Tight contact structures with no symplectic fillings

Non-simply connected symplectic fillings of lens spaces

We prove results exploring the relationship between the fundamental group and the second Betti number of minimal symplectic fillings of lens spaces. These results unify and generalize several disparate facts appearing in the literature. The Fibonacci numbers make a cameo appearance.

辛流形的单位余切丛的填充及其应用

Symplectic fillings of unit cotangent bundles of spheres and applications

该研究证明了奇数维球面的单位余切丛的辛填充在微分同胚意义下唯一，并探讨了其在辛cobordism和闭合辛流形中的应用。

Symplectic fillings of quotient surface singularities and minimal model program

We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic varieties.

Symplectic fillings of Seifert fibered spaces

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbing of spheres. In other cases, we produce new manifolds with convex symplectic boundary, thus yielding new cut-and-paste operations on symplectic manifolds containing certain configurations of symplectic spheres.

A Lefschetz fibration on minimal symplectic fillings of a quotient surface singularity

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution.

Symplectic fillings of lens spaces as Lefschetz fibrations

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.

关于小 Seifert 3-流形的辛填充

On symplectic fillings of small Seifert $3$-manifolds

本文分类了满足特定条件的小 Seifert 3-流形的最小辛填充，并证明它们可由加权齐次复曲面奇点最小解消经有理爆破收缩得到。

Symplectic fillings and positive scalar curvature

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…

Topologically Distinct Lagrangian and Symplectic Fillings

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-manifolds with topologically distinct symplectic fillings, and is generalized to higher dimensions.

Spine removal surgery and the geography of symplectic fillings

We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the authors with Van Horn-Morris.

透镜空间的辛填充的最小$b_2$

Minimum $b_2$ of symplectic fillings of lens spaces

该研究对标准接触结构的辛填充中具有典范负定型 plumbing 最小$b_2$的透镜空间分类，发现其分类需4个无限族禁子图，辛填充与光滑负定型填充存在差异。

On Lens Spaces and Their Symplectic Fillings

The standard contact structure on the three-sphere is invariant under the action of the cyclic group of order p yielding the lens space L(p,q). Therefore, every lens space carries a natural quotient contact structure Q. A theorem of Eliashberg and McDuff classifies the symplectic fillings of (L(p,1), Q) up to diffeomorphism. We announce a generalization of that result to every lens space. In particular, we give an explicit handlebody decomposition of every symplectic filling of (L(p,q), Q) for every p and q. Our results imply that: (a) there exist infinitely many lens spaces L(p,q) with q>1 su…

A topological characterization of symplectic fillings of Seifert 3-manifolds

In this paper, we investigate a relation between rational blowdown surgery and minimal symplectic fillings of a given Seifert 3-manifold with a canonical contact structure. Consequently, we determine a necessary and sufficient condition for a minimal symplectic filling of a Seifert 3-manifold satisfying certain conditions to be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity. Furthermore, as an application of the main results, we prove that every minimal symplectic filling of a large family of Seifert 3-mani…

On symplectic fillings

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 …

Some applications of Menke's JSJ decomposition for symplectic fillings

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 …

On geography of symplectic fillings of contact branched covers

In this paper, we determine the Euler characteristics and signatures of the exact symplectic fillings of the contact double, 3-fold or 4-fold cyclic covers of the standard contact 3-sphere branched over certain transverse quasi-positive links. These links include all quasi-positive knots with crossing numbers smaller than 11 and all quasi-positive links with crossing numbers smaller than 12 and nonzero nullity.

Symplectic fillings, contact surgeries, and Lagrangian disks

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.

The diffeomorphism type of symplectic fillings

We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We a…

On Symplectic Fillings of Lens Spaces

Let C be the contact structure naturally induced on the lens space L(p,q) by the standard contact structure on the three--sphere. We obtain a complete classification of the symplectic fillings of (L(p,q),C) up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.

Contact Dehn surgery, symplectic fillings, and Property P for knots

These are notes of a talk given at the Mathematische Arbeitstagung 2005 in Bonn. Following ideas of Ozbagci-Stipsicz, a proof based on contact Dehn surgery is given of Eliashberg's concave filling theorem for contact 3-manifolds. The role of that theorem in the Kronheimer-Mrowka proof of property P for nontrivial knots is sketched.

