aa r X i v : . [ m a t h . G T ] F e b CONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS
C¸ A ˘GRI KARAKURT
Abstract.
In this paper, we show that the Ozsv´ath-Szab´o contact invariant c + ( ξ ) ∈ HF + ( − Y ) of a contact 3-manifold ( Y, ξ ) can be calculated combinatorially if Y is theboundary of a certain type of plumbing X , and ξ is induced by a Stein structure on X .Our technique uses an algorithm of Ozsv´ath and Szab´o to determine the Heegaard-Floerhomology of such 3-manifolds. We discuss two important applications of this technique incontact topology. First, we show that it simplifies the calculation of the Ozsv´ath-Stipsicz-Szab´o obstruction to admitting a planar open book. Then we define a numerical invariantof contact manifolds that respects a partial ordering induced by Stein cobordisms. We doa sample calculation showing that the invariant can get infinitely many distinct values. Introduction
The last decade was the scene of many achievements in contact topology in dimensionthree. In year 2000, in his seminal work [11], Giroux established a one to one correspondencebetween contact structures and open book decompositions of closed oriented 3-manifolds.This allowed Ozsv´ath and Szab´o to find a Heegaard-Floer homology class that reflects cer-tain properties of a given contact structure, [23]. In another direction, based on Giroux’swork, Ozbagci and Etnyre [9] defined an invariant, the support genus, which is the mini-mal page genus of an open book decomposition compatible with a fixed contact structure.Previously, Etnyre [6] had found out that being supported by a genus zero open book putssome restrictions on intersection forms of symplectic fillings of a contact structure. Hisresult was later improved by Ozsv´ath, Stipsicz and Szab´o who showed that the image ofthe Ozsv´ath-Szab´o contact invariant in the reduced version of Heegaard-Floer homology isactually an obstruction to be supported by a planar open book. More precisely, they provedthe following.
Theorem 1.1. (Theorem 1.2 in [18]) Suppose that the contact structure ξ on Y is compatiblewith a planar open book decomposition. Then its contact invariant c + ( ξ ) ∈ HF + ( − Y ) iscontained in U d · HF + ( − Y ) for all d ∈ N .In spite of having useful corollaries, this theorem may not be easy to apply all the timebecause it involves calculation of the group HF + and identification of the contact invariantin this group. The former problem can be solved if we restrict our attention to a certainclass of manifolds. In [19], Ozsv´ath and Szab´o gave a purely combinatorial description ofHeegaard-Floer homology groups HF + of some 3–manifolds which are given as the boundary Date : October 9, 2018.1991
Mathematics Subject Classification. of certain plumbings of disk bundles over sphere. The present work is about pinning downthe contact element within this combinatorial object.To state our main results, we shall assume that G is a negative definite plumbing tree withat most one bad vertex. Let X ( G ) and Y ( G ) be the 4– and 3–manifolds determined by theplumbing diagram respectively. Denote the set of all characteristic co-vectors of the lattice H ( X ( G ) , Z ) by Char( G ). We form the group K + ( G ) = ( Z n ≥ × Char( G )) / ∼ where therelation ∼ is to be described in Section 3. Recall that the Heegaard-Floer homology group HF + of any 3-manifold is equipped with an endomorphism U . In [19] (see also Section 3below), Ozsv´ath and Szab´o established the following isomorphism.(1.1) Hom (cid:18) K + ( G ) Z > × Char( G ) , F (cid:19) ≃ Ker( U ) ⊂ HF + ( − Y ( G ))Recall that if ξ is a contact structure, its Ozsv´ath-Szab´o contact invariant c + ( ξ ) is ahomogeneous element in Ker( U ) ⊂ HF + ( − Y ( G )). It is also known that c + ( ξ ) is non-zeroif ξ is induced by a Stein filling. The following proposition pins down the image of contactinvariant under the above isomorphism. Proposition 1.2.
Let J be a Stein structure on X ( G ) and ξ be the induced contact structureon Y ( G ). Under the identification described in Equation 1.1, the contact invariant c + ( ξ ) isrepresented by the dual of the first Chern class c ( J ) ∈ H ( X, Z ) . Remark 1.3.
This proposition can be generalized in several different directions. First, wemay allow the graph G to have two bad vertices. In this case, the group on the left hand sideof Equation 1.1 gives only even degree elements in Ker( U ) ⊂ HF + ( − Y ( G )). Second, thegraph G which has at most one bad vertex could be semi-definite implying that b ( Y ) = 1,and we use the generalization of the Ozsv´ath-Szab´o algorithm given in [27]. Finally, keeping G negative definite, we may require J to be an ω − tame almost complex structure on X ( G )for some symplectic structure ω which restricts positively on the set of complex tangenciesof Y ( G ) (i.e. ( X ( G ) , ω ) forms a weak filling rather than a Stein filling for the correspondingcontact structure on the boundary).When combined with Theorem 1.1, Proposition 1.2 allows us to determine whether or notcertain contact structures admit planar open books. Recall that the correction term for anyspin c structure t of a rational homology 3-sphere Y is the minimal degree of any non-torsionclass in HF + ( Y, t ) coming from HF ∞ ( Y, t ). Theorem 1.4.
