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011 AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY BY HOMOTOPY

CLASSES OF ORIENTED 2-PLANE FIELDS

VINICIUS GRIPP AND YANG HUANG

ABSTRACT. For a closed oriented 3-manifoldY, we define an absolute grading on the HeegaardFloer homology groups ofY by homotopy classes of oriented 2-plane fields. We show that this ab-solute grading refines the relative one and that it is compatible with the maps induced by cobordisms.We also prove that ifξ is a contact structure onY, then the grading of the contact invariantc(ξ) isthe homotopy class ofξ.

1. INTRODUCTION

For a closed oriented 3-manifoldY, Ozsvath and Szabo [16] defined a collection of invariantsofY, the Heegaard Floer homology groupsHF◦(Y), whereHF◦(Y) denotes eitherHF(Y), HF+(Y),HF−(Y), or HF∞(Y). They showed thatHF◦(Y) splits into a direct sum by Spinc structures

HF◦(Y) =⊕

s∈Spinc(Y)

HF◦(Y, s).

For eachs ∈ Spinc(Y), they also defined a relative grading onHF◦(Y, s), that takes values inZ/d(c1(s)), whered(c1(s)) is the divisibility ofc1(s) ∈ H2(Y;Z), i.e. d(c1(s))Z = 〈c1(s),H2(Y)〉.

Moreover given a 4-dimensional compact oriented cobordismW : Y0 → Y1, i.e. ∂W =

−Y0 ∪ Y1 as oriented manifolds, and given a Spinc structuret on W, there is a natural mapFW,t : HF◦(Y0, t|Y0)→ HF◦(Y1, t|Y1) defined by Ozsvath-Szabo [19].

It has been shown that Heegaard Floer homology is isomorphicto two other homology theories:Seiberg-Witten Floer homology [10] and embedded contact homology (ECH) [5,7,8]. For a proofof the existence of these isomorphisms, see [1,11,21]. It isknown that both ECH [6] and Seiberg-Witten Floer homology [10] are absolutely graded by homotopy classes of oriented 2-plane fields,but no such absolute grading had been defined for Heegaard Floer homology. In this paper, weconstruct such an absolute grading for Heegaard Floer homology, which is compatible with therelative grading and cobordism maps discussed above.

We will now fix some notation that will be used in this paper. Let (Σ,α,β, z) be a Heegaarddiagram ofY. HereΣ is a genusg surface,α = (α1, . . . , αg) andβ = (β1, . . . , βg) are collections ofdisjoint circles onΣ and the basepointz is a point onΣ in the complement ofα1∪· · ·∪αg∪β1∪· · ·∪βg. We also require thatα andβ are linearly independent sets inH1(Y) and thatαi andβ j intersecttransversely for everyi and j. We consider the toriTα = α1 × · · · × αg andTβ = β1 × · · · × βg inthe symmetric product Symg(Σ). Recall that the Heegaard Floer chain complexCF(Y) is the freeabelian group generated by the intersection pointsx ∈ Tα ∩Tβ. If x andy are intersection points inthe same Spinc structure, we denote by gr(x, y) their relative grading, as defined in [16].

We denote byP(Y) the set of homotopy classes of oriented 2-plane fields onY. Each homo-topy class of oriented 2-plane fields belongs to a Spinc structure, as we will explain in Section 2.

1

2 VINICIUS GRIPP AND YANG HUANG

ThereforeP(Y) splits by Spinc structures as

P(Y) =∐

s∈Spinc(Y)

P(Y, s).

It turns out thatP(Y, s) is an affine space overZ/d(c1(s)). For each Spinc structures, we willconstruct an absolute gradinggr onCF(Y, s) with values inP(Y, s).

For a contact structureξ onY, Ozsvath-Szabo [17] defined the contact invariantc(ξ) ∈ HF(−Y).In [16], Ozsvath-Szabo showed that a Heegaard move induces an isomorphism on Heegaard Floerhomology.

Consider a compact oriented cobordismW : Y0 → Y1. Let ξ0 andξ1 be oriented 2-plane fieldsonY0 andY1 respectively. We say thatξ0 ∼W ξ1 if there exists an almost complex structureJ onWsuch that [ξ0] = [TY0∩ J(TY0)] and [ξ1] = [TY1∩ J(TY1)] as homotopy classes of oriented 2-planefields.

We can now state the main theorem of this paper.

Theorem 1.1. For every Heegaard diagram(Σ,α,β, z) of Y, there exists a canonical functiongr : Tα ∩ Tβ → P(Y) such that:

(a) If x, y ∈ Tα∩Tβ are in the same Spinc structures, thengr(x) andgr(y) belong toP(Y, s) andgr(x) − gr(y) = gr(x, y) ∈ Z/d(c1(s)). In particular, gr extends to the set of homogeneouselements ofCF(Y).

(b) Let ξ be a contact structure on Y, and let c(ξ) ∈ HF(−Y) be the contact invariant. Thengr(c(ξ)) = [ξ] as homotopy classes of oriented 2-plane fields.

(c) This absolute grading is invariant under the isomorphisms induced by Heegaard movesand hence it induces an absolute grading onHF(Y) which is independent of the Heegaarddiagram.

(d) Let W : Y0 → Y1 be a compact, oriented cobordism, and lett be a Spinc structure onW. Then the induced map FW,t : HF(Y0, t|Y0) → HF(Y1, t|Y1) respects the grading inthe sense thatgr(x) ∼W gr(y) for any homogeneous elementx ∈ HF(Y0, t|Y0) and anyy ∈ HF(Y1, t|Y1), which is a homogeneous summand of FW,t(x).

Remark1.2. Theorem 1.1(a) implies that we have the following decomposition by degrees.

(1.0.1) CF(Y; s) =⊕

ρ∈P(Y,s)

CFρ(Y; s).

HereCFρ(Y; s) is theZ-module generated by allx ∈ Tα ∩ Tβ with gr(x) = ρ.

Remark1.3. The generators ofHF∞(Y) are of the form [x, i], wherex ∈ Tα ∩ Tβ and i ∈ Z. Werecall thatZ acts onP(Y), sinceP(Y, s) is an affine space overZ/d(c1(s)). So we can define anabsolute grading onHF∞(Y), and hence onHF−(Y) andHF+(Y), by gr([x, i]) = gr(x) + 2i, for ahomogeneous elementx. It is easy to see that Theorem 1.1 implies that (a),(c) and (d) also holdfor HF∞(Y), HF−(Y) andHF+(Y).

Remark1.4. Using the absolute grading functiongr constructed in Theorem 1.1, one can recoverthe absoluteQ-grading forHF◦(Y, s) defined by Ozsvath-Szabo whenc1(s) ∈ H2(Y;Z) is a torsionclass. See Corollary 4.3 for details.

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 3

We can also generalize the absolute grading functiongr to the twisted Heegaard Floer homologygroups defined by Ozsvath-Szabo [15]. Recall that the twisted Heegaard Floer homology groupHF(Y, s) is the homology of the twisted Heegaard Floer chain complexCF(Y; s) ⊗ Z[H1(Y;Z)],where the (infinity version) differential is defined by

∂∞[x, i] =∑

y∈Tα∩Tβ

( ∑

φ∈π2(x,y)

#M(φ)eA(φ)[y, i − nz(φ)])

whereA : π2(x, y) → H1(Y;Z) is a surjective, additive assignment. See [15] for more details.Now we define the twisted absolute grading function by simplyignoring the twisted coefficient asfollows:

grtw : Z[H1(Y;Z)](Tα ∩ Tβ)→ P(Y)(1.0.2)

eξx 7→ gr(x),

whereξ ∈ H1(Y;Z) and we writeZ[H1(Y;Z)] multiplicatively.1 Using an obvious twisted versionof Theorem 1.1(b), we will prove the following corollaries in Section 3.

Let FY denote the set of homotopy classes (as 2-plane fields) of contact structures onY whichare weakly fillable.

Corollary 1.5 (Kronheimer-Mrowka [9]). FY is finite.

Corollary 1.6. If Y is an L-space, then|FY| ≤ |H1(Y;Z)|.Corollary 1.7 (Lisca [13]). If Y admits a metric of constant positive curvature, then|FY| ≤|H1(Y;Z)|.Remark1.8. Corollary 1.5 and Corollary 1.7 are previously proved usingthe relationship betweenSeiberg-Witten theory and contact topology.

Remark1.9. In fact the assertion in Corollary 1.5 holds for the set of homotopy classes of 2-planefields which support a tight contact structure by the work of Colin-Giroux-Honda [2]. But ourresult does not imply this generalization. In particular wedo not have an upper bound on|F (Y)|for tight contact structures.

The paper is organized as follows. In Section 2, we constructthe absolute grading onCF, whichrefines the relative grading defined in [16]. That proves part(a) of the Theorem. In Section 3, wecompute the absolute grading of the contact invariant and show that it is the homotopy class of thecontact structure, which proves part (b) of the Theorem. This fact is known, by construction, forthe absolute grading in ECH [6]. In Section 4, we prove part (d) at the chain level, showing thatgr is natural under cobordism maps, as stated in Theorem 4.1.This was shown for Seiberg-WittenFloer homology by Kronheimer-Mrowka [10]. In Section 5, we prove thatgr is preserved underHeegaard moves, see Theorem 5.1. That means that the decomposition (1.0.1) is preserved underHeegaard moves and therefore it also holds in the homology level. That implies that part (d) alsoholds in homology.

1The twisted absolute grading defined here does not refine the relativeZ-grading within each Spinc structure definedin [15]. A slightly more sophisticated construction of the twisted grading is needed to recover the relativeZ-grading.But since we do not need this refinement in this paper, we do notinclude the details here.

4 VINICIUS GRIPP AND YANG HUANG

Acknowledgements. We would like to thank Ko Honda and Michael Hutchings for suggesting thisproblem to us and for providing guidance throughout the course of this project. We also thank TyeLidman for pointing out applications of the absolute grading of the contact invariant to us. Thiswork started during our visit to the Mathematical Sciences Research Institute in 2009-2010, wherean excellent environment for math research was provided. The first author was partially supportedby NSF grant DMS-0806037.