Stein fillings of homology $3$-spheres and mapping class groups

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillings of a rational homology $3$-spheres into strongly symplectic fillings, we also show that a symplectically fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique symplectic filling up to diffeomorphism and blow-…

对棱镜空间的辛填充的有理吹脱图

Rational blowdown graphs for symplectic fillings of lens spaces

本文提出通过凸多边形三角剖分简化有理吹脱算法，将棱镜空间的最小辛填充组织为有向图，并给出有理吹脱深度的上界。

每个尖点奇点链环都拥有无穷多个强辛填充

Every cusp singularity link admits infinitely many strong symplectic fillings

本文证明孤立复曲面奇点的链环（若为Sol^3或SL(2;R)流形）具有无穷多个互不同胚且非爆破相关的强辛填充，进而推出所有尖点奇点链环都有无穷多个极小强辛填充。

Symplectic fillings of virtually overtwisted contact structures on lens spaces

Symplectic fillings of standard tight contact structures on lens spaces are understood and classified. The situation is different if one considers non-standard tight structures (i.e. those that are virtually overtwisted), for which a classification scheme is still missing. In this work we use different approaches and employ various techniques to improve our knowledge of symplectic fillings of virtually overtwisted contact structures. We study curves configurations on surfaces to solve the problem in the case of a specific family of lens spaces. Then we give general constraints on the topology …

On uniqueness of symplectic fillings of links of some surface singularities

We consider the canonical contact structures on links of rational surface singularities with reduced fundamental cycle. These singularities can be characterized by their dual resolution graphs: the graph is a tree, and the weight of each vertex is no greater than its negative valency. In a joint work with Starkston, we previously showed that if the weight of each vertex in the graph is at most -5, the contact structure has a unique symplectic filling (up to symplectic deformation and blow-up). The proof was based on a symplectic analog of de Jong-van Straten's description of smoothings of thes…

关于三叶结的手术与辛填充

Surgeries on the trefoil and symplectic fillings

本文确定了对右手三叶结进行特定手术后所得流形上接触结构的Stein可填充性，发现其可填充与否缺乏明显规律。

脊柱开卷分解与具有奇异纤维的辛填充

Spinal open books and symplectic fillings with exotic fibers

本文推广了Wendl的结果，证明平面脊柱开卷分解支撑的接触3-流形的最小辛填充形变等价于带边Lefschetz纤维化中正多截面的补集，并利用新定义的脊柱映射类群完成抛物环面丛的强填充分类。

A few remarks about symplectic filling

We show that any compact symplectic manifold (W,ω) with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane ξon dW which is weakly compatible with omega, i.e. the restriction ω|ξdoes not vanish and the contact orientation of dW and its orientation as the boundary of the symplectic manifold W coincide. This result provides a useful tool for new applications by Ozsvath-Szabo of Seiberg-Witten Floer homology theories in three-dimensional topology and has helped complete the Kronheimer-Mrowka proof of Property P for knots.

辛理性同调球填充的Seifert纤维空间

Symplectic rational homology ball fillings of Seifert fibered spaces

研究了Seifert纤维空间作为辛理性同调球边界条件，证明了特定棱柱空间的填充可能性，并探讨了Brieskorn球体与Stein领域的关系。

关于脊柱开放书分解的辛填充 I：几何构造

On symplectic fillings of spinal open book decompositions I: Geometric constructions

本文介绍了脊柱开放书与接触结构及辛或Stein结构的关系，提出新的辛cobordism构造脊柱移除手术，扩展了Eliashberg、Gay-Stipsicz和第三作者的构造，并应用该手术得到新的非强（有时非弱）辛填充的接触流形例子。

Homology spheres bounding acyclic smooth manifolds and symplectic fillings

In this paper, we collect various structural results to determine when an integral homology $3$--sphere bounds an acyclic smooth $4$--manifold, and when this can be upgraded to a Stein manifold. In a different direction we study whether smooth embedding of connected sums of lens spaces in $\mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.

Topological complexity of symplectic 4-manifolds and Stein fillings

We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by…

On Symplectic Cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are ``symplectic cobordism equivalent.''

切触手术图中的混合环面

Mixed tori in contact surgery diagrams

本研究提出将辛JSJ分解应用于三维切触流形切触手术图的图示框架，可推导原流形的恰当/弱辛填充，既重现已知分类结果，又给出大类配边三维流形的填充分类算法。

关于脊柱开书分解的辛填充 II：全纯曲线与分类

On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

本文证明当接触3流形具有平面页的统一脊柱开书分解时，其(弱、强和/或精确)辛和Stein填充可由Lefschetz纤维化的微分同胚类分类。进一步结果包括新的ECH接触不变量和SFT中代数扭力消失准则，非定向圈 bundles的填充分类，以及关于四维实数中Stein结构变形类的

Calabi-Yau Caps, Uniruled Caps and Symplectic Fillings

We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinitely family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admittin…

Lefschetz-Bott fibrations on line bundles over symplectic manifolds

We describe Lefschetz-Bott fibrations on complex line bundles over symplectic manifolds explicitly. As an application, we construct more than one strong symplectic filling of the link of the $A_{k}$-type singularity. In the appendix, we show that the total space of a Lefschetz-Bott fibration over the unit disk serves as a strong symplectic filling of a contact manifold compatible with an open book induced by the fibration.

