γ2 be the components ofΓ whichintersect∂A andδ1, . . . , δm be the components ofΓ which do not intersect∂A. For eachδi, there isa parallel copyδ′i on∂M ′. Moreover, there is a Legendrian arc fromδi to δ′i which has zero twistingnumber with respect to a surface parallel toΣ − Σ′

0. For eachγi, there are two componentsγ′i andγ′′i on ∂M ′ which share a parallel arc withγi. Hence there are Legendrian arcs fromγi to γ′i andfrom γi to γ′′i which have zero twisting number as well. The above Legendrian arcs constrainK0

so thatK ′ ∪K0 is isotopic toK. This provesΦ = Φ0.We next prove thatΦ0 is an isomorphism. According to Juhasz [Ju2], gluing alonga product

annulus gives an isomorphism of sutured Floer homology groups. Although our situation is slightlydifferent, the result is the same. By [HKM3, Theorem 6.2],V (Σ′, F ′) is a direct summand ofV (Σ, F ) since the dividing set onA is ∂-parallel. Now, according to Proposition 6.4,Φ0 is indeedthe direct summand map of [HKM3, Theorem 6.2]. More precisely, Φ0 induces an isomorphismonto the Spinc-direct summand corresponding to the∂-parallel dividing set with relative half-Eulerclassχ(R+) − χ(R−) = 2 − 0 = 2. To see thatV (Σ′, F ′) ≃ V (Σ, F ) under the mapΦ0, we usea rank argument. BothV (Σ′, F ′) andV (Σ, F ) are isomorphic toZr, wherer = 1

2(#F ) − χ(Σ).

SinceV (Σ′, F ′) is a direct summand ofV (Σ, F ) and they are both free with the same rank, itfollows thatV (Σ′, F ′) ≃ V (Σ, F ). �

As an application of Lemma 7.9, we give a sufficient conditionfor a dividing setK for (Σ, F ) tohavec(K) which is nonzero and primitive inV (Σ, F ), whenZ-coefficients are used. A connectedcomponent ofΣ −K which is not connected to∂Σ is called anisolated region ofK in Σ. We saythatK is isolating if there is an isolated region ofK in Σ, andnonisolatingif there is no isolatedregion. For example, ifK has a homotopically trivial closed curve, then it is isolating.

We then have the following:

Proposition 7.10. With Z-coefficients, the dividing setK has nonzero and primitivec(K) if K isnonisolating.

Proof. SupposeΣ is connected. (IfΣ is not, we consider each component ofΣ separately.) If(Σ, F ) = (D2, F ), then we are done by Proposition 7.5. Therefore, supposeΣ 6= D2. In viewof Lemma 7.9, it suffices to find a properly embedded arcγ ⊂ Σ which intersectsK exactlyonce, so that cutting along it increases the Euler characteristic of Σ by one. LetΣ0 be a connectedcomponent ofΣ − K which has Euler characteristic6= 1. SinceK is nonisolating,Σ0 mustnontrivially intersect∂Σ. It is then easy to find a properly embedded arcγ0 ⊂ Σ which lies inΣ0,and which is not∂-parallel inΣ0. We can isotop the endpoints ofγ0 along∂Σ so the resultingγintersectsK exactly once. �

We also have the following corollary of Lemma 7.9:

Proposition 7.11. With Z-coefficients,V (Σ, F ) is generatedc(K), whereK ranges over all di-viding sets for which∂K = F .

Proof. The assertion is clearly true whenΣ = D2 and#F = 2 or 4. Now, any(Σ, F ) can be splitalong an arcγ so that the resulting(Σ′, F ′) satisfiesχ(Σ′) = χ(Σ)+1 and#F ′ = #F +2, and sothatV (Σ′, F ′) ≃ V (Σ, F ). Once we reachΣ′ = D2, a good choice of splitting will decrease#F ′

of each component, until each component is(D2, F ) with #F = 2 or 4. The proposition followsby gluing. �


7.5. Analysis whenΣ is an annulus. SupposeΣ is an annulus. We consider the situation whereF consists of two points on each boundary component. The calculations will be done inZ-coefficients, but calculations in a twisted coefficient system will certainly yield more information.See for example [GH].

