Paolo Lisca, Andras I. Stipsicz
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.
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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 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.
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.
Tight contact structures without symplectic fillings are everywhere
该研究证明高维流形中同一殆切触类内存在紧但不可辛填充的切触结构，并构造了特定条件下Liouville可填充但Weinstein不可填充的切触结构，推进了切触拓扑与辛几何的关联研究。
As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.
Fillable structures on negative-definite Seifert fibred spaces
本文研究负定星形连接空间上的可填充接触结构，证明此类Seifert纤维空间具有唯一的负最大扭数，并通过晶格上同调的Alexander滤子计算该扭数，进一步展示这些流形上的负扭紧结构由复表面奇点的最小解析中的Stein结构诱导。
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…
Tight contact structures on a family of hyperbolic L-spaces
本文对Whitehead链环手术上的紧接触结构进行分类，首次给出无限族双曲L-空间的分类结果，并确定了Stein可填充与几乎过度扭转的结构。
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.
A note on Stein fillability of circle bundles over symplectic manifolds
本文证明了维数 $2n \geq 4$ 的闭积分辛流形上，当整数 $k>\int_Σω^{n}$ 时，$(Σ, kω)$ 上的Boothby-Wang丛不带有Stein可填充切触结构，从而否定地回答了Eliashberg的问题，并证明了某些孤立奇点的不可平滑性。
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 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.
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.
Fillability of small Seifert fibered spaces
本文研究小Seifert纤维空间上切触结构的填充性，通过分析平面开书的单值因子化，刻画了$M(-1;r_1,r_2,r_3)$上零扭转切触结构中的可填充结构。
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 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…
In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over Hirzebruch surfaces. Note that the contact structures in question are fillable. This may be contrasted with an earlier paper where we showed that there are infinitely many non-conjugate tori in the contactomorphism groups of certain overtwisted lens spaces (Contact cuts, Israel …
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.
We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots - namely, the range of slopes on which admissible transverse surgery preserves tightness - and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universall…
Contact domination
本文证明任意闭连通定向奇数维流形均可由同维紧且 Liouville 可填充的接触流形非零度数支配，并给出接触除子应用。
We show the existence of tight contact structures on infinitely many hyperbolic three-manifolds obtained via Dehn surgeries along sections of hyperbolic surface bundles over circle.
Non-fillability of overtwisted contact manifolds via polyfolds
本文通过polyfolds方法证明弱辛可填充切触流形是紧的，验证了特定过度扭转边界条件下的强Weinstein猜想，并推广至带边Legendrian开书情形。
We consider a large family F of torus bundles over the circle, and we use recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y in F we exhibit non-homotopy…
Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations. The proof expresses hypothesis and conclusion as differential inequalities involving functions on disks transverse to the fibres, and shows that one inequality implies the other.
It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston-Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space of the circle with its standard tight contact structure.
Tight surgeries on torus knots
本文对沿任意环面上纽结手术所得流形上的紧接触结构完成分类，还分类了相关曲面丛的接触结构，研究其辛可填充性并反驳了一项相关猜想。
Tight Contact Structures on Contact Mapping Tori and their Folded Sums
本文基于粘合过程提供了接触映射环面“折叠和”上接触结构的丛理论构造，并证明了在奇数维 $2n+1\geq 7$ 中，若公共接触折叠上的诱导接触形式无可缩 Reeb 轨道，则该折叠接触结构及接触映射环面是紧的。
Spaces of tight contact structures in dimension 3
该研究引入微纤维化方法，证明紧接触结构空间的弱同伦型在接触0-、1-、3-配柄附接下不变，还确定了多种接触结构空间及接触同胚群的弱同伦型，部分解决Giroux猜想并证明了Hatcher定理的类似结论。
It is a conjecture of Colin and Honda that the number of Reeb periodic orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic geometries. Along the line of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many non-isomorphic contact structures for which the number of Reeb periodic orbits of any non-degenerate Reeb vector field grows exponentially. Our result hinges on the e…
By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is null-bordant in the oriented bordism group $ΩSO^*(W)$ of any symplectically aspherical filling $W$, and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical b…
Geodesic vector fields, induced contact structures and tightness in dimension three
本文简化了Gluck-Harrison定理的证明，证明了在非负混合曲率条件下测地向量场诱导切触结构，并分类了具有等距Reeb流的闭切触3-流形，推广了紧致性结果。
We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic structures to the …
In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section~1, we prove that V carries a (positive) contact structure transverse to the fibers if and only if the Euler number of the fibration is less or equal to 2g-2. In Section~2, we show that, for any contact structure $ξ$ on V, one of the following properties holds: either $ξ$ is isot…
Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seif…
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