没有辛填充的切触紧结构无处不在

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。

Brieskorn球面与有理同调球的辛填充

Brieskorn spheres and rational homology ball symplectic fillings

本文确认了关于Brieskorn球面的辛填充存在的猜想，研究了不同情况下接触结构的填充可能性及同调球的性质。

Fillings of unit cotangent bundles of nonorientable surfaces

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism.

Symplectic fillability of tight contact structures on torus bundles

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.

A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact type embeddings into any closed symplectic 4-manifold, and has vanishing contact invariant in Embedded Contact Homology, and we give examples of contact manifolds that have planar k-torsion fo…

Embedding fillings of contact 3-manifolds

In this survey article we describe different ways of embedding fillings of contact 3-manifolds into closed symplectic 4-manifolds.

具有正手术但非弱辛可填充的 $L$-空间纽结

$L$-space knots with positive surgeries that are not weakly symplectically fillable

本文提出了一种检测 $L$-空间缺乏弱辛填充的一般策略，通过计算仅依赖于纽结类型和手术系数的几何不变量来阻碍可填充性，并给出了一个无穷双曲 $L$-空间族不承认弱辛填充的新例子。

Topological constraints for Stein fillings of tight structures on lens spaces

In this article we give a sharp upper bound on the possible values of the Euler characteristic for a minimal symplectic filling of a tight contact structure on a lens space. This estimate is obtained by looking at the topology of the spaces involved, extending this way what we already knew from the universally tight case to the virtually overtwisted one. As a lower bound, we prove that virtually overtwisted structures on lens spaces never bound Stein rational homology balls. Then we turn our attention to covering maps: since an overtwisted disk lifts to an overtwisted disk, all the coverings o…

Explicit concave fillings of contact three-manifolds

In this paper we give explicit, handle-by-handle constructions of concave symplectic fillings of all closed, oriented contact 3-manifolds. These constructions combine recent results of Giroux relating contact structures and open book decompositions of 3-manifolds, earlier results of the author on attaching 4-dimensional symplectic 2-handles along transverse links, and some tricks with mapping class groups of compact surfaces with non-empty boundary.

Stein fillings of contact 3-manifolds obtained as Legendrian surgeries

In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,ξ_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,ξ_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deforma…

相对汤姆猜想，辛结构与更广泛领域

Relative Thom Conjectures, symplectic and beyond

该研究通过建立准则证明辛表面在特定条件下最小化亏格，并探讨了接触结构紧致性与辛几何的关系。

Exactly fillable contact structures without Stein fillings

We give examples of contact structures which admit exact symplectic fillings, but no Stein fillings, answering a question of Ghiggini.

Strong symplectic fillability of contact torus bundles

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…

Circular spherical divisors and their contact topology

This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors $ D $ that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such $D$. When $ D $ is anticanonical and convex, we give explicit betti number bounds for Stein fillings…

Contact surgery on the Hopf link: classification of fillings

Let $H\subseteq S^3$ be the two-component Hopf link. After choosing a Legendrian representative of $H$ with respect to the standard tight contact structure on $S^3$ we perform contact $(-1)$-surgery on the link itself. The manifold we get is a lens space together with a tight contact strucure on it, which depends on the chosen Legendrian representative. We classify its minimal symplectic fillings up to homeomorphism (and often up to diffeomoprhism), extending the results of Lisca which covers the case of universally tight structures and the article of Plamenevskaya - Van Horn Morris which desc…

Strongly fillable contact 3-manifolds without Stein fillings

We use the Ozsvath-Szabo contact invariant to produce examples of strongly symplectically fillable contact 3-manifolds which are not Stein fillable.

夹层奇点与近Lefschetz纤维化

Sandwiched singularities and nearly Lefschetz fibrations

通过引入脊柱开书和近Lefschetz纤维化，研究夹层奇点的Milnor纤维与辛填充，对比代数几何形变理论与辛拓扑，发现一类夹层奇点链环存在非预期的Stein填充。