By Juhasz’ formula,V (Σ, F ) = Z2 ⊗ Z

2 = Z(2) ⊕ Z2(0) ⊕ Z(−2). One can easily see that

Z(2) is generated by a∂-parallelK+ with two positive∂-parallel arcs, andZ(−2) is generated by a∂-parallelK− with two negative∂-parallel arcs.

It remains to analyzeZ2(0). The nonisolating dividing setsK with nontrivialc(K) ⊂ Z2

(0) are thefollowing: K ′

0 andK ′1, which have two∂-parallel arcs of opposite sign and one closed curve,L0

consisting of two parallel arcs from one boundary componentto the other, as well asLj , obtainedfrom L0 by performingj positive Dehn twists about the core curve of the annulus. SeeFigure 18.The other possible dividing setsK, besides those with homotopically trivial components, haveat least two parallel closed curves. The corresponding contact structure will necessarily have atleast2π-torsion. It was proved in [GHV] that any contact structure with 2π-torsion has vanishingcontact invariant overZ.

K ′

0

+

K ′

1

L0 L1

+

FIGURE 18. The sides of each annulus are identified.

First consider the mapΦ : Z2(0) → Z(2), obtained by attaching an annulus with configuration

K+ from below. Since a homotopically trivial curve is created,we haveΦ(c(K ′0)) = {0}. Also,

Φ(c(K ′1)) = {0}, since the resulting dividing set will have two parallel closed curves. On the other

hand, thec(Li) all map to the generatorc(K+) of Z(2). Hence the mapΦ is surjective, and musthaveker Φ ≃ Z. Next, sinceK ′

0 andK ′1 are nonisolating,c(K ′

0) andc(K ′1) must be primitive;

this implies thatc(K ′0) = c(K ′

1) and generateker Φ. Now, one can make a coordinate change ifnecessary so thatc(L0) = {±(1, 0)} andc(K ′

i) = {±(0, 1)}.Next we computec(L1). For this, we use Lemma 7.4 and the following fact which follows from

the proof of Theorem 7.6: forΣ = D2 andn = 3, a representativec(K3) of c(K3) is a superpo-sition of representativesc(K1) andc(K2) of c(K1) andc(K2) with ±1 coefficients. Observe thatK1 is obtained fromK3 by applying a bypass attachment from the front, andK2 is obtained fromK3 by a bypass attachment to the back. It is easy to see from theΦ in the previous paragraph thatc(L1) = {±(1, n)} for some integern. Given the configurationK ′

0, take a bypass arc of attach-mentδ with endpoints on the two∂-parallel arcs and one other intersection point withK ′

0, namely


along the closed component. Take a small diskD2 aboutδ. Consider the gluing map

Ψ : V (D2, K ′0|∂D2) ⊗ V (Σ −D2, K ′

0|∂Σ−D2 ∪ F ) → V (Σ, F ).

By tensoring thec(Ki) with c(K ′0|Σ−D2), the equationc(K3) = ±c(K1) ± c(K2) becomes

c(K ′0) = ±c(L0) ± c(L1).

This means(0, 1) = ±(1, 0) ± (1, n). The only possible solutions are(0, 1) = (1, 0) − (1,−1)or (0, 1) = −(1, 0) + (1, 1). (The two possibilities are equivalent after a basis change.) Hencec(L1) = {±(1, 1)}, for example.

In general, we conjecture thatc(Ln) = {±(1, n)}. A proof of this conjecture requires a morecareful sign analysis than we are willing to do for the moment.

7.6. Determination of nonzero elementsc(K) in V (Σ, F ). In this section we prove Theorem 1.5,i.e., we determine exactly which elementsK have nonzero invariantsc(K) in V (Σ, F ) with Z/2Z-coefficients.

Proposition 7.12. If K is isolating, thenc(K) = 0 with Z/2Z-coefficients.