Let J be a Stein structure on X ( G ) and ξ be the induced contact structureon Y ( G ). Denote the correction term of the induced spin c structure t on Y ( G ) by d . Also,let d ( ξ ) be the 3-dimensional invariant of the contact structure ξ . Suppose that we haveeither d ( ξ ) = − d − / HF + d ( − Y ( G ) , t )) > ξ can not be supported by aplanar open book. ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 3
Note that checking the conditions stated in this theorem is simply a combinatorial matter,[18] (see also Section 3 below). Corollary 1.7 of [18], which holds for arbitrary rationalhomology 3-spheres, implies the above statement when d = − d ( ξ ) − /
2. This could betaken as an evidence to conjecture that Theorem 1.4 also holds for every rational homology3-sphere.
Remark 1.5.
There is another version of Ozsv´ath-Szab´o contact invariant c ( ξ ) which livesin d HF ( − Y ) and is related to c + ( ξ ) by ι ( c ( ξ )) = c + ( ξ ) where ι is the natural map ι : d HF ( − Y ) → HF + ( − Y ). The invariant c ( ξ ) can be calculated combinatorially as shown in[25] and [1]. However, for the present applications the usage of the c + is essential.The techniques of this paper can also be used to study a natural partial ordering on contact3 − manifolds up to some equivalence. Following [8] and [12], we write ( Y , ξ ) (cid:22) ( Y , ξ ) ifthere is a Stein cobordism from ( Y , ξ ) to ( Y , ξ ). Moreover, we write ( Y , ξ ) ∼ ( Y , ξ ) ifthese contact manifolds satisfy ( Y , ξ ) (cid:22) ( Y , ξ ) and conversely ( Y , ξ ) (cid:22) ( Y , ξ ). Clearly, ∼ defines an equivalence relation on the set of contact manifolds and (cid:22) is a partial orderingon the equivalence classes. One can define a numerical invariant of contact manifolds thatrespects this partial ordering. Namely, if we let σ ( Y, ξ ) = − max (cid:8) d : c + ( ξ ) ∈ U d · HF + ( − Y ) (cid:9) the naturality properties of the Ozv´ath-Szab´o contact invariant (c.f. Section 2 below) implythat we have σ ( Y , ξ ) ≤ σ ( Y , ξ ) whenever ( Y , ξ ) (cid:22) ( Y , ξ ). Note that σ invariant can beinfinite. In fact, σ ( Y, ξ ) = −∞ if ( Y, ξ ) admits a planar open book by Theorem 1.1. Clearly,if two contact manifolds have different σ -invariants, they lie in different equivalence classes.The following theorem tells that there are infinitely many such equivalence classes. Theorem 1.6.
Any negative integer can be realized as the σ invariant of a contact manifold.In fact, we are going to obtain some contact manifolds with distinct σ invariants by doingLegendrian surgery on certain stabilizations of some torus knots. See Theorem 7.1 below.After completing the first draft of this paper, the author found an explicit formula for the σ invariant of a contact manifold that is obtained by Legendrian surgery from 3-sphere ifthe knot has L –space surgery [14]. The formula depends only the Alexander polynomial,Thurston–Bennequin number and the rotation number of the surgery knot and it generalizesTheorem 7.1. The technique, however, is quite different than the one used here. Remark 1.7.
Recently Latschev and Wendl defined an analogous invariant of contact man-ifolds, which they call algebraic torsion , in arbitrary odd dimension within the context ofSymplectic Field Theory, [15]. In dimension 3, both invariants provide obstructions to exactsymplectic cobordisms, so one may wonder if these two are somehow related. So far, we cannot see an obvious relation, because Theorem 1.1 in [15] says that contact manifolds withalgebraic torsion are not strongly fillable whereas our examples with finite σ invariant areall Stein fillable. C¸ A ˘GRI KARAKURT
This paper is organized as follows. In Section 2, basic properties of Heegaard-Floer ho-mology and contact invariant are briefly reviewed. Section 3 is devoted to the algorithm ofOzsv´ath and Szab´o to determine the generators of Heegaard-Floer homology of 3-manifoldsgiven by plumbing diagrams. Remarks given at the end of the section allow us to find rela-tions easily by combinatorial means. We prove Proposition 1.2 and Theorem 1.4 in Section4. Some examples are given in Section 5. We discuss the planar obstruction in Section 6.Theorem 1.6 is proved in Section 7I would like thank my advisor Selman Akbulut for his patience and constant encourage-ment. I am grateful to Tolga Etg¨u, Matt Hedden and Yankı Lekili for helpful conversations.A special thanks goes to Burak Ozbagci for his careful revision of the first draft of this paper.This work is supported by a Simons postdoctoral fellowship.2.