2. THE ABSOLUTE GRADING

Let Y be an oriented closed 3-manifold and letP(Y) denote the set of homotopy classes oforiented 2-plane fields onY. Let us first recall that there is a surjectionψ : P(Y) → Spinc(Y).Also, for a fixed Spinc structures, we can endowψ−1(s) = P(Y, s) with the structure of an affinespace overZ/d(c1(s)), whered(c1(s)) is the divisibility of the first Chern class ofs. So, givenξ, η ∈ P(Y) mapping to the same Spinc structures, there is a well-defined differenceξ − η. Oneway of seeing this affine space structure is by using the Pontryagin-Thom construction, as follows.Eachξ ∈ P(Y) corresponds to a unique homotopy class of nonvanishing vector fields, which wedenote by [vξ]. Fixing a representativevξ and a trivialization ofTY, and after a normalization,we can think ofvξ as a mapY → S2. The preimage of a regular value of this map gives a linkand the preimage of the tangent plane to this regular point under the derivative map determines aframing of this link. We recall that two framed linksLO, L1 ⊂ Y are called framed cobordant, ifthere exists a framed surfaceS ⊂ Y × [0, 1], whose boundary is−LO × {0} ∪ L1 × {1} and suchthat the framing restricted to the boundary coincides with the initial framings onL0 and L1. Itfollows from Pontryagin-Thom theory that two nonvanishingvector fields are homotopic if andonly if the respective framed links are framed cobordant. Ifξ, η map to the same Spinc structure,then the respective links are cobordant and the difference of framings isξ − η ∈ Z/d(c1(s)). Thesign convention we are using here is that a left-handed twistincreases a framing by+1.

Now let (Σ,α,β, z) be a Heegaard diagram representingY, whereα = (α1, . . . , αg) andβ =(β1, . . . , βg). Recall that the generators ofCF(Y) are the intersection points of the toriTα andTβ inSymg(Σ). Our goal in this section is to construct a canonical mapTα ∩ Tβ → P(Y) that refines therelative grading, which we denote bygr, and the map that assigns a Spinc structure to a generator,which we denote bysz : Tα ∩ Tβ → Spinc(Y). For the definitions of these maps, see [16].

Theorem 2.1. There is a canonical mapgr : Tα ∩Tβ → P(Y), such that ifx, y ∈ Tα ∩Tβ are suchthat sz(x) = sz(y) = s, then

gr(x) − gr(y) = gr(x, y) ∈ Z/d(c1(s)).

2.1. The construction. We fix a self-indexing Morse functionf : Y → R compatible with(Σ,α,β). Let x ∈ Tα ∩ Tβ. Thenx corresponds tog pointsx1, . . . , xg onΣ, which give rise to flowlinesγx1, . . . , γxg connecting the index 1 critical points to the index 2 critical points. The basepointzdetermines a flow lineγ0 from the index 0 critical point to the index 3 critical point.We can choosea gradient-like vector fieldv, tubular neighborhoodsN(γxi ) of γxi and diffeomorphismsN(γxi ) � B3

such that, under these diffeomorphisms,v|N(γxi ): B3 → R3 is given byv(x, y, z) = (x,−y, 1− 2z2),

for i , 0 andv|N(γ0) : B3 → R3 is given byv(x, y, z) = (2xz, 2yz, 1 − 2z2). Figure 1(a) shows twocross-sections ofv|N(γxi )

, for i , 0. Figure 1(b) showsv|N(γ0) on any plane passing through the origincontaining thez-axis. Outside the union of the neighborhoodsN(γxi ), v is a nonvanishing vector

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 5

field. We will define a nonvanishing continuous vector fieldwx on Y that coincides withv in thecomplement of the neighborhoodsN(γxi ).

xz-plane yz-plane

(a) (b)

FIGURE 1.

For i , 0, on∂N(γxi ) � ∂B3, we note that

v(x, y, z) = (x,−y, 1− 2z2) = (x,−y, 2x2+ 2y2 − 1).

We definewx = (x,−y, 2x2+ 2y2− 1) in N(γi), see Fig 2(a). This is a nonzero vector field inN(γxi )

that coincides withv on∂N(γxi ). Also, on∂N(γ0), we see that

v(x, y, z) = (−2xz,−2yz, 1− 2z2) = (−2xz,−2yz, 2x2+ 2y2 − 1).

This new vector field is still zero on the circleC = {(x, y, z)|x2+y2= 1/2, z= 0}. A vertical section

of it in B3 is shown in Figure 2(b).So we definewx in N(γ0) by

wx(x, y, z) = (−2xz,−2yz, 2x2+ 2y2 − 1)+ φ(x, y, z)(y,−x, 0),

whereφ is a bump function aroundC (i.e. φ = 1 onC andφ = 0 in the complement of a smallneighborhood ofC). Thereforewx is a nonvanishing vector field onY that equalsv outside theunion of the neighborhoodsN(γxi ). We can perturbwx to a smooth vector field. Finally we definegr(x) to be the homotopy class of the orthogonal complement ofwx.

Remark2.2. We could use the gradient vector field itself instead of some other gradient-like vectorfield to define the absolute grading, but it would be harder to write down the formulas for the canon-ical modification of the gradient vector field in the neighborhoods of the flow lines. Nevertheless,we would obtain the same homotopy class.

2.2. The relative grading. This subsection is dedicated to proving that the absolute grading re-fines the relative grading. Given two intersection pointsx, y ∈ Tα ∩ Tβ such thatsz(x) = sz(y),there exists a Whitney diskA ∈ π2(x, y), as proven in [16]. This means thatA is a homotopy classof mapsϕ : D2 ⊂ C → Symg(Σ) taking i to x, −i to y, the semicircle with positive real part toTβ and the one with negative real part toTα. Let D1, . . . ,Dn denote the closures of the connectedcomponents ofΣ − α1 − · · · − αg − β1 − · · · − βg. We write D(A) =

∑nk=1 akDk, whereak is the

multiplicity of ϕ on eachDk. We can choose a Whitney diskA so thatak ≥ 0 for everyk.

6 VINICIUS GRIPP AND YANG HUANG

xz-plane yz-plane

(a) (b)

FIGURE 2.

We will now construct surfacesF1 ⊃ · · · ⊃ Fm, whose union projects to∑n

k=1 akDk = D(A) onΣ.We takeak copies of eachDk and we glue them along their boundaries in the following way:weconstructF1 by gluing one copy of eachDk with ak > 0. Then we constructF2 by gluing one copyof eachDk such thatak − 1 > 0. Inductively we construct surfacesF1, . . . , Fm, wherem= maxak.So the union of the surfacesFl can be identified withD(A). (Similar constructions can be foundin [12,16,20]).

The Euler measure of a surface with cornersS, denoted bye(S), is defined to beχ(S) − p4 +

q4,

wherep is the number of convex corners ofS andq is the number of concave corners ofS. Ifw ∈ αi ∩ β j, for somei, j, then a small neighborhood ofw, when intersected with the complementof the union of theα and theβ curves, gives rise to four regions. We definenw(Dk) to be 1/4 timesthe number of those regions contained inDk. We extendnw linearly to theZ-module generated bythe domainsDk. Now we definenx to be the sum of allnxi , for i = 1, . . . , g. For example, a convexcornerxi of Fl contributes tonx(Fl) with 1/4 and a concave cornerxi with 3/4. Similarly we defineny. By Lipshitz [12], the Maslov index of the Whitney diskA, denoted byµ(A), is given by

µ(A) = ind(A) = e(D(A)) + nx(D(A)) + ny(D(A)) =m∑

l=1

(e(Fl) + nx(Fl) + ny(Fl)

).

For eachDk, we definenz(Dk) to be 0 ifz < Dk and 1 ifz ∈ Dk, and we extendnz linearly to sumsof Dk. The relative grading was defined by Ozsvath-Szabo [16] tobe

gr(x, y) = µ(A) − 2nz(D(A)) ∈ Z/d,whered is the divisibility ofc1(s(x)). So we need to show that

gr(x) − gr(y) =m∑

l=1

(e(Fl) + nx(Fl) + ny(Fl) − 2nz(Fl)

)∈ Z/d.

Step 1: We first assume thatm = 1 and thatnz(F1) = 0. Recall that a cornerxi is calleddegenerate ifxi = yj for somej. We also assume that there are no degenerate corners.

We will now choose a convenient trivialization ofTY in order to apply the Pontryagin-Thomconstruction. Letf be a self-indexing Morse functionf , which is compatible with (Σ,α,β). Let

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 7

F := F1. Let pi be the index 1 critical point corresponding toαi andq j the index 2 critical pointcorresponding toβ j. Each edge of the boundary ofF is part of anαi or aβ j. So each edge of∂Fdetermines a surface by flowing downwards or upwards towardsa pi or q j, respectively, and, byaddingpi andq j, we get a compact surface with corners. This surface has typically three cornersunless it corresponds to an edge starting at a boundary degenerate corner in which case, this edgeis actually a circle and the surface corresponding to it is a disk. We callAi andB j the surfacescorresponding to the edges contained inαi andβ j, respectively. We note that the flow we considerhere is the one generated by a gradient-like vector fieldv compatible with the Morse functionf .

Let C be the union ofF and the surfacesAi andB j. We will first choose a trivialization ofTYonC. We start by defining a unit vector fieldE1, which is tangent toF. The orientation ofΣ inducesan orientation onF. We setE1 to be the positive unit tangent vector along∂F, with respect to itsboundary orientation, outside a small neighborhood of the corners. At a neighborhood of a corner,we defineE1 on∂F by keeping it tangent toF and rotating it by the smallest possible angle. Thatmeans that once we start rotating,E1 will not be tangent to∂F at any point. In other words, eachconnected component of the set of points of∂F at whichE1 is not tangent to∂F contains exactlyone corner ofF. We also have to choose a corner to rotate an extra 2πχ(F) clockwise. That allowsus to extendE1 to F. We now defineE1 on eachAi andB j to be an extension ofE1 on∂F such thatit is tangent toAi andB j everywhere outside small neighborhoods of the cornersxi andyj and suchthat it is always transverse to the flow linesγxi andγyj . In particularE1 is tangent toAi nearpi andto B j nearq j. Near the cornersxi andyj, we requireE1 to never be tangent toAi andB j, similarlyto how we definedE1 on F. We defineE3 on F to be the positive normal vector field toF, and weextend it toAi andB j so that{E1,E3} is an oriented orthonormal frame on the respective tangentspaces, except maybe outside a small neighborhood of∂F. In this neighborhood, we require thateach connected component of the set of points whereE3 is not tangent toAi or B j intersectsF.Now we takeE2 to be the unit vector field onC orthogonal toE1 andE3 such that{E1,E2,E3}is an oriented basis ofTY. So mappingEi to ei ∈ R3, we get a trivialization ofTY alongC. Weextend this trivialization to a neighborhood ofC in such a way thatE1 andE3 are still tangent to thecorresponding unstable and stable surfaces near the critical pointspi andq j and thate1 is a regularvalue ofwx andwy when seen as mapsY→ S2. Now, since there are no degenerate points,C doesnot contain anα or β curve. Therefore there is no obstruction to extending this trivialization to allof Y. So we choose one of those extensions.

Now we defineK′x = w−1x (e1) andK′y = w−1

y (e1) as framed links. We note that inside neighbor-hoods of the flow linesγxi andγyi , these are one stranded braids contained in the correspondingunstable or stable surface, except that near each corner ofF, this braid rotates around the respectiveflow line as much asE1 restricted to this flow line does, but in the opposite direction. This is shownin Figure 3(a). It follows from the way that we chose the trivialization onC thatK′x andK′y do notintersectC outside of those neighborhoods.

