We give examples of contact structures which admit exact symplectic fillings,
but no Stein fillings, answering a question of Ghiggini.Comment: 6 pages; Erroneous Lemma 2.7 removed and Section 2 shortened
  significantly; updated references and other minor edits (to appear in Algebr.
  Geom. Topol.

Bowden, J.

English

arXiv

MPG.PuRe

arXiv:1201.6550v3  [math.GT]  5 Jul 2012EXACTLY FILLABLE CONTACT STRUCTURES WITHOUT STEINFILLINGSJONATHAN BOWDENAbstract. We give examples of contact structures which admit exact symplectic fillings,but no Stein fillings, answering a question of Ghiggini.1. IntroductionIt is a fundamental problem is contact topology to determine which contact 3-manifoldsadmit symplectic fillings. There are many varieties of symplectic fillings. In addition to weakand strong fillings it is natural to consider contact manifolds that are exactly fillable and ifthe filling has a complex structure, then the natural class of symplectic fillings are those thatare Stein. The relationship between these various notions is depicted in following sequenceof inclusions:{Stein fillable} ⊂ {Exactly fillable} ( {Strongly fillable} ( {Weakly fillable} ( {Tight}.Let us emphasise that a symplectic filling is always required to have connected boundary.Examples of strongly fillable contact structures that are not exactly fillable were found byGhiggini, [8]. Examples of weakly fillable contact structures that admit no strong fillingswere first discovered by Eliashberg, [4]. Finally, Etnyre and Honda, [5] showed that thereexist tight contact structures that are not weakly fillable. So all these inclusions are strictexcept possibly for the first. The main result of this paper is that the first inclusion is alsostrict.Theorem 1.1. There exist exactly fillable contact structures that admit no Stein fillings.This answers a question raised by Ghiggini whilst studying the relationship between strongand Stein fillability ([8], p. 1686). The contact structures of Theorem 1.1 are obtained by us-ing the fact that the Brieskorn spheres Σ(2, 3, 6n+5) considered in [8] can be realised as cover-ings of Seifert fibred manifolds that are compact quotients of PSL(2,R). One then constructsan exactly fillable contact structure on the connected sum Σ(2, 3, 6n + 5)#Σ(2, 3, 6n + 5)using the PSL(2,R)-structure in an explicit way. By a result of Eliashberg a connected sumof contact manifolds is Stein fillable if and only if each of the summands is Stein fillable.However, the contact structure on Σ(2, 3, 6n + 5) is Ghiggini’s non-Stein fillable contactstructure, which is in fact not even exactly fillable.This then exhibits the failure of Eliashberg’s result for exactly fillable contact structuresand also implies that the notion of exact fillability is strictly weaker than that of exactsemi-fillability, where one allows fillings with disconnected boundary. This latter fact isperhaps surprising since the notions of strong and weak fillability do not depend on whetherDate: July 6, 2012.2010 Mathematics Subject Classification. Primary 57R17, 53D10; Secondary 32Q28.12 JONATHAN BOWDENone requires the boundaries of the fillings to be connected or not. Moreover, since thecontact structures in question are perturbations of taut foliations, we further deduce thatperturbations of taut foliations on homology spheres are not necessarily Stein fillable.Acknowledgments: The author would like to thank T. Vogel and P. Ghiggini for helpfulsuggestions. He would also like to thank K. Cieliebak for pointing an error in an earlierversion of this article and for providing an advanced version of [1]. The hospitality of theMax Planck Institut fu¨r Mathematik in Bonn, where part of this research was carried out,is gratefully acknowledged.Conventions: All manifolds are smooth, oriented and connected and all contact structureswill be assumed to be oriented and positive.2. Stein fillings of non-prime ManifoldsIn [3], Eliashberg states a result about decomposing symplectic fillings of non-prime mani-folds. For our purposes we will be content with the case of Stein fillings, for which a detailedproof can now be found in [1].Theorem 2.1 ([3], Section 8, [1], Theorem 16.7). Let X be a Stein filling with boundaryM = (M1, ξ1)#(M2, ξ2) that decomposes as a non-trivial connected sum of contact manifolds.Then X decomposes as a boundary connect sum X = X1#∂X2, where X1, X2 are Stein fillingsof (M1, ξ1) and (M2, ξ2) respectively.The connected sum operation is well-defined