Heegaard-Floer homology and contact invariant
Let Y be a closed oriented 3-manifold and t be a spin c structure on Y . In [21] and[22], Ozsv´ath and Szab´o define four versions of Heegaard-Floer homology groups d HF ( Y, t ), HF + ( Y, t ), HF − ( Y, t ),and HF ∞ ( Y, t ). These groups are all smooth invariants of ( Y, t ).When Y is a rational homology sphere, they admit absolute Q -gradings. The groups HF + , HF − , and HF ∞ are also Z [ U ] modules where multiplication by U decreases degree by2. Any spin c cobordism ( X, s ) between ( Y , t ) and ( Y , t ) induces a homomorphism welldefined up to sign F ◦ X, s : HF ◦ ( Y , t ) → HF ◦ ( Y , t )Here HF ◦ stands for any one of d HF , HF + , HF − , or HF ∞ . We work with F = Z / Z coefficients in order to avoid sign ambiguities. Also, we drop the spin c structure from thenotation when we direct sum over all spin c structures.Given any contact structure ξ on Y , Ozsv´ath and Szab´o associate an element c ( ξ ) ∈ d HF ( − Y ) which is an invariant of isotopy class of ξ [23]. In this paper we are interested inthe image c + ( ξ ) of c ( ξ ) in HF + ( Y ) under the natural map. We list some of the propertiesof this element below.(1) c + ( ξ ) lies in the summand HF + ( − Y, t ) where t is the spin c structure induced by ξ .(2) c + ( ξ ) = 0 if ξ is overtwisted.(3) c + ( ξ ) = 0 if ξ is Stein fillable.(4) c + ( ξ ) ∈ Ker( U ).(5) c + ( ξ ) is homogeneous. When Y is a rational homology sphere, it has degree − d ( ξ ) − /
2, where d ( ξ ) is the 3-dimensional invariant of ξ .(6) c + ( ξ ) is natural under Stein cobordisms: If W is a compact Stein manifold, ∂W = Y ′ ∪ − Y , and ξ ′ and ξ are the induced contact structures, we can regard W as acobordism from − Y ′ to − Y and the induced map satisfies F + W ( c + ( ξ ′ )) = c + ( ξ ). ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 5
The contact invariant c + ( ξ ) is studied by Plamenevskaya in [24]. The following result isto be used later in this paper when we prove our main theorem. We state it in a slightlymore general form than in [24] but Plamanevskaya’s proof is valid for our case as well. Theorem 2.1. (Theorem 4 in [24]) Let X be a smooth compact 4-manifold with boundary Y = ∂X . Let J be a Stein structure on X that induces a spin c structure s on X andcontact structure ξ on Y . Let s be another spin c structure on X that does not necessarilycome from a Stein structure. Suppose that s | Y = s | Y , but the spin c structures s , s arenot isomorphic. We puncture X and regard it as a cobordism from Y to S . Then(1) F + X, s ( c + ( ξ )) = 0(2) F + X, s ( c + ( ξ )) is a generator of HF +0 ( S ).Note that in this theorem if the spin c structures s | Y and s | Y are not the same then con-clusion (1) follows trivially. Remark 2.2.
Theorem 2.1 was later generalized by Ghiggini in [13] where he requires J to be only an ω -tame almost complex structure for some symplectic structure ω on X ( G )that gives a strong filling for the boundary contact structure. In this paper, we work withrational homology spheres. For these manifolds, any weak filling can be perturbed into astrong filling, [17]. 3. The Algorithm
In this section, we review Ozsv´ath–Szab´o’s combinatorial description of Heegaard Floerhomology of plumbed 3–manifolds given in [19] to set our notation. Proof of our maintheorem heavily relies on the understanding the algebraic structure of their combinatorialdescription. Particularly one should understand the U –action in this combinatorial object.We shall describe this action in Equation 3.2. Strictly speaking, Ozsv´ath and Szab´o’s algo-rithm determines only the part of the Heegaard-Floer homology group that lies in the kernelof U map. In order to determine the full group, one should find all the minimal relations.Although these relations can be found in some special cases, no general technique is knownto find all of these relations. Towards the end of the section we discuss a systematic methodto find some (not necessarily minimal) relations. These relations will turn out to be minimalin the cases of interest (Example 6.4).Let G be a weighted graph. For every vertex v of G , let m ( v ) and d ( v ) denote the weightof v and the number of edges connected to v respectively. A vertex v is said to be a badvertex if m ( v ) + d ( v ) >
0. Enumerating all vertices of G , one can form the intersectionmatrix whose i th diagonal entry is m ( v i ) and i − j th entry is 1 if there is an edge between v i and v j , and is 0 otherwise. Throughout, we assume that G satisfies the following conditions.(1) G is a connected tree.(2) The intersection matrix of G is negative definite.(3) G has at most one bad vertex. C¸ A ˘GRI KARAKURT
There is a 4-manifold X ( G ) obtained by plumbing together disk bundles D i , i = 1 · · · | G | over sphere where D i is plumbed to D j whenever there is an edge connecting v i to v j . Let Y ( G ) be the boundary of X ( G ). In [19], Ozsv´ath and Szab´o give a purely combinatorialdescription of Heegaard-Floer homology group HF + ( − Y ( G )). From now on, we identifyspin c structures on 4–manifolds with their first Chern classes. Since all our 4–manifoldsare simply connected and have non-empty boundary, this does not cause any ambiguity.However, we should be careful in the 3–manifold level when 2–torsion exists in the firsthomology. We will deal with such an example in Section 6(see Remark 6.5).The second homology H ( X ( G ) , Z ) is a free module generated by vertices of G . LetChar( G ) be the set all characteristic(co)vectors of this module, i.e. every element K ofChar( G ) satisfies h K, v i = m ( v ) (Mod 2) for every vertex v . Let T + be the graded alge-bra F [ U, U − ] /U F [ U ] where the formal variable U has degree −