We can isotopeK′x in neighborhoods of eachγxi in the following way. Near each corner, thislink is rotating aroundγxi . We isotope a neighborhood of this part of the link to the segment ofthe flow line about which it is rotating fixing the endpoints. Outside of this neighborhood of thecorner, but still inside the neighborhood of the flow line, the link is contained in the correspondingunstable or stable surface. We will call this new linkKx. We can think of the framing of a link as aunit normal vector field to the link. So the framing onKx induced from this isotopy can be seen bya vector field that is normal to the stable and unstable surfaces away from the corners and rotates

8 VINICIUS GRIPP AND YANG HUANG

with respect to the stable surface as much asK′x rotates about the flow line, as seen in Figure 3(b).We denote this framing byτx. We note that once we fix which of the two unit normal vector fieldsto the stable surface we choose, the unit normal vector field to the unstable surface is determined.

We can do the same forK′y and defineKy with framing denoted byηy. Figure 3(c) shows apicture of bothKx andKy at a neighborhood of a flow lineγxi . Now we modifyC in the followingway. For each edge ofF, we substitute the correspondingAi or B j by the region on the unstable orstable surface bounded by the corresponding edge ofF and the segments ofKx andKy, see Figure3(c). We smooth the edges of this surface and denote byC this smooth surface with boundary,which has cusps. We note thatC gives rise to a cobordismS ⊂ Y × [0, 1] betweenKx × {0} andKy × {1} that is trivial whereKx andKy coincide.

γxi K′x

β

α

γxi Kx

β

αKx

Ky

Ky

(a) (b) (c)

FIGURE 3.

If we are given a link cobordism between two links and a framing of one, then it induces aframing of the other. Soτx induces a framingτy of Ky. The Pontryagin-Thom construction tells usthatgr(x) − gr(y) equalsτy − ηy. We will now compute this difference. SinceKx andKy coincideas framed links outside ofC, we only need to do this calculation in a neighborhood ofC. To doso, we take a normal vector fieldN to C and extend it arbitrarily toKx ∩ Ky. SoN gives rise to aframing ofS, which we callν. We denote byνx andνy the restrictions ofν to Kx andKy, resp. Wewill compute the difference between the framings by first comparing them withν and then usingthe fact that

τy − ηy = (τy − νy) − (ηy − νy) = (τx − νx) − (ηy − νy).

We will look at a neighborhood of the corners ofF. In fact we only need to compute how manytimesτx rotates with respect toνx, whereKx coincides with eachγxi and similarly forηy. We calla nondegenerate corner ofF convex2 if it is a corner of someDk ⊂ F for only onek andconcave1

if it is a corner of someDk ⊂ F for three values ofk. For convex vertices, the difference is 0 forboth anxi and ayj. For concave vertices, it is+1 for anxi and−1 for ayj, as shown in Figure 4.In this picture, the orientation of the link is pointing down, so a counterclockwise turn counts as

2Some authors use the adjectives acute and obtuse to denote convex and concave, respectively.

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 9

τxνx τxνx

ηy νy ηy νy

convexxi concavexi convexyj concaveyj

β

α

β

α

α

β

α

β

FIGURE 4.

a+1, since that is a left-handed twist. At the distinguished corner, we rotatedE1 by an additional2πχ(F) clockwise. If this is anxi it accounts forχ(F) in τx − νx and if it is ayj, it accounts for−χ(F) in ηy − νy. Soτy − ηy = χ(F) + q, whereq is the number of concave corners.

Now if we denote byp the number of convex corners, by Lipshitz’s formula,

ind(F) = e(F) + nx(F) + ny(F)

= χ(F) − 14 p+ 1

4q+ 14 p+ 3

4q

= χ(F) + q = τy − ηy.

Sincenz(F) = 0, we conclude thatgr(x) − gr(y) = τy − ηy = µ(A) = gr(x, y).Step 2: We will now prove a technical lemma that will be useful in thegeneral case.Given two linksK1 andK2 in Y that belong to the same homology class, letS be an immersed

cobordism between them. That means thatS is an immersed oriented compact surface inY× [0, 1]that is embedded near its boundary and such that∂S = K1 × {1} ∪ (−K2) × {0}. Since an immersedsurface also has a normal bundle, we can ask whether framingsof K1 andK2 extend to a framingof S. So given a framing ofK1, the surfaceS induces a framing ofK2. The induced framing ofK2 depends heavily onS. In fact, if we denote the signed number of self-intersections of S byδ(S), we have the following lemma. Here we orientY × [0, 1] by declaring that{∂t,E1,E2,E3} isan oriented basis, where{E1,E2,E3} is an oriented basis forTY andt is the coordinate function on[0, 1].

Lemma 2.3. Let K1 and K2 be links in Y that belong to the same homology class and let S andS′ be immersed cobordisms between them, which are in the same relative homology class. Givena framing of K1, let ζS and ζS′ be the framings induced on K2 by S and S′, respectively. ThenζS − ζS′ = 2(δ(S) − δ(S′)).

To prove that, we will use another lemma, which is a standard result in Differential Topology.

Lemma 2.4. LetΣ be a closed oriented surface immersed into a closed oriented4-manifold X. Lete(NΣ) be the Euler class ot the normal bundle ofΣ with the orientation induced by the orientationof X. Then

[Σ] · [Σ] = e(NΣ) + 2δ(Σ).

Proof of Lemma 2.3.We are givenS,S′ ⊂ Y × [0, 1] such that∂S′ = ∂S = K1 × {1} ∪ (−K2 × {0})and such thatS′ − S vanishes inH2(Y × [0, 1]). Now we take two copies ofY × [0, 1], switchthe orientation of one of them and glue along their common boundaries. We can think of this asY× [−1, 1] with the obvious identification ofY× {−1} andY× {1}, which gives usY×S1. We can

10 VINICIUS GRIPP AND YANG HUANG

also glueS ⊂ Y×[0, 1] to−S′ ⊂ Y×[−1, 0] and we get a closed surface that we callΣ. Now we canassume that inY× [−ε, ε], the surfaceΣ is K2 × [−ε, ε], for ε small. We useS to get a framing onK2 ⊂ Y×{ε} andS′ to get a framing onK2 ⊂ Y×{−ε}. These are exactlyζS andζS′, respectively. Itfollows that the relative Euler class of the normal bundle ofΣ restricted toK2 × [−ε, ε] given thesetwo framings isζS′ − ζS. Thereforee(NΣ) = ζS′ − ζS. Now, if we think ofS, S′ andΣ as chains inY× S1, we can writeΣ = S − S′. SoΣ − (K1 × S1) vanishes inH2(Y × S1). Hence

[Σ] · [Σ] = [K1 × S1] · [K1 × S1] = 0.

Therefore, by Lemma 2.4,ζS − ζS′ = 2δ(Σ) = 2(δ(S) − δ(S′)).

�

Step 3: We now proceed to the general case. We had writtenD(ϕ) as a union of surfacesFl ⊂ Σ,which can be seen as 2-chains inΣ. We need to show that

gr(x) − gr(y) =m∑

l=1

(e(Fl) + nx(Fl) + ny(Fl) − 2nz(Fl)

).

Let γa be the projection toΣ of the image of∂D2∩{z; Re(z) ≤ 0} underϕ andγb be the projectionof the image of∂D2∩ {z; Re(z) ≥ 0}. Thenγa−γb = ∂D(A) =

∑l ∂Fl. We observe that the a corner

of Fl can either be anxi, a yj or neither. If it is neither of the two, then the interiors ofγa andγb

intersect at that point. We call this point an auxiliary corner and denote each of them bywk forsomek. Now fix and auxiliary cornerwk. Let r be the multiplicity ofγa ands be the multiplicityof γb in a neighborhood ofwk and assumer < s, see Figure 5(a). We might also have an extrat tothe multiplicity of all the four regions. But that will not affect the calculations. So, for simplicity,we can assume thatt = 0. We get a convex corner forr of theFl ’s and a concave one forr of theFl ’s. For (s− r) of theFl ’s, this point lies on the boundary and is not a corner. We denote byγwk

the flow line passing throughwk. We say thatwk is positive if it behaves as a convexxi (i.e γwk ispositively oriented) and as a concaveyj (i.e γwk is negatively oriented), and thatwk is negative ifthe opposite happens, as shown in Figure 5(b).

The orientations onγa and−γb give rise to an orientation of∂Fl. That is also the orientationinduced fromΣ, sinceA ≥ 0. Now we need to define{E1,E2,E3}. We want to defineE1 on Fl inthe same way as we did when we had only oneFl. But we have to be more careful since we mayhaveα andβ curves contained on the surfaceFl. This can happen in three different ways: there isa boundary degenerate corner, an interior degenerate corner or a pair of nondegenerate corners thatare on∂Fl but are not corners of∂Fl for somel. Figure 6 shows an example of each of those case.

For eachFl, we can defineCl, just as we did to defineC in Step 1, except that when one ofthe edges ofFl is a circle, we will attach a disk to it, not a triangular surface. We will first defineE1 on Fm. For each edge ofFm that is not a circle, we defineE1 to be the positive unit tangentvector to∂Fm outside neighborhoods of the corners. Along an edge that is acircle, we defineE1

to be any vector field whose rotation number along this circleis 0. We note that nondegeneratecorners along this circle, e.g. Figure 6, cannot happen forFm. If we have anα or β circle containedin the interior ofFm, then we defineE1 along this circle such that its rotation number is 0. In a

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 11

s

rr + s

positivewk negativewk

β β α

α α β

(a) (b)FIGURE 5.

0 0

1 1

1 1

1 1

00 0

21 1

boundary degenerate interior degenerate pair of nondegenerate

FIGURE 6.

neighborhood of each corner including the auxiliary ones, we rotateE1 as least as possible, as wedid in Step 1. We also need to choose some nondegenerate corners, i.e. not auxiliary corners, torotate a total ofχ(Fm) + d(Fm), whered(Fm) denotes the number of boundary degenerate cornersof Fm. After doing that, we can now extendE1 to a vector field onFm. Now we extend it to thetriangular surfaces belonging toCm just as we did in Step 1. For each circle on∂Fm, we extendE1

to the attaching disk by requiring that it is tangent to the surface f −1(t), for every 3/2 ≤ t ≤ 2, ifthe circle is aβ j and for every 1≤ t ≤ 3/2 if the circle is anαi. We note thatE1 is not tangent tothis disk at any point except for the corresponding criticalpoint, i.e whent = 1 or 2, and onΣ.