on tight contact manifolds by [2]. Moreover,if both (M1, ξ1) and (M2, ξ2) are Stein fillable, then attaching a Stein 1-handle yields a Steinfilling of the contact connected sum. In this way Theorem 2.1 implies the following as acorollary.Corollary 2.2. A connected sum of contact manifolds (M1, ξ1)#(M2, ξ2) is Stein fillable ifand only if (M1, ξ1) and (M2, ξ2) are Stein fillable.3. Non-Stein Exact Symplectic FillingsWe use a construction which goes back to McDuff, [13] to construct many examples ofexact symplectic fillings (X, dλ) with H3(X) 6= 0. Since the third homology is non-trivial,these fillings, although exact, cannot be Stein. The starting point for the construction is anexact symplectic filling of the form (M × [0, 1], dλ) both of whose ends are convex, whichcan, for example, be obtained by considering compact quotients of PSL(2,R). This was firstobserved by Geiges and independently by Mitsumatsu.Example 3.1 ([7], [15]). Let psl2 denote the lie algebra of PSL(2,R) and choose the followingbasis:h =12(1 00 −1), l =12(0 11 0), k =12(0 −11 0).We identify psl∗2 with the space of left-invariant 1-forms on PSL(2,R) and define a linkingpairing byLK(α, β) = α ∧ dβ.With respect to this pairing the ordered basis {h∗, l∗, k∗} is orthogonal andLK(h∗, h∗) = LK(l∗, l∗) = −1 and LK(k∗, k∗) = 1.EXACTLY FILLABLE CONTACT STRUCTURES WITHOUT STEIN FILLINGS 3Any non-zero 1-form α ∈ psl∗2 then defines a positive resp. negative contact structure or ataut foliation, depending on whether LK(α, α) is positive resp. negative or zero. We let Γbe a co-compact lattice in PSL(2,R) and consider M = PSL(2,R)/Γ. If we setλ = tk∗ + (1− t)h∗on M × [0, 1], then the pair (M × [0, 1], dλ) is a symplectic filling with convex ends.Other examples of symplectic structures on M × [0, 1] with convex ends are given by T 2-bundles over S1 with Anosov monodromy or by smooth volume preserving Anosov flows (cf.[15]). It is now easy to construct examples of non-Stein exact fillings: one simply attaches asymplectic 1-handle to (M × [0, 1], dλ) with ends in each component of the boundary.Proposition 3.2 ([13]). There exist exact, non-Stein symplectic fillings.Proof. Let (M × [0, 1], dλ) = (X,ω) be an exact symplectic filling with convex ends. Attacha symplectic 1-handle to obtain a filling of the connected sum (M, ξ0)#(M, ξ1). We denotethis new filling by X˜ = X ∪ e1, where e1 denotes a topological 1-handle. The symplecticform on X˜ restricts to ω on X . Thus by the long exact sequence in cohomology of the pair(X˜,X) we see that X˜ is an exact filling and the hypersurface M × 12⊂ X˜ is non-separating,whence H3(X˜) 6= 0 and X˜ cannot be Stein. The manifolds (N, ξ) = (M, ξ0)#(M, ξ1) in Proposition 3.2 are always exactly fillable, buttheir natural fillings are not Stein. This raises the question of whether they are always Steinfillable or not. Or equivalently whether (M, ξ0) and (M, ξ1) are always Stein fillable. Wewill answer this question, by considering various Brieskorn spheres, which can be realised asfinite covers of compact quotients of PSL(2,R).Note that it is not sufficient to take McDuff’s original examples where M = ST ∗Σ is theunit cotangent bundle of a surface of genus at least 2. For in this case the contact structuresone obtains on M resp. M are isotopic to the canonical contact structure ξcan on M and ahorizontal contact structure ξLC given by the Levi-Civita connection of a hyperbolic metricon Σ. The canonical contact structure always admits Weinstein, and hence Stein, fillings([1], Example 11.12) and by the classification of contact structures on S1-bundles (see [10],[11]) the same is true of ξLC. Thus the resulting contact structure on the connected sum isalso Stein fillable.4. Brieskorn Spheres and PSL(2,R)-structuresWe consider the Brieskorn spheres Σ(2, 3, 6n + 5) taken with the opposite orientation tothat given by their description as the link of the complex singularity z21 + z32 + z6n+53 = 0.These are Seifert fibred homology spheres, whose quotient orbifolds are hyperbolic for anynatural number n greater than one.The manifold Σ(2, 3, 6n+5) admits a contact structure ηtan that is tangential to the Seifertfibration (cf. [14], p. 1764). This contact structure has the property that it is isotopic to itsconjugate ηtan, which denotes the same contact structure taken with the opposite orientation,and we note this in the following proposition.Proposition 4.1. For any natural number n ≥ 1 the manifold Σ(2, 3, 6n + 5) admits