2. Form the set H + ( G ) ⊂ Hom(Char( G ) , T + ) where any element φ of H + ( G ) satisfies the following property; If K isa characteristic vector, v is a vertex, and n is an integer such that h K, v i + m ( v ) = 2 n, we have U m + n φ ( K + 2PD( v )) = U m φ ( K ) if n > , or U m φ ( K + 2PD( v )) = U m − n φ ( K ) if n < . The set of spin c structures on Y ( G ) gives rise to a natural splitting for H + ( G ). For, if t is a spin c structure on Y ( G ), one can consider the subset Char t ( Y ( G )) consisting of thosecharacteristic vectors whose restriction on Y ( G ) are t . The set H + ( G, t ) is the set of allmaps in H + ( G ) with support Char t . It is easy to see that H + ( G ) = L t H + ( G, t ) .The group H + ( G ) is graded in the following way. An element φ ∈ H + ( G ) is said to behomogeneous of degree d if for every characteristic vector K with φ ( K ) = 0, φ ( K ) ∈ T + isa homogeneous element with deg( φ ( K )) − K + | G | d. We are ready to describe the isomorphism relating H + ( G ) to the Heegaard-Floer homologyof Y ( G ). Fix a spin c structure t on − Y ( G ). Let K be a characteristic vector on Char t ( G ).Puncture X ( G ) and regard it as a cobordism form − Y ( G ) to S . It is known that X ( G )and K induce a homomorphism F X ( G ) ,K : HF + ( − Y ( G ) , t ) → HF + ( S ) ≃ T + . Now the map T + : HF + ( − Y ( G ) , t ) → H + ( G, t ) is defined by the rule T + ( ξ )( K ) = F X ( G ) ,K ( ξ ) Theorem 3.1. (Theorem 2.1 in [19]) T + is a U -equivariant isomorphism preserving theabsolute Q -grading. ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 7
To simplify the calculations, we work with the dual of H + ( G ). Let K + be the quotient set Z ≥ × Char( G ) / ∼ , where the equivalence relation ∼ is defined as follows. Denote a typicalelement of Z ≥ × Char( G ) by U m ⊗ K (We drop U m ⊗ from our notation if m = 0). Let v be a vertex and n be an integer such that2 n = h K, v i + m ( v )then we have U m + n ⊗ ( K + 2PD( v )) ∼ U m ⊗ K if n ≥ U m ⊗ ( K + 2PD( v ) ∼ U m − n ⊗ K if n < . Define a pairing K + ( G ) × H + ( G ) → Z by ( φ, U m ⊗ K ) → ( U m φ ( K )) where () denotesthe projection to the degree 0 subspace of T + . It is possible to show that this pairing iswell defined and non-degenerate and hence it defines an isomorphism between H + ( G ) andHom( K + ( G ) , Z ). Using the duality map and isomorphism T + one can identify ker U n +1 ⊂ HF + ( − Y ( G )) as a quotient of K + ( G ) for every n ≥ Lemma 3.2. (Lemma 2.3 in [19]) Let B n denote the set of characteristic vectors B n = { K ∈ Char(G) : ∀ v ∈ G, |h K, v i| ≤ − m ( v ) + 2 n } . The quotient map induces a surjection from n [ i =0 U i ⊗ B n − i onto the quotient space K + ( G ) Z >n × Char( G ) . In turn, we have an identification(3.1) Hom (cid:18) K + ( G ) Z >n × Char(G) , F (cid:19) ≃ ker U n +1 ⊂ H + ( G ) . One should regard the above isomorphism as one between F [ U ] modules where the U action on the left hand side of equation 3.1 is defined by the following relation.(3.2) U. ( U p ⊗ K ) ∗ ( U r ⊗ K ′ ) = U p ⊗ K ∼ U r +1 ⊗ K ′ U p ⊗ K U r +1 ⊗ K ′ Where ( U p ⊗ K ) ∗ denotes the dual of U p ⊗ K .Lemma 3.2 gives us a finite model for ker U n +1 for every n ≥
0. It is known that thesegroups stabilize to give HF + . Therefore, one can understand HF + by studying the quotients K + ( G ) / Z ≥ n × Char( G ) for all n ≥
0. The first quotient is well understood thanks to analgorithm of Ozsv´ath and Szab´o . Below, we describe the algorithm and discuss a possibleextension.
C¸ A ˘GRI KARAKURT
A characteristic vector K is called an initial vector if for every vertex v , we have(3.3) m ( v ) + 2 ≤ h K, v i ≤ − m ( v )Start with an initial vector K . Form a sequence ( K , K , · · · , K n ) of characteristic vectorsas follows: K i +1 is obtained from K i by adding 2PD( v ) where v is a vertex with h K i , v i = − m ( v ). The terminal vector K n satisfies one of the following.(1) m ( v ) ≤ h K n , v i ≤ − m ( v ) − v .(2) h K n , v i > − m ( v ) for some v .The sequence ( K , K , · · · , K n ) is called a full path , and characteristic vector K n is calledthe terminal vector of the full path. We say that a full path is called good if its terminalvector satisfies property (1) above and it is bad if the terminal vector satisfies (2). We listsome of the properties of full paths, the reader can consult [19](especially proposition 3.1 in[19]) for proofs. • Two characteristic vectors in B are equivalent in K + ( G ) if and only if there is a fullpath containing both of them where the set B is defined as in lemma 3.2. • If an initial vector K has a good full path then any other full path starting with K is good. • If K and K ′ are initial vectors having good full paths and K = K ′ then K K ′ in K + ( G ). • A terminal vector K n of a bad full path is equivalent to U m ⊗ K ′ in K + ( G ) for some m > K ′ ∈ Char( G ). A terminal vector of good full path can not be equivalentto such an element of H + ( G ).Note that these properties allow us to find the generators of ker U ; They are simply theinitial vectors having good full paths. In other words, we know the generators of the lowestgrade subgroup of HF + ( − Y ( G )). Recall from [20] that the lowest degree d ( Y, t ) of non-torsion elements in HF + ( Y, t ) is called the correction term for a spin c manifold ( Y, t ). Thealgorithm above provides us an efficient method to calculate the correction term d ( − Y ( G ) , t )for any spin c structure t (see Corollary 1.5 of [19])(3.4) d ( − Y ( G ) , t ) = min − K + | G | c structure t .The whole group H + ( G ) ≃ HF + ( − Y ( G )) is determined by the relations amongst thegenerators of Ker( U ). Given two characteristic vectors K i , K j admitting good full pathsand inducing the same spin c structure on Y ( G ), a relation between K and K is a pairof integers ( n, m ) satisfying U n ⊗ K ∼ U m ⊗ K . If the non negative integers ( n, m ) areminimal with that property, we call the corresponding relation minimal . Here we describea systematic method to find relations. Say K is a characteristic vector and n is a positive ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 9 integer. We want to understand the equivalence class in K + ( G ) containing U n ⊗ K . Wedefine three operations that do not change this equivalence class. (R1) U n ⊗ K ′ → U n ⊗ K where K ′ is obtained from K by applying the algorithm to findfull paths. (R2) U n ⊗ K → U n − ⊗ ( K + 2PD( v )) where v is a vertex with h K, v i + m ( v ) = − (R3) U n ⊗ K → U n +1 ⊗ ( K + 2PD( v )) where v is a vertex with h K n , v i + m ( v ) = 2Now assume that K is a characteristic initial vector which admits a good full path. In orderto find particular representatives with small U -depths for the equivalence class containing U n ⊗ K we apply R1 then apply R2 if possible else R3 . Then we repeat the same proceduretill it terminates at an element U r ⊗ K ′ . We call the vector part of this element as a rootvector (the exponent m is determined by n and degrees of K and K ′ ). A root vector isnot unique, it depends upon choices we made along the way; like the choice of the vertexat which we apply R2 or R3 is applied. However, the set of root vectors is a finite setwhich can be found easily and it can be used to establish relations amongst the generatorsof Ker( U ). This simple observation will be useful when we do our calculations. Proposition 3.3.
Let K and K be two characteristic initial vectors admitting good fullpaths. Suppose n and m are non-negative integers such that the root vector sets of U n ⊗ K and U m ⊗ K intersect non trivially. Then we have U n ⊗ K ∼ U m ⊗ K . Proof.
Follows from the definitions. (cid:3) Main theorem
Proof of Proposition 1.2.
Let s be the canonical spin c structure and s ′ be anyother spin c structure on X ( G ). Note that c ( s ) c ( s ′ ). Recall that the isomorphismKer( U ) ≃ Hom (cid:18) K + ( G ) Z > × Char( G ) , F (cid:19) is given by means of the pairing P : Ker( U ) × Hom (cid:18) K + ( G ) Z > × Char( G ) , F (cid:19) → F P ( a, L ) = ( F + X ( G ) ,L ( a )) In view of this observation, it is enough to show the following two equations hold.( F + X ( G ) , s ( c ( ξ ))) = 1(4.1) ( F + X ( G ) , s ′ ( c ( ξ )) = 0(4.2)These are simply the conclusions of Theorem 2.1. ✷ Proof of Theorem 1.4.
Let K = c ( J ). By Theorem 1.1 and Proposition 1.2, itis enough to show that K ∗ / ∈ Im( U k ) for some k ∈ N . To do that we will use the theidentification in Equation 3.1, keeping in mind that the U action is determined by Equation3.2. Let { K , K , · · · , K r } be the set of characteristic initial vectors admitting good paths such that deg( K ∗ i ) ≤ deg( K ∗ ) = − d ( ξ ) − / K i | Y ( G ) = t for all i = 1 · · · r . Basicproperties of the contact invariant imply that this set is not empty if one of the assumptionsis satisfied. It is known that on any rational homology sphere and for any spin c structure,the Heegaard-Floer homology decomposes as HF + = T + ⊕ HF red . This decomposition tellsthat in large even degrees the Heegaard-Floer homology is generated by a single element.So, one can find integers n , n , · · · , n r such that U n ⊗ K ∼ U n ⊗ K ∼ · · · ∼ U n r ⊗ K r . Moreover, by choosing these numbers large enough, we can guarantee that the dual of U n ⊗ K is the unique generator of the degree − d ( ξ ) − / n subspace of HF + ( − Y ).Then by Equation 3.2, U n ( U n ⊗ K ) ∗ = K ∗ + ( U n − n ⊗ K ) ∗ + · · · + ( U n r − n ⊗ K r ) ∗ . Therefore
K / ∈ Im( U n ). ✷ Examples
In this section, we shall discuss two examples. These examples have no particular impor-tance on their, own but they are simple enough to give a clear explanation of the ideas usedin this paper.
Example 5.1.
Let G be the graph indicated in Figure 1. Index the vertices so that thecentral one comes first. Our aim is to find all the characteristic co–vectors in the intersectionlattice of X ( G ) which admit good full path. We will denote each K ∈ H ( X ( G )) as a rowvector [ h K, v i , · · · , h K, v i ], where v i is the homology class of the sphere corresponding tothe i th vertex for all i = 1 , · · · ,
4. If K is characteristic and satisfies Inequality 3.3 then h K, v i i = 0 , or 2 for every i . So we need to find out which of the possible 16 co-vectorsadmit good full paths. To represent full paths, we indicate the index of the vertex whosetwice Poincare dual is added to the characteristic vector. The algorithm terminates at thevery first step for K = [0 , , , K = [0 , , , , , , , ,
2. By symmetry, K = [0 , , ,
0] and K = [0 , , ,
2] also admit good full path.For [2 , , , , , , h K, v i i = 2 for more than one i values then K admits a bad full path. Therefore K , · · · , K are the only characteristic co–vectors admitting good full path. -2 -2-2-2 Figure 1.