Now we want to extendE1 to Fm−1 ⊃ Fm. We first defineE1 on ∂Fm−1. We can do it the sameway as we did for∂Fm except near the intersection of∂Fm−1 andFm, whereE1 is already defined.This can only happen in two cases. The first one is when they intersect at an auxiliary corner.In this case we just rotateE1 along∂Fm−1 as least as possible, so that it coincides withE1 at thecorner. The second case is when there is a circle inFm−1 that contains two nondegenerate corners.In this case,E1 is already defined in the segment connecting the two nondegenerate corners. Sowe extend it to all of this circle in such a way that its rotation number is 0. After doing that, wecan extendE1 to Cm−1 just as we did forCm. Proceeding by induction, we defineE1 on Cl, forl = m,m− 1, . . . , 1.

We can defineE3 onCl as we did before, but when we have a circle on∂Cl , we extendE3 to thecorresponding disk by requiring thatE3 is normal tof −1(t) for everyt. Now we defineE2 such that{E1,E2,E3} is an orthonormal basis forTY alongCl for all l.

For everyα or β circle contained inF1, either we have attached the corresponding disk to it insomeCl or it contains an interior degenerate corner, in which case,we have also required that therotation number ofE1 along this circle is 0. So in the latter case, we can extendE1 andE3 as wedid when the circle was in the boundary. Now, there is no obstruction to extending the orthonormalframe{E1,E2,E3} to all ofY and, as before, that determines a trivialization by sendingEi to ei ∈ R3.

12 VINICIUS GRIPP AND YANG HUANG

Again, we takeK′x = w−1x (e1) andK′y = w−1

y (e1). We can isotope them the same way as beforeto getKx andKy so that they contain segments ofγxi andγyi near the respective corners. We alsodefine the surfacesCl in the same fashion as we did in Step 1. Now, to compute the difference oftheir framings, we will use several immersed cobordisms. Westart fromKy. We useC1 to definean immersed cobordism. This cobordism exchanges segments of the flow linesγyj correspondingto cornersyj of F1 with segments of someγxi corresponding to cornersxi of F1 and possiblysegments of someγwk, corresponding to concave auxiliary cornerswk. The next step is to useC2

to construct an immersed cobordism which exchanges segments of someγyi by segments of someγxi , possibly involves auxiliary corners and keeps the rest of the link fixed. We can continue thisconstruction inductively and define immersed cobordisms for C1, . . . , Cm. Every time we obtain aγwk, it will first appear as a concave corner and later as a convex corner. Ifwk is positively oriented,then it will appear as a positive concave angle and a negativeconvex angle, which means that theyjust cancel, when we stack the immersed cobordisms. Ifwk is negatively oriented, then it willappear as a negative concave corner first and as a positive convex corner later. In this case, we addtrivial cobordisms to the immersed cobordisms where the segment ofγwk appears and to all of theones in between. After stacking all those, the auxiliary corners cancel and we obtain an immersedcobordism fromKy to Kx. Similarly to the case when we had only oneFl, we conclude that thedifference of the framings using the cobordism induced byCl is χ(Fl) + d(Fl) + q(Fl) for eachl,whereq(Fl) is the number of concave corners ofFl, not counting the auxiliary corners. Moreoverfor each auxiliary cornerwk, the difference of framings is+1 if wk is positive, and−1 if wk isnegative. So using this immersed cobordism fromKy to Kx, the difference between the framings is∑m

l=1

(χ(Fl) + d(Fl) + q(Fl)

)plus the signed count of the auxiliary corners.

We know that there is an embedded link cobordism fromKy to Kx in the same relative homologyclass as the immersed cobordism we were considering. So, by Lemma 2.3,τy−ηy equals the differ-ence obtained using the immersed cobordism minus twice the signed number of self-intersectionsof the immersed cobordism, since the self-intersection number of an embedded cobordism is 0.We now need to consider three cases.

(i) There are boundary degenerate corners or a pair of nondegenerate corners on anα or βcurve contained in some∂Fl.

(ii) There are interior degenerate corners(iii) There are nondegenerate corners in the interior of some Fl.(iii) The basepointz in in the interior ofF1.

In case (i), self-intersections could exist ifKx or Ky intersectsCl for l such thatCl contains thedisk we attach to the correspondingα or β circle. Letxi andyj be the corresponding corners. ThenCl dividesN(γxi ) in two disconnected components and we can see thatKx enters and exitsN(γxi )in the same component. Similarly foryj. Therefore the signed number of intersections withCl is0. In this case,nxi + nyj = 1. But this+1 appears in the difference of framings when we addedd(Fl) turns toE1 near a nondegenerate corner.

In case (ii), letxi = yj be the interior degenerate corner. So,nxi + nyj = 2. Also, Kx = Ky inN(γxi ). Also, Kx intersectsCl negatively at only one point. Therefore, by Lemma 2.3, we have twoadd +2 to the difference of the framings.

In case (iii), sinceFi ⊃ F j, for i < j, and the cobordism corresponding toCi is taken before theone corresponding toC j, only the nondegenerateyj ’s which are in the interior of anF j correspond

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 13

to intersections. So, by Lemma 2.3, we have to add twice the number of interior nondegenerateyj ’s. On the other hand, if we had built our immersed cobordismsin the opposite order, i.e. startingwith Fm and going all the way toF1, then we would get the same result, except that we wouldbe counting twice the number of interior nondegenerate corners xi, but in this case the sign ofthe auxiliary corners are switched. Since the two calculations have to coincide, it follows that thenumber of interior nondegenerate cornersxi plus the number of positive auxiliary corners equalsthe number of interior nondegenerate cornersyj plus the number of negative auxiliary corners.So twice the number of interior nondegeneratexi ’s plus the signed count of the auxiliary cornersequals the total number of interior nondegenerate corners.That is exactly what we were missingto get the fullnx(Fl) andny(Fl). Therefore, combining cases (i),(ii) and (iii), we conclude that thedifference of the framings is

∑ml=1

(e(Fl) + nx(Fl) + ny(Fl)

), which is equal toµ(A).

In case (iv), thenKx = Ky nearγz. If Kx intersectsFl, then it does so positively. Hence, byLemma 2.3, we get an extra−2

∑l nz(Fl) in the difference of framings. Therefore

gr(x) − gr(y) = τy − ηy = µ(A) − 2nz(A) = gr(x, y).

3. THE ABSOLUTE GRADING OF THE CONTACT INVARIANT

In [17], Oszvath-Szabo defined the contact classc(ξ) ∈ HF(−Y) for a contact 3-manifold (Y, ξ),and they showed that it is an invariant ofξ. Later, Honda-Kazez-Matic [4] gave an alternativedefinition ofc(ξ) using an open book decomposition adapted toξ. In this section, we compute theabsolute grading of the contact invariantc(ξ).

3.1. Contact topology and open book decompositions. Let Y be a closed oriented 3-manifold.A contact structureξ is a maximally non-integrable co-oriented 2-plane field, i.e. there exists a1-formλ such thatλ ∧ dλ > 0 andξ = kerλ. We call suchλ a contact formof ξ. TheReeb vectorfield Rλ associated withλ is the unique vector field which satisfies (i)Rλ y dλ = 0, (ii) Rλ y λ = 1.Although the dynamics ofRλ depend heavily on the choice ofλ, its homotopy class is an invariantof ξ. In fact, two contact structures are homotopic if and only iftheir associated Reeb vector fieldsare homotopic.

Now recall that anopen book decompositionof Y is a pair (S, h), whereS is a compact, orientedsurface of genusg with boundary,h : S→ S is a diffeomorphism which is the identity on∂S, andY is homeomorphic to (S × [0, 1])/ ∼. The equivalence relation∼ is defined by (x, 1) ∼ (h(x), 0)for x ∈ S and (y, t) ∼ (y, t′) for y ∈ ∂S andt, t′ ∈ [0, 1]. Given a contact structureξ on Y, an openbook (S, h) is adapted toξ if there exists a contact formλ for ξ such thatRλ is positively transverseto int(S) and positively tangent to∂S.

Fix an adapted open book (S, h) of (Y, λ). Following [4], let {a1, · · · , a2g} be a set of pairwisedisjoint, properly embedded arcs onS such thatS\⋃2g

i=1 ai is a single polygon. We call{a1, · · · , a2g}a basisfor S. Next letbi be an arc which is isotopic toai by a small isotopy so that the followinghold:

(1) The endpoints ofai are isotoped along∂S, in the direction given by the boundary orienta-tion of S.

(2) ai andbi intersect transversely in one pointxi in the interior of S.(3) If we orientai, andbi is given the induced orientation from the isotopy, then the sign of the

intersectionai ∩ bi is+1.

14 VINICIUS GRIPP AND YANG HUANG

See Figure 7.

ai bi

xiS

FIGURE 7. The arcsai andbi onS.

Observe that (S, h) naturally induces a Heegaard splitting ofY by lettingH1 = (S× [0, 1/2])/ ∼andH2 = (S× [1/2, 1])/ ∼. This gives a Heegaard decomposition ofY of genus 2g with HeegaardsurfaceΣ = ∂H1 = −∂H2. By choosing a basis{a1, · · · , a2g} for S and following the constructionsabove, we obtain two collections of simple closed curvesα = {α1, · · · , α2g} andβ = {β1, · · · , β2g}onΣ, whereαi = ∂(ai × [0, 1/2]) andβi = ∂(bi × [1/2, 1]) for i = 1, · · · , 2g. Then one can properlyplace the basepointz and reverse the orientation ofY to obtain a weakly admissible Heegaarddiagram (Σ,β,α, z) for −Y. It is observed in [4] thatx = (x1, · · · , x2g) ∈ CF(Σ,β,α, z) defines acycle, wherexi = ai ∩ bi ∈ αi ∩ βi, i = 1, · · · , 2g.

Theorem 3.1 (Honda-Kazez-Matic [4]). The class[x] ∈ HF(−Y) represented byx ∈ CF(Σ,β,α, z)from above is an invariant ofξ and it is equal to c(ξ) defined in [17].

Remark3.2. In light of Theorem 3.1, in order to prove Theorem 1.1(b), it suffices to show

(3.1.1) gr(x) = [ξ]

as homotopy classes of oriented 2-plane fields.

3.2. Proof of Theorem 1.1(b). Throughout this section, we fix a contact formλ and an adaptedopen book decomposition (S, h) of (Y, λ). Note that the contact invariant is presented as an inter-section pointx in CF(−Y). The plan is to use the Pontryagin-Thom construction to show that thevector field constructed in Section 2 to definegr(x) is homotopic to the Reeb vector fieldRλ.

