We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-fillings. The argument uses results of Mrowka, Ozsvath and Yu on the translation-invariant solutions to the Seiberg-Witten equations on cylinders and the non-triviality of the Kronheimer-Mrowka monopole invariants of symplectic fillings.
In this short note, we exhibit an infinite family of hyperbolic rational homology $3$--spheres which do not admit any fillable contact structures. We also note that most of these manifolds do admit tight contact structures.
Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.
In this paper, we find infinite hyperbolic 3-manifolds that admit no weakly symplectically fillable contact structures, using tools in Heegaard Floer theory. We also remark that part of these manifolds do admit tight contact structures.
We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.
Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are non-fillable. Using contact surgery on his examples we create on every sphere S^{2n-1}, n>1, an exotic contact structure ξ_- that also contains a plastikstufe. As a consequence, every closed contact manifold M (except S^1) can be converted into a contact manifold that is not (semi-positively) fillable by taking the connected sum of M with (S^{2n-1},ξ_-).
We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the 5-sphere the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.
We introduce a new method to obstruct Liouville and weak fillability. Using this, we show that various rational homology 3-spheres admit strongly fillable contact structures without Liouville fillings, which extends the result of Ghiggini on a family of Brieskorn spheres. We also make partial progress on a conjecture of Ghiggini and Van-Horn-Morris.
We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot $K$ in a contact 3-manifold $Y$, gives rise to a weakly fillable contact structure. We show that this happens if and only if $Y$ itself is weakly fillable, and $K$ is isotopic to the boundary of a properly embedded symplectic disk inside a filling of $Y$. Moreover, if $Y'$ is a contact manifold arising from positive contact surgery along $K$, then any filling of $Y'$ is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of $Y$. Using this resu…
We define symplectic fractional twists, which generalize Dehn twists, and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show that left-handed fractional twists often give rise to non-fillable contact manifolds. These manifolds are in fact "algebraically overtwisted", yet they do not seem to contain bLobs, nor are they directly related to negative stabilizations. We also show that the Weinstein conjecture…
We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations. This answers two questions of Etnyre.
We study the existence of positive loops of contactomorphisms on a Liouville-fillable contact manifold $(Σ,ξ=\ker(α))$. Previous results show that a large class of Liouville-fillable contact manifolds admit contractible positive loops. In contrast, we show that for any Liouville-fillable $(Σ,α)$ with $\dim(Σ) \geq 7$, there exists a Liouville-fillable contact structure $ξ'$ on $Σ$ which admits no positive loop at all. Further, $ξ'$ can be chosen to agree with $ξ$ on the complement of a Darboux ball.
We continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 admits a weakly fillable contact structure provided M admits a weak symplectic filling. We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are …
A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse statement holds, that is if C(S,h) is Stein fillable then h is positive. On the other hand, results by Wand and by Baker, Etnyre and Van Horn-Morris imply the existence of counterexamples to the converse statement with S of arbitrary genus strictly greater than one. The main purpo…
In this article we provide an infinite family of weakly symplectically fillable contact structures with trivial Ozsvath-Szabo contact invariants over Z/2Z. As a consequence of this fact, we show how Heegaard-Floer theory can distinguish between weakly and strongly fillable contact structures.
We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many Reeb orbits. This property is used to generalize a famous result by Ustilovsky: We show that in a large class of manifolds (including all unit cotangent bundles and all Weinstein fillable contact manifolds with torsion first Chern class) each carries infinitely many exactly fillable contact structures. These are all different from the ones constructed recen…
A geometric obstruction, the so called "plastikstufe", for a contact structure to not being fillable has been found by K. Niederkruger. This generalizes somehow the concept of overtwisted structure to dimensions higher than 3. This paper elaborates on the theory showing a big number of closed contact manifolds with a "plastikstufe". They are the first examples of non-fillable contact closed high-dimensional manifolds. In particular we show that $S^3 \times \prod_j Σ_{j}$, for $g(Σ_j)\geq 2$, possesses this kind of contact structure and so any connected sum with those manifolds also does it.
We say that a contact manifold is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x). Generalizing results of Milnor and Giroux, we associate to each holomorphic function f defined on X, with isolated singularity at x, an open book which supports the contact structure. Moreover, we prove that any 3-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. * * * * * * * * In the first version of the paper, we showed that the open book associated to f carries the contact structure onl…
In this note we make several observations concerning symplectic fillings. In particular we show that a (strongly or weakly) semi-fillable contact structure is fillable and any filling embeds as a symplectic domain in a closed symplectic manifold. We also relate properties of the open book decomposition of a contact manifold to its possible fillings. These results are also useful in proving property P for knots [P Kronheimer and T Mrowka, Geometry and Topology, 8 (2004) 295-310, math.GT/0311489] and in showing the contact Heegaard Floer invariant of a fillable contact structure does not vanish …
We prove that $(\mathbb{RP}^{2n-1},ξ_{std})$ is not exactly fillable for any $n\ne 2^k$ and there exist strongly fillable but not exactly fillable contact manifolds for all dimension $\ge 5$.