atangential contact structure ηtan. Moreover, any tangential contact structure is isotopic toits conjugate and is universally tight.4 JONATHAN BOWDENProof. We let B denote the quotient orbifold of Σ(2, 3, 6n+5) given by the Seifert fibration.A tangential contact structure ηtan induces a fibrewise cover Σ(2, 3, 6n+ 5)→ ST∗B to theunit cotangent bundle of the orbifold B so that ηtan is the pullback of the canonical contactstructure ξcan on ST∗B by ([14], Proposition 8.9). By assumption B is a hyperbolic orbifoldand hence ST ∗B is a compact quotient of PSL(2,R) by a discrete lattice. Furthermoreξcan comes from a left-invariant contact structure on PSL(2,R), which is the kernel of someleft-invariant 1-form, where we have identified PSL(2,R) with ST ∗H2 using the action ofPSL(2,R) on H2 via Mo¨bius transformations.Using the notation of Example 3.1, any left-invariant 1-form that is tangential lies in thespan of h∗ and l∗, since k generates the circle action. Moreover, since the linking form isnegative definite on the span of h∗ and l∗, any non-zero form that is tangential determinesa contact structure. Hence the space of tangential PSL(2,R)-invariant contact structuresis connected, so in particular ξcan is isotopic to its conjugate and the same then holds forηtan by taking pullbacks. In general any tangential contact structure can be perturbed to ahorizontal contact structure, which is then universally tight by ([14], Theorem A). Ghiggini has shown that Σ(2, 3, 6n+5) admits a contact structure which is strongly fillable,but admits no Stein fillings, when n is even ([8], Theorem 1.5). The only properties of thecontact structures used in Ghiggini’s proof of non-Stein fillability is that they are isotopic totheir conjugates and that their d3-invariant is −32. However, it follows from the classificationof [9] that all tight contact structures on Σ(2, 3, 6n+5) satisfy this latter constraint, thus inview of Proposition 4.1 we deduce the following.Theorem 4.2 ([8]). If n is even, then a tangential contact structure ηtan on Σ(2, 3, 6n+ 5)does not admit any Stein fillings.Remark 4.3. It is possible to deduce the fact that d3(ηtan) = −32without using the full forceof the classification given in [9]. One only needs the classification of contact structures withtwisting number −5, for which the Heegaard-Floer computations used in [8] suffice, and thefact that at least one of these must be horizontal ([14], p. 1764).One can further show that the contact structure ηtan corresponds to ηn−1,0 in terms of theclassification of tight contact structures on Σ(2, 3, 6n+5) given in [9]. In this way, Theorem4.2 also holds for n odd and at least 2 by ([12], Theorem 1.8).As pointed out to us by C. Wendl, the fact that (Σ(2, 3, 6n+ 5), ηtan) can be realised as aboundary component of an exact filling with disconnected boundary shows that the notionof exact semi-fillability is strictly weaker than that of exact fillability, unlike for its weak andstrong counterparts. We note this in the following proposition.Proposition 4.4. There exist infinitely many contact manifolds that are exactly semi-fillablebut admit no exact fillings.With these preliminaries we may now construct examples of exactly fillable contact structuresthat admit no Stein fillings.Theorem 4.5. There exist infinitely many exactly fillable contact manifolds that do notadmit Stein fillings.Proof. We consider the fibrewise covering Σ(2, 3, 6n + 5) → ST ∗B given in the proof ofProposition 4.1. Since ST ∗B admits a PSL(2,R)-structure, the product ST ∗B × [0, 1] canEXACTLY FILLABLE CONTACT STRUCTURES WITHOUT STEIN FILLINGS 5be made into an exact symplectic filling with convex ends as in Example 3.1 and by takingpullbacks the same is true of Σ(2, 3, 6n+ 5)× [0, 1].We let ξ = ξ0#ξ1 be the contact structure on Σ(2, 3, 6n + 5)#Σ(2, 3, 6n + 5) given byattaching a symplectic 1-handle as in Proposition 3.2 and note that ξ0 is tangential byconstruction. Then by Corollary 2.2 we have that ξ is Stein fillable if and only if ξ0 andξ1 are Stein fillable. However, if n is even the contact structure ξ0 is not Stein fillable byTheorem 4.2 and it follows that ξ is exactly fillable, but not Stein fillable. In fact, the argument used to show that ηtan is not Stein fillable actually shows that it is noteven exactly fillable ([8], p. 1685). In view of the examples used in Theorem 4.5 this thenexhibits the failure of the analogue of Corollary 2.2 for exactly fillable contact structures.Corollary 