Proof of Theorem 1.1(b).Let f be a Morse function adapted to our special Heegaard diagram(Σ,α,β, z), whereΣ = (S × {0}) ∪ (S × {1/2}). Note that one needs to reverse the orientationof Y to define [x] = c(ξ). Equivalently, we shall consider, for the rest of the proof, the same Hee-gaard diagram (Σ,α,β, z), but with the downward gradient vector field−∇ f . All the constructionsof the absolute grading function carry over by simply reversing the direction of all vector fields.Let vx be a nonvanishing vector field, which is a modification of−∇ f , as defined in Section 2. Inparticular, the homotopy class of the orthogonal complement of vx equalsgr(x). Let S ⊂ int(S)be a closed subsurface such thatS deformation retracts ontoS, and assume thath is supported inS × {1}. It is easy to see that−∇ f is homotopic toRλ by linear interpolation in a small neigh-borhoodN(S × {1}) of S × {1} in M because they are both positively transverse toS × {1}. LetH = Y \ N(S × {1}) be the genus 2g handlebody3. So it suffices to show thatvx|H is homotopic toRλ|H relative to∂H.

3In fact H is a handlebody with corners, but this is irrelevant here because we are considering continuous vectorfields.

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 15

To do so, consider a closed collar neighborhoodai × [−1, 1] ⊂ S×{1/2} of ai on the middle pagesuch that it containsbi in the interior, fori = 1, · · · , 2g. Let Bi = (ai × [−1, 1] × [0, 1]) ∩ H ⊂ Hbe a 3-ball (with corners) inH, which containsai andbi in the interior. See Figure 8 for picturesof the vector fieldsRλ|Bi and−∇ f |Bi .

aibi

xiS S

(a) (b)

FIGURE 8. (a) The Reeb vector fieldRλ restricted toBi. (b) The downward gradientvector field−∇ f restricted toBi.

Claim: There exists a non-singular vector fieldR′λ on H, homotopic toRλ relative to∂H, such that(i) R′

λ|∂Bi = vx|∂Bi , (ii) R′

λ|Bi is homotopic tovx|Bi relative to∂Bi, for i = 1, · · · , 2g.

Proof of Claim.Let Dl = (ai × {−1} × [0, 1]) ∩ H andDr = (ai × {1} × [0, 1]) ∩ H be the left andright disk boundaries ofBi, respectively. Observe thatRλ = vx on∂Bi \ (Dl ∪ Dr) by construction.We shall consider a collar neighborhoodN(Dl) = (ai × [−1− δ,−1+ δ] × [0, 1])∩H of Dl for somesmallδ > 0, and homotopeRλ to R′λ with the desired properties withinN(Dl). Note that the sameconstruction can be carried over to a collar neighborhood ofDr .

We construct a model vector fieldVl on D2 × [−1, 1] in steps. First letF0 be a singular foliationon D2 which has two elliptic singularities as depicted in Figure 9(a). Letγ ⊂ D2 × [−1, 0] be aproperly embedded, boundary parallel arc such that∂γ is exactly the union of the two singularitiesof F0 on D2 × {−1}. Then there exists a foliationF by disks onD2 × [−1, 0] such that for anyleaf F of F , we have∂F ∩ int(D2 × [−1, 0]) = γ, and∂F ∩ (D2 × {−1}) is a leaf ofF0. Let V′lbe a non-singular vector field onD2 × [−1, 0] such that it is positively tangent toγ and positivelytransverse to the interior of all leaves ofF as depicted in Figure 9(b). Up to homotopy, we canassume thatV′l |D2×{0} = vx|Dl as vector fields on a disk. By fixing a trivialization of the tangentbundleT(D2 × [−1, 1]) using the standard embeddingD2 × [−1, 1] ⊂ R3, we define the vector fieldVl on D2 × [−1, 1] by

Vl(x, t) =

V′l (x, t) if − 1 ≤ t ≤ 0,

V′l (x,−t) if 0 ≤ t ≤ 1.

wherex ∈ D2 is any point. IdentifyD2 × [−1, 1] with N(Dl ) by rescaling in the [−1, 1]-directionsuch thatDl is identified withD2× {0}, N(Dl) \ Bi is identified withD2 × [−1, 0], andN(Dl )∩ Bi isidentified withD2× [0, 1]. It is easy to see thatRλ|N(Dl ) is homotopic toVl as vector fields onN(Dl)relative to the boundary. Similarly, one can define a non-singular vector fieldVr on N(Dr ) suchthatRλ|N(Dr ) is homotopic toVr as vector fields onN(Dr ) relative to the boundary. By applying theabove homotopy, which is supported inN(Dl ) ∪ N(Dr ), to Rλ, and repeat this process for everyBi,i = 1, · · · , 2g, we obtain a new non-singular vector fieldR′

λ. Observe thatR′

λsatisfies condition (i)

by construction.

16 VINICIUS GRIPP AND YANG HUANG

(a) (b)

γ

FIGURE 9. (a) The singular foliation onD2. (b) The vector fieldV′l on a leaf ofFin D2 × [−1, 0].

To show thatR′λ satisfies condition (ii), we use the Pontryagin-Thom construction. Trivialize thetangent bundleT Bi by embeddingBi ⊂ R3 such thatDl (or Dr) is parallel to thexz-plane, and the[−1, 1]-direction is parallel to they-axis. Consider the associated Gauss mapsGvx |Bi : Bi → S2

andGR′λ|Bi : Bi → S2. Without loss of generality, we assume thatGvx |Bi andGR′

λ|Bi are smooth, and

p = (0, 1, 0) ∈ S2 is a common regular value. Letp′ = (ǫ,√

1− ǫ2, 0) ∈ S2 be a nearby commonregular value which keeps track of the framing, whereǫ > 0 is small. It is now a straightforwardcomputation that the Pontryagin submanifoldsG−1

vx(p) andG−1

R′λ

(p) are both framed cobordant to theframed arc depicted in Figure 10 relative to the boundary. HenceR′λ|Bi is homotopic tovx|Bi relativeto ∂Bi, for all i = 1, · · · , 2g. This finishes the proof of the claim. �

FIGURE 10. A framed arc inBi, where the framing is indicated by the green arc.

It remains to show thatR′λ

is homotopic tovx on H \ (⋃2g

i=1 Bi) relative to the boundary. Let(D2, id) be the trivial open book ofS3, andD ⊂ int(D2) be a slightly smaller disk. LetH denote

H \ (⋃2g

i=1 Bi) and observe that it is naturally identified with (D2 × [0, 1] \ ((D × [0, ǫ)) ∪ (D ×(1 − ǫ, 1])))/ ∼ by construction. On the one hand, it is easy to see thatR′λ|H is homotopic to therestriction of the Reeb vector field compatible with the openbook (D2, id). On the other hand, notethatH is nothing but a neighborhood of the gradient trajectory which connects the index 0 criticalpoint to the index 3 critical point. Hence it follows immediately from our construction ofgr(x) thatvx|H is also homotopic to the Reeb vector field compatible with (D2, id). This finishes the proof ofTheorem 1.1(b). �

Now we compute the twisted absolute grading of the twisted contact invariant defined in [14].Let x ∈ Tα ∩ Tβ be the generator inCF(−Y), which defines the usual contact invariant as before.

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 17

Let Z[H1(Y;Z)]× denote the set of invertible elements inZ[H1(Y;Z)]. First recall that the twistedcontact invariantc(ξ) associated with the contact structureξ is defined by

c(ξ) = [u · x] ∈ HF(−Y)/Z[H1(Y;Z)]×

whereu ∈ Z[H1(Y;Z)]×. Althoughc(ξ) is only well-defined up to a unit inZ[H1(Y;Z)], the twistedabsolute gradinggrtw(c(ξ)) defined by (1.0.2) still makes sense. The following resultis immediate.

Corollary 3.3. If ξ is a contact structure on Y, thengrtw(c(ξ)) = [ξ] ∈ P(Y).

Proof. This follows immediately from (1.0.2) and Theorem 1.1(b). �

Now we are ready to prove the corollaries given in Section 1.

Proof of Corollary 1.5.If (Y, ξ) is strongly fillable, thenc(ξ) , 0 ∈ HF(−Y) according to [17].Since HF(−Y) is a finitely generated Abelian group, there can be only finitely many absolutegradings, i.e., homotopy classes of 2-plane fields, that support strongly fillable contact structures.

Now if (Y, ξ) is weakly fillable, thenc(ξ) , 0 ∈ HF(−Y)/Z[H1(Y;Z)]× according to [14]. SinceHF(−Y) is finitely generated as aZ[H1(Y;Z)] module, the same argument as above together withCorollary 3.3 implies that there can be only finitely many homotopy classes of 2-plane fields inYthat support weakly fillable contact structures. �

Proof of Corollary 1.6.By definition if Y is anL-space, thenHF(−Y) is a free Abelian group ofrank |H1(Y;Z)|. Therefore there are at most|H1(Y;Z)|-many homotopy classes of 2-plane fieldsthat support strongly fillable contact structures. To get the same result for weakly fillable contactstructures, it suffices to observe that sinceY is a rational homology sphere by assumption, we have

HF(−Y) ≃ HF(−Y) ⊗ Z[H1(Y;Z)].

HenceHF(−Y) is a freeZ[H1(Y;Z)] module of rank|H1(Y;Z)|, and therefore the conclusion fol-lows as before. �

Proof of Corollary 1.7.It suffices to note that according to [18], ifY admits a metric of constantpositive curvature, thenY is anL-space. �

4. 4-DIMENSIONAL COBORDISM AND ABSOLUTEQ-GRADING

Let W be a connected compact oriented 4-dimensional cobordism between two connected ori-ented 3-manifoldsY0 andY1 such that∂W = −Y0 ∪ Y1. Fixing a Spinc structuret on W, Ozsvath-Szabo [19] constructed a mapFW,s : HF◦(Y0, t|Y0)→ HF◦(Y1, t|Y1) between Heegaard Floer homol-ogy groups by choosing a handle decomposition ofW, and counting holomorphic triangles. It turnsout thatFW,t is an invariant ofW, i.e., it is independent of the choice of a handle decomposition ofW. Throughout this section we fix a Heegaard diagram (Σ,α,β) for Y0 and a handle decompositionof W. Let (Σ,α,γ) be the associated Heegaard diagram forY1 as constructed in [19]. We considerthe associated chain mapFW,t : CF(α,β, t|Y0)→ CF(α,γ, t|Y1).

Observe thatFW,t : CF(α,β, t|Y0) → CF(α,γ, t|Y1) is a linear map between graded vectorspaces. However, according to Theorem 1.1(a),CF(α,β, t|Yi) is graded by the set of homotopyclasses of oriented 2-plane fieldsP(Yi), i = 0, 1, so it is not possible to define an integer degree ofFW,t. There is a weaker notion which is applicable here. Namely, let W : Y0 → Y1 be a cobordism

18 VINICIUS GRIPP AND YANG HUANG

andξi be an oriented 2-plane field onYi, for i = 0, 1. We sayξ0 ∼W ξ1 if and only if there exists analmost complex structureJ onW such that [ξi] = [TYi ∩ J(TYi)], for i = 0, 1, as homotopy classesof oriented 2-plane fields.