Next we claim that each one of K , · · · , K restricts to a different spin c structure t , · · · , t on the boundary. One way of seeing this is to apply the criterion mentioned in Remark 6.5.Another way is the following: Recall that the set of spin c structures on any 3–manifold can ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 11 be identified with its first homology. In this case the first homology of Y ( G ) is given by Z / Im I ( G ), where I ( G ) is the intersection matrix. Observe that det( I ( G )) = 4, so − Y ( G )has 4 spin c structures. Each one of these spin c structures are torsion so the Heegaard-Floer homology of − Y ( G ) is non-trivial in the corresponding component. Since we haveexactly 4 co–vectors contributing the Heegaard-Floer homology they must lie in differentspin c components. This shows − Y ( G ) (and hence Y ( G )) is an L –space (i.e. its Heegaard-Floer homology is the same as a Lens space).Let us calculate degree of each K i . In the formula deg( K ) = ( K + | G | ) /
4, the inverseof the intersection matrix should be used when squaring K . We see that deg( K ) = 1and deg( K j ) = 0 for j = 2 , ,
4. Since the isomorphism given in Equation 3.1 is given interms of dual co–vectors, we should take the negative of the degrees when we think of K i ’sas elements of the Heegaard-Floer homology. As a result, HF + ( − Y ( G ) , t ) = T +( − and HF + ( − Y ( G ) , t i ) = T +(0) , for i = 2 , , X ( G ) withthe obvious Stein structure J : First make the attaching circles of handles corresponding tothe vertices Legendrian unknot with tb = −
1, see Figure 2. Since each handle is attachedwith framing tb −
1, the unique Stein structure on the 4–ball extends across these handles,[4]. To identify the contact invariant, we need to determine the Chern class of J . The valueof c ( J )( v i ) is given by the rotation number of the corresponding Legendrian unkot. In thiscase the rotation numbers are all 0, so c ( J ) = K . Hence the Ozsv´ath-Szab´o invariant ofthe induced contact structure is the unique generator of HF + ( − Y ( G ) , t ) in in degree − U k for every k , so we do not get any obstructionto planarity. Figure 2.
Example 5.2.
This example is a follow up of the calculation of the Heegaard Floer homologyof the Brieskorn sphere Σ(3 , ,
7) given in [19]. This 3–manifold is given by the plumbinggraph G which we indicate in Figure 3. We order the vertices so that the central node comesfirst, the − K = (0 , − , , , , , , , , , , K = (0 , , , , , , , , , , , K = (0 , , , , , , , , , , , − K = (0 , , , , , − , , , , , , We have deg( K ) = deg( K ) = 0, and deg( K ) = K = 2. So the correction term for theunique spin c structure is −
2. Next we consider the Stein structures on X ( G ). We makethe each unknot Legendrian as before, but this time we should do a stabilization for the − tb −
1. Depending on how we do the stabilization, weobtain two Stein structures J , J whose Chern classes are given by K and K . Let ξ and ξ respectively denote the induced contact structures on the boundary. Since − deg( K i ) isnot minimal, neither contact structure is compatible with a planar open book by Theorem1.4. This also can be seen by using simple criteria found by Ozsv´ath,Stipsicz and Szab´o, seeTheorem 6.2 and 6.3.Finally, we would like to show why c + ( ξ i ) is not in the image of U , for i = 1 ,
2. ByTheorem 1.2 the contact invariant c + ( ξ i ) is represented by the dual K ∗ i . It was shown in[19] that the minimal relations are given as follows U ⊗ K ∼ U ⊗ K U ⊗ K ∼ U ⊗ K ∼ U ⊗ K Therefore ( U ⊗ K ) ∗ is the unique generator of degree 2 and U ( U ⊗ K ) ∗ = ( U ⊗ K ) ∗ + K ∗ + K ∗ , by Equation 3.2. So neither K ∗ nor K ∗ is in the image of U . -3 Figure 3.
Plumbing graph for Σ(3 , , − Planar Obstruction
In this section, we shall illustrate an application Theorem 1.4 and show that certain Steinfillable contact structures do not admit planar open books. Obstructions to being supportedby planar open books were known to exist before. Some of these obstructions can be checkedby using simple criteria. The importance of our examples is that no other simple criterion issufficient to prove their non-planarity. Before discussing our examples we shall give a briefexposition on what is known about obstruction to planarity.The first known obstruction to planarity was found by Etnyre. It puts some restrictionson intersection forms of symplectic fillings of planar open books.
Theorem 6.1. (Theorem 4.1 in [6]) If X is a symplectic filling of a contact 3-manifold( Y, ξ ) which is compatible with a planar open book decomposition then b ( X ) = b ( X ) = 0,the boundary of X is connected and the intersection form Q X embeds into a diagonalizablematrix over integers.Ozsv´ath Szab´o and Stipsicz found another obstruction in [18]. Their obstruction is a conse-quence of Theorem 1.1 above though its statement has no reference to Floer homology. ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 13
Theorem 6.2. (Corollary 1.5 of [18]) Suppose that the contact 3-manifold (
Y, ξ ) with c ( s ( ξ )) = 0 admits a Stein filling ( X, J ) such that c ( X, J ) = 0. Then ξ is not supportedby a planar open book decomposition.Yet another criterion is stated in [18]. It partially implies Theorem 1.4 above. Theorem 6.3. (Corollary 1.7 of [18]) Suppose that Y is a rational homology 3-sphere. Thenumber of homotopy classes of 2-plane fields which admit contact structures which are bothsymplectically fillable and compatible with planar open book decompositions is boundedabove by the number of elements in H ( Y ; Z ). More precisely, each spin c structure s isrepresented by at most one such 2 − plane field, and moreover, the Hopf invariant of thecorresponding 2 − plane field must coincide with the correction term d ( − Y, s ).Below, we give examples of non-planar Stein fillable contact structures on a Seifert fiberedspace. Non-planarity of some of our examples do not follow from Theorem 6.1, Theorem 6.2or Theorem 6.3.