In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually…
This thesis considers fillable contact structures on odd-dimensional manifolds. For that purpose, Rabinowitz-Floer homology (RFH) is used which was introduced by Cieliebak and Frauenfelder in 2009. A major part of the thesis is devoted to technical problems in the definition of RFH. In particular, it is shown that the moduli spaces involved are cut out transversally. Moreover, it is proved that RFH is essentially invariant under subcritical handle attachment. Finally, RFH is calculated for some Brieskorn manifolds. The obtained results are then used to show for every manifold, which supports f…
We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of the 3-torus similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplecto…
According to Lerman, compact connected toric contact 3-manifolds with a non-free toric action whose moment cone spans an angle greater than $π$ are overtwisted, thus non-fillable. In contrast, we show that all compact connected toric contact manifolds in dimension greater than three are weakly symplectically fillable and most of them are strongly symplectically fillable. The proof is based on the Lerman's classification of toric contact manifolds and on our observation that the only contact manifolds in higher dimensions that admit free toric action are the cosphere bundle of $T^d, d\geq3$ $(T…
For any integer $n\geq 2$, we construct an infinite family of Stein fillable contact $(4n-1)$-manifolds each of which admits infinitely many pairwise homotopy inequivalent Stein fillings.
We construct a family of Stein fillable contact homology 3-spheres such that each contact structure of the family is supported by an open book with planar page, and a Stein filling of the contact manifold is of Mazur type.
Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of w…
In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.
In this note we observe that one can contact embed all contact 3-manifolds into a Stein fillable contact structure on the twisted $S^3$-bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.
Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obtained by the Eliashberg-Gompf construction.For example we prove the Weinstein conjecture for the Brieskorn homology spheres $Σ(2,3,6n-1)$, $n\geq2$, oriented as the boundary of the corresponding Milnor fibre. Furthermore, for tight contact structures on odd lens spaces, non-contra…
We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded two-spheres of square zero. This, together with standard gauge theory, is used to show that any contact three-manifold of non-zero torsion (in the sense of Giroux) cannot be strongly symplectically fillable. John Etnyre pointed out to the author that the same argument together wi…
Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted contact structures and is non-zero for Stein fillable ones. The construction uses of Giroux's interpretation of contact structures in terms of open book decompositions, and the knot Floer homologies introduced in math.GT/0209056.
This paper completely answers the question of when contact (r)-surgery on a Legendrian knot in the standard contact structure on the 3-sphere yields a symplectically fillable contact manifold for r in (0,1]. We also give obstructions for other positive r and investigate Lagrangian fillings of Legendrian knots.
We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.
Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at least n pairwise non-isomorphic tight, not fillable contact structures.
Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.
Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khovanov-homological reasons, and show that some of these branched covers are not Stein fillable.
We prove several results on weak symplectic fillings of contact 3-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable - this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori - this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fil…
We characterize which Legendrian $4$-plat knots in the standard contact $3$-space have exact orientable Lagrangian fillings. As a corollary, we show that the underlying smooth knot types of fillable Legendrian $4$-plats are positive.
Let a contact 3-manifold $(Y, ξ_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $ξ_0$. We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant $c^+(ξ_0)\in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. It follows that Karakurt's "height of $U$-tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of $U$-tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between th…
We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.
We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology o…
According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an application, we derive classification results for subcritically Stein fillable contact 5-manifolds and characterise the standard contact structure on the 5-sphere in terms of such fillings. This characterisation is discussed in the context of the Andrews-Curtis conjecture concern…
We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds -- with or without boundary -- defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifol…
We give a possible generalization of Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We moreover show that $R^{2n+1}$ has at least three distinct contact structures. This version of the paper contains both the texts of the published version of the paper together with an Erratum to the published version appended to the end.
We obtain several results for (iterated) planar contact manifolds in higher dimensions: (1) Iterated planar contact manifolds are not weakly symplectically semi-fillable. This generalizes a 3-dimensional result of Etnyre to a higher-dimensional setting. (2) They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds. This generalizes a result by Albers-Bramham-Wendl. (3) They satisfy the Weinstein conjecture, i.e. every contact form admits a closed Reeb orbit. This is proved by an alternative approach as that of [Acu], and is a higher-dimensional generaliz…
We give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifolds with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.
Let X be a 4-manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b_2^+(X)>0 or the boundary of X is disconnected. As an application we show that the Poincare homology 3-sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semi-fillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3-manif…