The main goal of this section is to prove Theorem 1.1(d) on thechain level, which we formalizein the following theorem for the reader’s convenience.

Theorem 4.1. Let W : Y0 → Y1 be a compact oriented cobordism with a fixed handle decom-position, t ∈ Spinc(W) a Spinc structure on W, and FW,t : CF(α,β, t|Y0) → CF(α,γ, t|Y1) theassociated cobordism map as discussed above. Thengr(x) ∼W gr(y) for any homogeneous gener-ator x ∈ Tα ∩ Tβ in CF(α,β, t|Y0), and any homogeneous summandy of FW,t(x).

Before we give the proof of Theorem 4.1, we take a step back andlook at the Heegaard FloerhomologyHF◦(Y, s) for a torsion Spinc structures. By [19], there is an absoluteQ-grading ofHF◦(Y, s) which lifts the relativeZ-grading. We shall see that our construction indeed generalizestheir absoluteQ-grading. To do so, recall the following construction due toR. Gompf [3]. Letξ bean oriented 2-plane field on a closed, oriented 3-manifoldY. Then there exists a compact, almostcomplex 4-manifold (X, J) whosealmost-complex boundaryis (Y, ξ), i.e. Y = ∂X (as orientedmanifolds) andξ = TY∩ J(TY) with the complex orientation. Ifc1(ξ) is a torsion class, then letθ(ξ) = (PD c1(X))2−2χ(X)−3σ(X) ∈ Q, whereχ is the Euler characteristic andσ is the signature.Observe thatθ(ξ) is independent of the choice of the capping almost complex 4-manifold (X, J)because the quantity (PD c1(X))2 − 2χ(X) − 3σ(X) vanishes for a closedX.

Let s ∈ Spinc(Y) be a Spinc structure such thatc1(s) is a torsion class, and letU be the setof homogeneous elements inCF(Y, s). We define an absolute grading functiongr0 : U → Q bygr0(x) = (2+θ(gr(x)))/4 ∈ Q for anyx ∈ U. This induces an absolute grading function onCF∞(Y, s)by gr0([x, i]) = 2i + gr0(x), and hence on the sub- and quotient-complexesCF−(Y, s) andCF+(Y, s).

For reader’s convenience, we recall the following theorem/definition of the absoluteQ-gradingdue to Ozsvath-Szabo [19].

Theorem 4.2 (Ozsvath-Szabo). There exists an absolute grading functiongr : U → Q satisfyingthe following properties:

(1) The homogeneous elements of least grading inHF(S3, s0) have absolute grading zero.(2) The absolute grading lifts the relative grading, in the sense that ifx, y ∈ U, thengr(x, y) =

gr(x) − gr(y).(3) If W is a cobordism from Y0 to Y1 endowed with a Spinc structuret whose restriction to Yi

is torsion for i= 0, 1, then

gr(FW,t(x)) − gr(x) =(PD c1(t))2 − 2χ(W) − 3σ(W)

4for anyx ∈ U.

We have the following corollary:

Corollary 4.3. The functiongr0 described above defines an absoluteQ-grading for HF◦(Y, s),which coincides with the absoluteQ-gradinggr defined above.

Proof. We use the Pontryagin-Thom construction. By fixing a trivialization ofTY, the homotopyclasses of oriented 2-plane fields onY are 1-1 correspondent to the framed cobordism classes

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 19

of framed links inY. The first assertion of the corollary follows from Theorem 1.1(a) and theobservation that adding a right-handed full twist toξ is equivalent to decreasingθ(ξ) by 4.

It follows from the proof of Theorem 4.1 that ift be a Spinc structure onW whose restrictionto Yi is torsion, fori = 0, 1, thenFW,t(x) is homogeneous for every homogeneous elementx ∈ U.Since we have shown in Theorem 2.1 that our absolute gradinggr refines the relative grading, inorder to show thatgr0 coincides with the absoluteQ-grading defined in [19], it suffices to verifythe following two conditions:

(1) (Normalization) For the standard contact 3-sphere (S3, ξstd), gr0(c(ξstd)) = 0.(2) (Cobordism formula) LetW : Y0 → Y1 be a cobordism, andt be a Spinc structure onW

whose restriction toYi is torsion,i = 0, 1. Then

gr0(FW,t(x)) − gr0(x) =(PD c1(t))2 − 2χ(W) − 3σ(W)

4for any homogeneousx ∈ U.

To prove (1), note that it follows from the fact that (S3, ξstd) is the almost complex boundary ofthe standard unit 4-ballB4 ⊂ C2.

To prove (2), let (X, J) be an almost complex 4-manifold with almost complex boundary (Y0, gr(x)).By Theorem 4.1, there exists an almost complex structureJ′ on W such that bothgr(x) andgr(FW,t(x)) are J′-invariant with the complex orientation. We obtain a new almost complex 4-manifold with almost complex boundary (X ∪Y0 W, gr(FW,t(x))) by gluing (X, J) and (W, J′) alongY0. Recall the following theorem on the signature of 4-manifolds due to Novikov:

Theorem 4.4 (Novikov). Let M be an oriented 4-manifold obtained by gluing two 4-manifolds M1

and M2 along some components of their boundaries. Then the signature is additive:

σ(M) = σ(M1) + σ(M2).

We therefore calculate as follows:

gr0(FW,t(x)) − gr0(x) =θ(gr(FW,t(x))) − θ(gr(x))

4

=(PD c1(W, J′))2 − 2χ(W) − 3σ(W)

4

=(PD c1(t))2 − 2χ(W) − 3σ(W)

4,

This finishes the proof of the second assertion of the corollary. �

The proof of Theorem 4.1 occupies the rest of this section. Weshall follow the construction ofFW,t given in [19].

Proof of Theorem 4.1.We fix a handle decomposition ofW, and study the 2-handle attachmentsand 1- and 3-handle attachments inW separately.

CASE 1. SupposeW is given by 2-handle attachments along a framed linkL ⊂ Y0. Let ∆ denotethe two-simplex, with verticesvα, vβ, vγ labeled clockwise, and letei denote the edgevj to vk,where{i, j, k} = {α, β, γ}. Recall that given a Heegaard triple (Σ,α,β,γ), one can associate to it a4-manifold

20 VINICIUS GRIPP AND YANG HUANG

(4.0.1) Wα,β,γ =(∆ × Σ) ∐(eα × Uα)

∐(eβ × Uβ)

∐(eγ × Uγ)

(eα × Σ) ∼ (eα × ∂Uα), (eβ × Σ) ∼ (eβ × ∂Uβ), (eγ × Σ) ∼ (eγ × ∂Uγ)

whereUα (resp. Uβ, Uγ) is the handlebody determined by theα (resp.β, γ) curves. LetYα,β =Uα ∪Uβ, Yβ,γ = Uβ ∩Uγ, andYα,γ = Uα ∪Uγ be the 3-manifolds obtained by gluing theα-, β- andγ-handlebodies alongΣ in pairs. After smoothing the corners, we have

∂Wα,β,γ = −Yα,β − Yβ,γ + Yα,γ

as oriented manifolds. See Figure 11.

eα

eβ eγ

Yα,β Yα,γ

Yβ,γ

FIGURE 11. The 4-manifoldWα,β,γ associated with a Heegaard triple (Σ,α,β,γ).

According to [19], ifW is obtained by attaching 2-handles along a framed linkL, then thereexists a triple Heegaard diagram (Σ,α,β,γ, z) such thatYα,β = Y0, Yβ,γ = #n(S1 × S2) for somen ≥ 1, andYα,γ = Y1. Moreover, after filling in the boundary componentYβ,γ by the boundaryconnected sum #nb(S

1×B3), we obtain the original cobordismW. Fix a Spinc structuret onW withsi = t|Yi , i = 0, 1. LetΘ ∈ CF(#n(S1 × S2)) be the top dimensional generator and letx ∈ Tα ∩ Tβ.By definition, the image ofx under the cobordism mapFW,t : CF(Y0, s0) → CF(Y1, s1) is a lin-ear combination of the generatorsy ∈ Tα ∩ Tγ with coefficients being the count of Maslov index 0holomorphic triangles connectingx,Θ andy. Let y be a generator appearing inFW,t with a nonzerocoefficient. We prove the following claim.

x

Θ

yα

β γ

FIGURE 12. A holomorphic triangle onΣ which connectsx,Θ, andy.

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 21

Claim: There exists an almost complex structureJ on Wα,β,γ such thatgr(x) ∈ P(Y0), gr(Θ) ∈P(#n(S1 × S2)), andgr(y) ∈ P(Y1) are allJ-invariant with the complex orientation.

Proof of Claim.We first assume thaty is the intersection point as shown in Figure 12, which isconnected tox andΘ by the obvious (embedded) holomorphic triangle. We begin byconstructinga 2-plane field oneα ×Uα, and note that the same construction carries over toeβ ×Uβ andeγ ×Uγ.

For simplicity of notations, we assumeg(Σ) = 1, so, for instance,x ∈ Tα ∩ Tβ is just one pointinstead of ag-tuple of points. The same argument applies to Heegaard surfaces of arbitrary genuswithout difficulty. Let Vα be the gradient flow onUα compatible with theα-curve so that it ispointing out along∂Uα. Let p ∈ Uα be the index 1 critical point ofVα andw ∈ Uα be the index0 critical point ofVα. Identify the edgeeα ⊂ ∆ with the subarc of theα-curve fromx to y, whichis an edge of the holomorphic triangle, such thatvγ is identified withx andvβ is identified withy. Abusing notations, we shall not distinguish a point oneα and the corresponding point on theα-curve under the above identification. For anyq ∈ eα, let γ0 andγ1 be the gradient trajectorieswhich connectw to zandp to q respectively. LetN(γi) be a tubular neighborhood ofγi as depictedin Figure 13, fori = 0, 1. By restricting the construction of the absolute grading in Section 2.1 toUα, we obtain a non-vanishing vector fieldV′α,q on Uα which depends on the choice ofq ∈ eα asdepicted in Figure 14. Thus we have constructed a 2-plane field ξα(q, x) = (V′α,q(x))⊥3 on eα × Uα,for anyq ∈ eα andx ∈ Uα. Here⊥3 denotes taking the orthogonal complement ofV′α,q within TUα.

γ0 γ1

w

z

p

q

w

z

p

q

(a) (b) (c)

FIGURE 13. (a) Theα-handlebodyUα and tubular neighborhoods of the gradienttrajectoriesγ0 andγ1. (b) The gradient vector fieldVα|N(γ0) in N(γ0). (c) The gradi-ent vector fieldVα|N(γ1) in N(γ1).

z q

(a) (b)

FIGURE 14. (a) The non-vanishing vector fieldV′α,q restricted toN(γ0). (b) Thenon-vanishing vector fieldV′α,q restricted toN(γ1).