Example 6.4.
Consider the star shaped plumbing graph consisting of eight vertices wherethe central vertex has weight −
4, a neighboring vertex has weight − − Y is the Seifert fibered space M ( − , , · · · , | {z } ,
13 ). The reason why we have so many self intersection − L -spaces where Theorem 1.1 does not provide an obstruction to admit-ting a planar open book. For the topological characterization of L -spaces among Seifertfibered spaces see [16]. It can be shown that the corresponding intersection form is negativedefinite,and has determinant 128. Moreover it can be embedded into a symmetric matrixwhich is diagonalizable over integers. To see this, index the vertices so that the central onecomes first and the weight − e , e , · · · , e be a basis for R suchthat e i · e i = − i = 1 , , · · · ,
11. The embedding is defined by the following set ofequations v → − e − e − e − e v → e − e v → e + e v → e − e v → e + e v → e − e v → e + e v → e + e + e First, we calculate HF + ( − Y, t ) for every spin c structure t . For similar calculations, see[3], [26], [27], and Section 3.2 of [19]. As before, we write any characteristic vector K in theform K = [ h K, v i , · · · , h K, v i ]. There are 768 characteristic vectors satisfying equality 3.3,and 138 of them have good full paths. When we distribute these to spin c structures of Y , -4 -3 Figure 4.
Plumbing description of Y. All unmarked vertices have weight -2.we see that for 10 spin c structures Ker( U ) has rank 2, and the rank is 1 for the rest. Table1 shows HF + for these 10 spin c structures. Remark 6.5.
As pointed out in [19], the set of spin c structures on Y can be identified with2 H ( X ( G ) , ∂X ( G )) orbits in Char( G ). Therefore two characteristic vectors K , K , inducethe same spin c structure on the boundary, if and only if all the entries of (1 / I ( G ) − ( K − K ) are integer where I ( G ) is the intersection matrix.Next, we consider the obvious Stein structures that arise from the handlebody diagramassociated to G . Following Eliashberg, we isotope the attaching circles of 2-handles intoLegendrian position so that their framing become one less than the Thurston-Bennequinframing. For − v and v take Legendrian isotopes with rotation numbers i and j respectively where i = − , ,
2, and j = − ,
1. Call the resulting Stein structure as J i,j andthe induced contact structure by ξ i,j , see figure 5 for a picture of J , − . Note that the firstChern class of J i,j is given by the characteristic vector K i,j = [ i, , , , , , , j ]. It is easyto verify that d ( ξ i,j ) + 1 / K i,j + | G | ) / K i,j ). According to Theorem 1.4, thecontact structures ξ ± , ± do not admit planar open books. By the algorithm given in [10]these contact structures do admit genus one open books, so their support genera are all one.One can not use Theorem 6.2 directly to get this conclusion because the Chern classes ofthe corresponding spin c structures are all of order 4. Though Theorem 6.3 also implies ourconclusion for ξ , and ξ − , − , it doesn’t apply to ξ , − or ξ − , . So the latter two are thecontact structures we promised at the beginning of the example. Figure 5.
Legendrian handlebody diagram giving J , − . The curve on leftcorresponds v and the other represents v . They are both oriented counterclockwise. We omit the other unknots linking to v in order not to complicatethe picture. Remark 6.6.
The main result of [9] implies that the support genera of plumbings withat most two bad vertices are at most one. On the other hand the algorithm of Ozsv´ath
ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 15 and Szab´o does not work if the number of bad vertices is greater than two. Therefore, thetechniques used in this paper do not seem to be sufficient to find an example of a contactstructure with support genus strictly greater than one. We are planning to turn this problemin a future project using a different approach.7.
Calculation of σ In this section we shall prove Theorem 1.6 by calculating explicitly the σ invariant of afamily of contact 3-manifolds. Our argument is based on a previous work of Rustamov, [26].For every positive integer n , consider the contact manifold ( Y n , ξ n ) obtained from ( S , ξ std )by doing Legendrian surgery on the (2 , n + 1) torus knot L n stabilized 2 n − L n is zero so the topologicalsurgery coefficient is negative one. In fact, the 3-manifold Y n is the Brieskorn homologysphere Σ(2 , n + 1 , n + 3). nn − Figure 6. K n Theorem 7.1. σ ( Y n , ξ n ) = − ( p n −
1) where p n is the n th element of the sequence1 , , , , , , · · · . Clearly, Theorem 7.1 implies Theorem 1.6. Another immediate application of Theorem7.1 is that ( Y n , ξ n ) can not be supported by a planar open book. This was first pointed outin [18]. Finally, combining this theorem with the fact that σ invariant respects the partialordering coming from Stein cobordisms we have the following corollary. Corollary 7.2.
There is no Stein cobordism from ( Y n , ξ n ) to ( Y m , ξ m ) if n > m + 1. Inparticular, one can not obtain ( Y m , ξ m ) from ( Y n , ξ n ) via Thurston-Bennequin minus one( tb −
1) surgery on a Legendrian link.The above corollary should be compared to a classical result of Ding and Geiges in [2]where it was proved that any two contact manifolds can be obtained from each other viaa sequence of ( tb −
1) or ( tb + 1) contact surgeries. In fact, one can always choose such asequence which contains at most one tb + 1 surgery. Therefore, the corollary tells us thatthe existence of ( tb + 1) surgery is essential even though the contact manifolds in both endsare Stein fillable. Proof of Theorem 7.1.