4.6. There exist infinitely many contact manifolds (M1, ξ1), (M2, ξ2) such thatthe contact connected sum (M1#M2, ξ1#ξ2) is exactly fillable but (M1, ξ1) is not.Since the contact structure ηtan is isotopic to a deformation of a taut foliation, we furtherdeduce the following.Corollary 4.7. There exist infinitely many contact structures that are deformations of tautfoliations on homology spheres, which are not Stein fillable.The non-Stein fillable contact structure ηtan is defined as the pullback of a tangentialcontact structure. This is completely analogous to the examples of Eliashberg in [4], whoshowed that the pullbacks of the standard contact structure on T 3 = ST ∗T 2 under suitablecoverings admit weak, but not strong, symplectic fillings. This leads to the following question,a negative answer of which would provide a very large class of symplectically fillable contactstructures without exact or Stein fillings.Question 4.8. Let M be a non-trivial fibrewise cover of ST ∗Σ, where Σ is a closed, hy-perbolic surface. Is the pullback of the canonical contact structure exactly or even Steinfillable?One may also formulate this question for the cotangent bundle of any hyperbolic orbifold.However, here the situation appears to be more subtle, since for example Σ(2, 3, 11) carriesa unique tight contact structure, which is both Stein fillable and a fibrewise pullback of thecanonical contact structure under a non-trivial covering map. A natural way of excluding thisexample would be to assume that the twisting number of the contact structure on the finitecover is not minimal amongst those contact structures that are isotopic to horizontal ones.This would also fit in with similar phenomena that occur in the classification of horizontalcontact structures on S1-bundles (cf. [10], [11]).We finally remark that in the case of coverings of ST ∗T 2 the obstruction to the existenceof a strong filling can be seen as given by the Giroux torsion of the contact structure (cf. [6]).Thus one might hope that there is some similar type of obstruction for covers of the unitcotangent bundle of a higher genus surface or even on more general Seifert fibred spaces.References[1] K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back: Symplectic geometry of affine complexmanifolds (to appear AMS).[2] V. Colin, Chirurgies d’indice un et isotopies de sphe`res dans les varie´te´s de contact tendues, C. R. Acad.Sci. Paris Se´r. I Math. 324 (1997), no. 6, 659–663.6 JONATHAN BOWDEN[3] Y. Eliashberg, Filling by holomorphic discs and its applications, Geometry of Low-Dimensional Manifolds:2, Proc. Durham Symp. 1989, London Math. Soc. Lecture Notes, 151, 1990, 45–67.[4] Y. Eliashberg, Unique holomorphically fillable contact structure on the 3-torus, Internat. Math. Res.Notices (1996), 77–82.[5] J. Etnyre and K. Honda, Tight contact structures with no symplectic fillings, Invent. Math. 148 (2002),no. 3, 609–626.[6] D. Gay, Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol. 10 (2006), 1749–1759.[7] H. Geiges, Examples of symplectic 4-manifolds with disconnected boundary of contact type Bull. LondonMath. Soc. 27 (1995), no. 3, 278–280.[8] P. Ghiggini, Strongly fillable contact 3-manifolds without Stein fillings, Geom. Topol. 9 (2005), 1677–1687.[9] P. Ghiggini and J. Van Horn-Morris, Tight contact structures on the Brieskorn spheres −Σ(2, 3, 6n− 1)and contact invariants, preprint: arXiv:0910.2752v3, 2010.[10] E. Giroux, Structures de contact sur les varie´te´s fibre´es en cercles audessus d’une surface, Comment.Math. Helv. 76 (2001), no. 2, 218–262.[11] K. Honda, On the classification of tight contact structures II, J. Differential Geom. 55 (2000), no. 1,83–143.[12] P. Lisca and A Stipsicz, Contact Ozsva´th-Szabo´ invariants and Giroux torsion, Algebr. Geom. Topol. 7(2007), 1275–1296.[13] D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991), 651–671.[14] P. Massot, Geodesible contact structures on 3-manifolds, Geom. Topol. 12 (2008), no. 3, 1729–1776.[15] Y. Mitsumatsu, Anosov flows and non-Stein symplectic manifolds, Ann. Inst. Fourier (Grenoble) 45(1995), no. 5, 1407–1421.Mathematisches Institut, Universita¨t Augsburg, Universita¨tsstr. 14, 86159 Augsburg,GermanyCurrent address : Max-Planck-Institut fu¨r Mathematik, Vivatsgasse 7, 53129 Bonn, GermanyE-mail address : jonathan.bowden@math.uni-augsburg.de

Exactly fillable contact structures without Stein fillings

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