22 VINICIUS GRIPP AND YANG HUANG

Similarly one constructs 2-plane fieldsξβ andξγ oneβ ×Uβ andeγ ×Uγ, respectively. However,note that the boundary componentYα,β = (vγ × Uα) ∪ (vγ × Uβ) of Wα,β,γ is a 3-manifold withcorners, and the 2-plane fieldsξα andξβ do not agree alongvγ × Σ because they are tangent tothe α- and β-handlebodies which intersect each other in an angle. To smooth the corners, wereplace the triangle∆ in (4.0.1) with a hexagonH with right corners and attachα, β, andγ handlesaccordingly as depicted in Figure 15. In this way we obtain a smooth cobordism which we stilldenote byWα,β,γ : Y0

∐(S1 × S2) → Y1, whereY0 = (vγ × Uα) ∪ ([0, 1] × Σ) ∪ (vγ × Uβ), Y1 =

(vβ × Uα) ∪ ([0, 1] × Σ) ∪ (vβ ×Uγ), andS1 × S2= (vα ×Uβ) ∪ ([0, 1] × Σ) ∪ (vα × Uγ) are smooth

3-manifolds. We construct a 2-plane fieldξ on (eα × Uα) ∪ (eβ × Uβ) ∪ (eγ × Uγ) ∪ ∂Wα,β,γ byextendingξα, ξβ, andξγ to the three copies of [0, 1] × Σ such that it is translation invariant in the[0, 1]-direction on each copy. By construction, it is easy to seethatξ|Y0 ≃ gr(x), ξ|S1×S2 ≃ gr(Θ),andξ|Y1 ≃ gr(y).

eα

eβ eγ

H

Y0 Y1

S1 × S2

FIGURE 15. The smooth cobordismWα,β,γ : Y0∐

(S1 × S2)→ Y1.

Let D1 ⊂ Σ be a closed neighborhood ofz, andD2 ⊂ Σ be a closed neighborhood of the holo-morphic triangle so that the non-vanishing vector fieldV′i,q is transverse toTΣ alongΣ \ (D1 ∪ D2)for any i ∈ {α, β, γ}, q ∈ ∂∆. We extendξ to the metric closure ofH × (Σ \ (D1 ∪ D2)) by lettingξ(x, y) = TyΣ for any x ∈ H, andy ∈ Σ \ (D1 ∪ D2). We construct an almost complex structureJon a subset ofWα,β,γ by askingξ andξ⊥4 to be complex line bundles, where⊥4 denotes taking theorthogonal complement inTWα,β,γ. In fact J is defined everywhere onWα,β,γ except finitely many4-balls (with corners), namely,H × D1 andH × D2. To extendJ to the wholeWα,β,γ, we round thecorners of∂(H × Di), i = 1, 2, in two steps.

Step 1. To round the corners of∂H × D1 and∂H × D2 near each vertex ofH, we first constructa local model for corner-rounding as follows.

Let (x1, y1, x2, y2) be coordinates onR2 ×R2 with the Euclidean metric. Consider a non-singularvector field

v(x1, y1, x2, y2) = f (x2, y2)∂

∂y1+ g(x2, y2)

∂

∂x2+ h(x2, y2)

∂

∂y2

onR2×R2, namely,f , g andh cannot be simultaneously zero. Observe thatv is everywhere tangentto R3 ≃ {(x1, y1, x2, y2) | x1 = constant}. Definev⊥3 to be the pointwise orthogonal complement to

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 23

v insideR3 ≃ {(x1, y1, x2, y2) | x1 = constant}. Let J be an almost complex structure onR2 × R2

which preserves the metric and satisfies:

• J( ∂∂x1

) = v||v|| ,

• J(v⊥3) = v⊥3.

LetL = {(x1, 0) | x1 ≥ 0} ∪ {(0, y1) | y1 ≥ 0} ⊂ R2 be aL-shaped broken line with a corner at theorigin. We round the corner ofL by considering

Lr = {(x1, 0) | x1 ≥ 1} ∪ {(0, y1) | y1 ≥ 1} ∪ {(x1 − 1)2 + (y1 − 1)2 = 1 | 0 ≤ x1 ≤ 1, 0 ≤ y1 ≤ 1}.Consider the smooth submanifoldL = Lr × R2 in R2 × R2. We compute the complex line

distributionTL ∩ J(TL) on TL with respect toJ. To do so, identifyL with (−∞,∞) × R2 suchthat{(0, y1) | y1 ≥ 1} is identified with (−∞, 0] ×R2, {(x1, 0) | x1 ≥ 1} is identified with [1,∞)×R2,and{(x1−1)2+(y1−1)2 = 1 | 0 ≤ x1 ≤ 1, 0 ≤ y1 ≤ 1} is identified with [0, 1]×R2. Letφt : R3→ R3

be the clockwise rotation about thex-axis byχ(t)π/2, where (x, y, z) are coordinates onR3 and

χ(t) =

0 if t ≤ 0,

t if 0 ≤ t ≤ 1,

1 if t ≥ 1.

Lemma 4.5. The 2-plane field TL ∩ J(TL) on L ≃ (−∞,∞) × R2 is the orthogonal complementof the non-singular vector fieldµ(t, x2, y2) = φt(v(x2, y2)).

Proof of Lemma 4.5.We first computeJ( ∂∂y1

) as follows. Note that

v⊥3 =

span{ ∂

∂x2, ∂∂y2} if g = h = 0,

span{g ∂∂y2− h ∂

∂x2, ∂∂y1− f g

λ2∂∂x2− f h

λ2∂∂y2} otherwise.

whereλ =√

g2 + h2. Since we assume thatJ preserves the Euclidean metric, we have

(4.0.2)

J( ∂

∂x2) = ∂

∂y2if g = h = 0,

J(g ∂∂y2− h ∂

∂x2) = λ2√

f 2+λ2( ∂∂y1− f g

λ2∂∂x2− f h

λ2∂∂y2

) otherwise.

It follows from (4.0.2) and the equationJ( ∂∂x1

) = v||v|| that

J( ∂∂y1

)=

1√f 2 + λ2

(− f

∂

∂x1− g

∂

∂y2+ h

∂

∂x2

).

It is easy to see thatTL ∩ J(TL) restricted to{t} × R2, t ≥ 1, is the orthogonal complement ofJ( ∂

∂y1) = µ(1, ·) up to positive rescaling withinTL. Moreover observe thatTL ∩ J(TL) restricted

to {t} × R2, for 0 ≤ t ≤ 1, is the orthogonal complement ofJ(t ∂∂y1+ (1 − t) ∂

∂x1), which is exactly

µ(t, ·) up to positive rescaling. �

Without loss of generality, letq be a vertex ofH whose adjacent edges areeα and [0, 1], where[0, 1] is an edge ofH connectingα- andβ-handlebodies. Take a small neighborhoodN(q) of q inH. Identify N(q) with a small neighborhood of the origin inR2 restricted to the first quadrant suchthateα ∪ [0, 1] is identified withL. We can further assume thatJ is defined onN(q)×Di by takingN(q) sufficiently small, and that it is invariant under translation in any direction tangent toN(q).

24 VINICIUS GRIPP AND YANG HUANG

Hence we can apply Lemma 4.5 to compute the complex line distribution onLr ×Di ⊂ N(q) ×Di,i = 1, 2, with respect toJ. By rounding all the corners ofH and applying Lemma 4.5, we concludethat:

(1) The complex line distributionT(∂H × D1) ∩ JT(∂H × D1) on∂H × D1 is, up to homotopyrelative to the boundary, the orthogonal complement of the non-singular vector fieldv1,wherev1|{p}×D1 is shown on Figure 16(a). In particularv1 is defined to be invariant in thedirection of∂H.

(2) Let θ ∈ [0, 2π) be the coordinate on∂H with the boundary orientation andψ : ∂H × D2 →∂H × D2 be a diffeomorphism defined byψ(θ, z) = (θ, eiθz). The complex line distributionT(∂H × D2) ∩ JT(∂H × D2) on ∂H × D2 is, up to homotopy relative to the boundary, theorthogonal complement of the non-singular vector fieldv2 = ψ∗(v′2), wherev′2 is invariantin the direction of∂H and its restriction top× D2, p ∈ ∂H, is shown on Figure 16(b).

∂H × D1 ∂H × D2

(a) (b)

FIGURE 16.

Step 2. Now we round the corners of∂(H×Di) = (∂H×Di)∪(H×∂Di), which is the union of twosolid tori meeting each other orthogonally. Note that the 2-plane fieldT(H × ∂Di) ∩ JT(H × ∂Di)on H × ∂Di is everywhere tangent toH by our choice ofDi ⊂ Σ, for i = 1, 2. Abusing notations,we still denote by∂(H ×Di) the smooth 3-sphere obtained by rounding the corners in thestandardway. Let ξi denoteT(∂(H × Di)) ∩ JT(∂(H × Di)), for i = 1, 2. Soξ1 and ξ2 are oriented 2-plane fields. Using the Pontryagin-Thom construction, we see thatξ1 is homotopic to the negativestandard contact structure onS3, while ξ2 is homotopic to the positive standard contact structureon S3. EmbedH × Di = B4 ⊂ C2 such thatH andDi are contained in orthogonal complex planesrespectively. Let

J0 =

(i 00 i

), J′0 =

(i 00 −i

)

be complex structures onC2. Then it is standard to check thatξ1 ≃ TS3 ∩ J′0TS3 and ξ2 ≃TS3 ∩ J0TS3 as oriented 2-plane fields, whereS3

= ∂B4 ⊂ C2. Hence we can extendJ to thewholeWα,β,γ satisfying all the desired properties.

Now we turn to the general case. Lety′ ∈ Tα ∩ Tγ be another intersection point inFW,t, i.e.there exists a holomorphic triangleψ′ ∈ π2(x,Θ, y′) such that the Maslov indexµ(ψ′) = 0. Lety ∈ FW,t(x) be the intersection point as shown in Figure 12 andψ ∈ π2(x,Θ, y) be the obviousholomorphic triangle of Maslov indexµ(ψ) = 0. Sinceψ andψ′ induces the same Spinc structure

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 25

t on W, we haveψ′ = ψ + φ1 + φ2 + φ3 for φ1 ∈ π2(x, x), φ2 ∈ π2(Θ,Θ), andφ3 ∈ π2(y, y′). Thisimplies

µ(ψ′) = µ(ψ) + µ(φ1) + µ(φ2) + µ(φ3).