Let V be the 4-manifold obtained by attaching a Weinstein2-handle to a 4-ball along L n . Eliashberg’s theorem [4] tells that V admits a Stein structure.Let s be the canonical spin c structure on V , and denote the homology class determined bythe 2-handle (for some orientation of L n ) by h . The way we stabilize L n ensures that.(7.1) c ( s )( h ) = rot( L n ) = ± L n ) stands for the rotation number of L n . Note that the sign of the rotationnumber depends on how we orient L n . Next, V is blown-up n + 2 times, and we do thehandleslides indicated in figure 8. We see that the resulting 4-manifold is given by theplumbing graph G in figure 7. The manifold X ( G ) is no longer Stein but it does admit asymplectic structure. Let s ′ be the canonical spin c structure on this symplectic manifold.Let e i denote the homology class of the i th exceptional sphere. We have(7.2) c ( s ′ )( e i ) = 1 i = 1 , , · · · , n + 2 . n-1 − − − n − − − − − Figure 7.
A Plumbing graph for Brieskorn homology sphere Σ(2 , n + 1 , n + 3)Order the vertices of G so that first four are the ones with weight − , − , − − n − −
2s on right are ordered accordingto the distance from the root starting with the closest one. In [26], Rustamov proves that HF + ( − Y n ) = T +0 ⊕ F p n (0) ⊕ n − M i =1 ( F p i q n − i ⊕ F p i q n − i )where q i = i ( i + 1) and F r ( k ) = F [ U ] /U r F [ U ] and U r − lies in degree k . More precisely, heshows that Ker U ⊂ HF + ( − Y n ) is generated by the characteristic vectors K i = (1 , , − , − n − i, , , · · · , , i = 1 , , · · · , n. He also proves that the minimal relations are given as follows: U p i ⊗ K i ∼ U p i + q n − i ⊗ K n +1 (7.3) U p i ⊗ K n + i ∼ U p i + q n − i ⊗ K n (7.4)where i = 1 , , · · · , n . Note that the characteristic vectors K n and K n +1 are in the bottomlevel, and their degree is zero. ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 17
Our aim is to pin down the contact invariant c + ( ξ n ) in HF + ( − Y n ). Note that Proposition1.2 in the stated form can not be applied directly as it concerns Stein fillings of plumbed man-ifolds. However, as indicated in Remark 1.3 it is also true for strong symplectic fillings. Theonly difference in the proof is that one uses Ghiggini’s generalization [13] of Plamenevskaya’stheorem [24]. Alternatively, one can use the blow-up formula and handleslide invariance forthis particular case to see that equations 4.1 and 4.2 hold. In any case, we see that thecontact invariant c + ( ξ n ) is represented by the first Chern class c ( s ′ ) of the canonical spin c structure. In figure 8, we keep track of the homology classes in order to pin down the firstChern class. By Equations 7.1 and 7.2, we have c ( s ′ ) = K n or K n +1 depending on theorientation of L n , but the contact invariant is in the image of U p n − in any case. ✷ References [1] J. Baldwin and O. Plamenevskaya.
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ONTACT STRUCTURES ON PLUMBED 3-MANIFOLDS 19 n + 1 h h − e − e − · · · − e n e e e n h − e − e − · · · − e n e e − e e − e e n − e n − h − e − e − · · · − e n e e − e e − e e n − e n − h − e − e − · · · − e n − e n +1 − e n +2 e − e n +1 − e n +2 e − e e − e e n − e n − e − e e n +1 e n +2 e − e n +1 − e n +2 e − e e − e e n − e n − e − e e n +1 e n +2 − e n +1 Blow-up IsotopyBlow-upHandleslide h − e − e − · · · − e n − e n +1 − e n +2 Handleslides e − e Figure 8.
Sequence of Blow-ups from K n to plumbing Spin c Characteristic Vectors Degree Relation HF + ( − Y )1 [2 , , , , , , , −
1] 7 / U ⊗ K = U ⊗ K T − ⊕ F − [ − , , , , , , ,
3] 7 /
82 [ − , , , , , , ,
1] 7 / U ⊗ K = U ⊗ K T − ⊕ F − [0 , , , , , , ,
3] 7 /
83 [ − , , , , , , , − − / U ⊗ K = U ⊗ K T − ⊕ F [0 , , , , , , ,
1] 15 /
84 [2 , , , , , , , − / U ⊗ K = U ⊗ K T − ⊕ F [0 , , , , , , , −
1] 15 /
85 + j [ − , , · · · , |{z} j +2 , · · · , , −
1] 3/4 U ⊗ K = U ⊗ K T − ⊕ F − [0 , , · · · , |{z} j +2 , · · · , ,
1] 3/4 j = 0 , · · · , c Root Vectors1 [2 , , · · · , , − |{z} i , , · · · , , − i = 2 , · · · ,
72 [0 , , , , , , , − , [0 , , · · · , , − |{z} i , , · · · , , ,i = 2 , · · · ,
73 [0 , , · · · , , − |{z} i , , · · · , , − ,i = 2 , · · · ,
74 [ − , , , , , , , − j [2 , , · · · , − |{z} i , · · · , − |{z} j +2 , · · · , , − i = 2 , · · · , j = 0 , · · · , Table 1. HF + of M ( − , , · · · , ,
13 ) for 10 spin cc