Proof. Suppose first that there is an isolated regionΣ0 which is an annulus. In that case, take anarc of attachmentδ of a bypass which intersects the two boundary components ofΣ0, and someother component ofK, in that order. By the TQFT property applied to a small neighborhoodD ofδ andΣ−D, we see that ifK ′ (resp.K ′′) is obtained fromK by applying a bypass from the front(resp. bypass to the back), thenc(K) = c(K ′) + c(K ′′), since the corresponding fact is true onD.One easily sees thatK ′ andK ′′ are isotopic, and isK with ∂Σ0 removed. WithZ/2Z-coefficients,then,c(K) = 2c(K ′) = 0.

Next suppose thatΣ0 has more than one boundary component, and is outermost amongallisolated regions, in the sense that one boundary componentγ of Σ0 is adjacent to a componentΣ1

whose boundary intersects∂Σ. Also suppose thatΣ0 is not an annulus. Take an arc of attachmentδ which begins onγ, intersectsγ after traveling insideΣ0, and ends on an arc component ofK on∂Σ1. Chooseδ so thatΣ0 − δ has two components, one which is an annulus and the other whichhas Euler characteristic> χ(Σ0). Then apply the bypass attachments from the front and to theback to obtainK ′, K ′′ as in the previous paragraph. Now,c(K) = c(K ′) + c(K ′′), and one ofc(K ′) or c(K ′′) is zero, since it possesses an annular isolated region. Thisreduces the number ofcomponents of∂Σ0.

Finally suppose that∂Σ0 is connected. IfΣ0 bounds a surface of genusg > 1, then the aboveprocedure can splitc(K) = c(K ′) + c(K ′′), where bothc(K ′) andc(K ′′) have isolated regionswith connected∂Σ0 and strictly smaller genus. Hence suppose thatΣ0 bounds a once-puncturedtorus. Also, by cutting along arcs as in Proposition 7.10, wemay assume thatΣ itself is a once-punctured torus with one∂-parallel arc and one closed curve parallel to the boundary.Chooseδ asgiven in Figure 19, namely,δ begins on the∂-parallel arc and intersects∂Σ0 twice, and restrictsto a nontrivial arc onΣ0. The resultingK ′ andK ′′ are the center and right-hand diagrams. Nowcut along the properly embedded, non-boundary-parallel arc τ which intersects each ofK ′ andK ′′ exactly once. Applying Lemma 7.9, we see thatc(K ′) = c(K ′′) if and only if the cut-opendividing curvesK ′

0 andK ′′0 have equal invariants in the cut-open surface. Finally, observe that, on

the cut-open surface (an annulus),c(K ′0) = c(K ′′

0 ) since they correspond toK0 andK1, discussedin Subsection 7.5. �


δ

τ τ

FIGURE 19.

Propositions 7.12 and 7.10, together give Theorem 1.5.In the case ofZ-coefficients we expect the following to hold:

Conjecture 7.13.OverZ-coefficients, the following are equivalent:(1) c(K) 6= 0;(2) c(K) is primitive;(3) K is nonisolating.

The difficulty comes from not being able to determine whetherc(K) is divisible by 2 withZ-coefficients, which in turn stems from our±1 difficulty in Subsection 7.3. When twisted coeffi-cients are used, the result is quite different, and will yield substantially more information [GH].

Acknowledgements.We thank John Etnyre, Andras Juhasz, and Andras Stipsiczfor helpful dis-cussions. KH thanks Francis Bonahon and Toshitake Kohno forhelpful discussions on TQFT. KHalso thanks Takashi Tsuboi and the University of Tokyo for their hospitality; much of the writingof this paper was done during his five-month stay in Tokyo in the summer of 2007.

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UNIVERSITY OF SOUTHERN CALIFORNIA , LOS ANGELES, CA 90089E-mail address: [email protected]: http://rcf.usc.edu/˜khonda

UNIVERSITY OF GEORGIA, ATHENS, GA 30602E-mail address: [email protected]: http://www.math.uga.edu/˜will

UNIVERSITY OF GEORGIA, ATHENS, GA 30602E-mail address: [email protected]: http://www.math.uga.edu/˜gordana

Documents

Ko Honda, William H. Kazez and Gordana Matic- Contact Structures, Sutured Floer Homology and TQFT