Thereforeµ(φ1) − 2nz(φ1) = −(µ(φ3) − 2nz(φ3)),

becauseµ(ψ) = µ(ψ′) = nz(ψ) = nz(ψ′) = µ(φ2) − 2nz(φ2) = 0. Since we have shown thatthere exists an almost complex structureJ on Wα,β,γ such thatgr(x) ∈ P(Y0), gr(y) ∈ P(Y1) andgr(Θ) ∈ P(#n(S1×S2)) are allJ-invariant with the complex orientation, it is easy to show that thereexists another almost complex structureJ′ onWα,β,γ such thatgr(x)+µ(φ1)−2nz(φ1), gr(y)−(µ(φ3)−2nz(φ3)), andgr(Θ) are allJ′-invariant with the complex orientation. Here we are using theZ-actionas explained in Remark 1.3. Now it remains to observe thatgr(x) = gr(x)+µ(φ1)−2nz(φ1) ∈ P(Y0)sinceµ(φ1) − 2nz(φ1) is an integral multiple of the divisibility ofc1(gr(x)) ∈ H2(Y0;Z), and that

gr(y′) = gr(y) − gr(y, y′) = gr(y) − (µ(φ3) − 2nz(φ3)).

�

It remains to show thatJ can be extended toW. Recall thatW = Wα,β,γ ∪ #nb(S

1 × B3). We needto show that there exists an almost complex structure on #n

b(S1 × B3) such that its restriction to

#n(S1 × S2) = ∂(#nb(S

1 × B3)) coincides withJ|#n(S1×S2). Note that [Θ] ∈ HF(−#n(S1 ×S2)) definesthe contact invariant of the standard contact structure on #n(S1 × S2), which is holomorphicallyfillable. Hence the conclusion follows immediately from Theorem 1.1(b). We finish the proof ofCase 1.

CASE 2. SupposeW is given by attaching 1- and 3-handles. By duality, it suffices to considerthe case thatW consists of 1-handle attachments. Let (Σ,α,β, z) be a Heegaard diagram ofY0

and (Σ0,α0,β0, z0) a standard Heegaard diagram of #n(S1 × S2). We obtain a Heegaard diagram(Σ′,α′,β′, z′) = (Σ,α,β, z)#(Σ0,α0,β0, z0) of Y1. There is an associated map between the Hee-gaard Floer homology groups

FW,t : CF(Σ,α,β, z, t|Y0)→ CF(Σ′,α′,β′, z′, t|Y1)

which is induced byFW,t(x) = x ⊗ Θ, wherex ∈ Tα∩Tβ is a generator in the Spinc structuret|Y0, andΘ ∈ CF(#n(S1 × S2)) is the top dimensional generator. Now the existence of an almost complexstructureJ onW with desired properties follows from Theorem 1.1(b) and thefact that the standardcontact structure on #n(S1 × S2) is fillable by (#n

b(S1 × B3), J′) for some almost complex structure

J′. So Case 2 is also proved. �

5. THE INVARIANCE UNDER HEEGAARD MOVES

Our aim for this section is to show that the absolute grading is an invariant of the 3-manifold.That means that if we have two different Heegaard diagrams for the same 3-manifold, then theabsolute grading is preserved under the isomorphism between the Floer homologies defined in [16].It is shown in [16] that any two Heegaard diagrams for the samemanifold differ by a sequence ofHeegaard moves, i.e. isotopies, handleslides, stabilizations and destabilizations. Every Heegaardmove gives rise to a chain map between the Floer complexes, which induces an isomorphism in

26 VINICIUS GRIPP AND YANG HUANG

homology. It is easy to see that these chain maps take homogeneous elements to homogeneouselements. We will show the following theorem.

Theorem 5.1. Let(Σ,α,β, z) be a Heegaard diagram for Y and(Σ′,α′,β′, z′) a Heegaard diagramobtained by a Heegaard move from(Σ,α,β, z). LetΓ : CF(Σ,α,β, z)→ CF(Σ′,α′,β′, z′) be thechain map defined in [16]. Ifx ∈ Tα ∩ Tβ, thengr(x) = gr(Γ(x)).

Remark5.2. Theorem 5.1 gives the invariance we wanted and implies that the following decom-position is independent of the Heegaard diagram.

HF(Y; s) =⊕

ρ∈P(Y,s)

HFρ(Y; s),

To prove Theorem 5.1, we will consider each type of Heegaard move at a time.

5.1. Isotopies. Let (Σ,α,β, z) be a Heegaard diagram forY and letα′ be given by movingα1

to α′1 by a Hamiltonian isotopy without passing throughz. Then there is a continuation mapΓ : CF(Σ,α,β, z) → CF(Σ,α′,β, z) defined by counting Maslov index 0 holomorphic diskswith dynamic boundary conditions, as defined in [16]. If thisisotopy does not create or destroyintersections betweenα andβ curves, then it corresponds to isotoping the Morse functionwithoutintroducing or removing any critical point. In this case it is clear thatΓ is an isomorphism and thatit preserves the absolute grading.

FIGURE 17.

A finger move is a Hamiltonian isotopy that creates a canceling pair of intersections, as shown inFigure 17. We only need to show thatΓ is invariant when the isotopy introduces or eliminates onefinger move and the general isotopy invariance follows from that. First assume thatα′1 is obtainedfrom α1 by introducing one finger move. Letx = (x1, . . . , xg) ∈ Tα ∩ Tβ, wherexi ∈ αi ∩ βσ(i), forsome permutationσ. Thenx1 is moved to a pointx′1 ∈ α′1 ∩ βσ(1). We note thatx′1 is never one ofthe two new intersection points. It is easy to see an index 0 holomorphic disk fromx1 to x′1, whichis actually just a flow line alongβσ(1). So if we takex′ = (x′1, x2, . . . , xg), thenx′ is one of the termsin Γ(x). It is easy to see thatgr(x) = gr(x′). ThereforeΓ preserves the absolute grading. Now weassume thatα′1 is obtained fromα1 by eliminating a finger move. It remains to see what happenswhenx1 is one of the two points that disappears. So we assume thatx1 is one of those two points,such thatx = (x1, . . . , xg) ∈ CF(Σ,α,β, z). If Γ(x) = 0, then there is nothing to prove. AssumethatΓ(x) , 0. So we can take a termx′ in Γ(x). Then since we only isotopedα1, none of the pointsxi, for i > 1, have moved. So we can writex′ = (x′1, x2, . . . , xg), wherex′1 ∈ α′1 ∩ βσ(1). That meansthat there exists a Maslov index 0 holomorphic diskϕ from x1 to x′1. Now undoing this isotopyand introducing the finger move again,x′1 corresponds to an intersectionx′′1 ∈ α1 ∩ βσ(1) and thereis a Maslov index zero holomorphic diskψ from x′1 to x′′1 . We now observe that the compositionϕ ∗ ψ is homotopic to a Whitney disk fromx1 to x′′1 with stationary boundary conditions, i.e. there

AN ABSOLUTE GRADING ON HEEGAARD FLOER HOMOLOGY 27

exists a Whitney disk fromx1 to x′′1 with its boundary mapping toα1 ∪ βσ(1). Therefore there is anindex zero Whitney disk fromx1 to x′′1 . So, since the absolute grading refines the relative gradinginCF(Σ,α,β, z), it follows thatgr(x) = gr(x′′), wherex′′ = (x′′1 , x2, . . . , xg), and hencegr(x) = gr(x′).That implies thatΓ preserves the absolute grading when a finger move is undone.

5.2. Handleslides. Let (Σ,α,β, z) be a Heegaard diagram forY and letβ′1 be the closed curveobtained by handleslidingβ1 over β2. Now we defineβ′ = (β′1, β2, . . . , βg). This handleslidegives rise to a trivial cobordismW = Y × [0, 1], which can also be obtained from the Heegaardtriple diagram (Σ,α,β,β′) by attachingg copies ofS1 × D3, as explained in [16]. LetFW :CF(Σ,α,β, z)→ CF(Σ,α,β′, z) be the induced chain map. Then, it follows from Theorem 1.1(c)that gr(x) ∼W gr(FW(x)). That means that there exists an almost-complex structure J on W suchthat [T(Y × {0}) ∩ J(T(Y × {0}))] = gr(x) and [T(Y × {1}) ∩ J(T(Y × {1}))] = gr(FW(x)). Now letξt = T(Y × {t}) ∩ J(T(Y × {t})), for 0 ≤ t ≤ 1. Under the canonical identificationY ≃ Y × {t}, {ξt}gives a homotopy betweenT(Y × {0}) ∩ J(T(Y × {0})) andT(Y × {1}) ∩ J(T(Y × {1})). Thereforegr(x) = gr(FW(x)).

5.3. Stabilization. Given a Heegaard diagram (Σ,α,β, z) we stabilize it by taking the connectedsum with a two-torus and introducing a new pair ofα andβ curves in this two-torus that intersectat exactly one point. This is equivalent to taking the connect sum ofY with anS3, that is endowedwith the standard genus one Heegaard decomposition. We can write (Σ′,α′,β′, z′) for the Hee-gaard diagram of the stabilization. HereΣ′ = Σ#E, for a two-torusE, α′ = (α1, . . . , αg, αg+1),β′ = (β1, . . . , βg, βg+1) andz′ ∈ Σ′ is naturally associated withz, assuming that the connected sumremoves a ball fromΣ that does not containz. Let w be the unique point inαg+1 ∩ βg+1. It isclear thatΓ : CF(Σ,α,β, z) → CF(Σ′,α′,β′, z′), which takes (x1, . . . , xg) to (x1, . . . , xg,w), is anisomorphism. Is is also shown in [16] that this map gives riseto an isomorphism in homology. Weneed to show that the absolute grading is invariant underΓ. Let x = (x1, . . . , xg) ∈ CF(Σ,α,β, z).In the definition ofgr(x) we modify a gradient-like vector field in neighborhoods of the flow linesγxi andγ0 to get a nonzero vector field. We can write

Y#S3= (Y \ Bε) ∪φ (S3 \ BR),

whereBε is a small ball,BR is a large ball andφ : ∂Bε → ∂BR is a diffeomorphism. We can see thesame neighborhoodsN(γxi ) ⊂ Y andN(γ0) ⊂ Y in Y#S3. Now we take a gradient-like vector fieldv for a Morse function compatible with (Σ′,α′,β, z′). The definition ofgr(Γ(x)) clearly impliesthat the vector fieldwΓ(x) is homotopic towx in Y \ Bε. So it remains to show thatwx andwΓ(x) arealso homotopic inS3 \ BR. We can think ofS3 \ BR as a small ballBδ in R3, wherewx is very closeto being constant with respect to the standard trivialization. We note thatv has only two criticalpoints inBδ. It is easy to homotopewx in a neighborhood ofBδ so that it coincides withv on∂Bδ.It is also easy to see that after we modifyv in N(γxg+1), the vector field we obtain is homotopic towx in Bδ. That concludes the proof of Theorem 5.1.

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UC BERKELEY, BERKELEY, CA 94720, USAE-mail address: vinicius@math.berkeley.edu

UNIVERSITY OF SOUTHERN CALIFORNIA , LOS ANGELES, CA 90089, USAE-mail address: huangyan@usc.edu