On symplectic fillings of virtually overtwisted torus bundles

Symplectic fillings and cobordisms of lens spaces

We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling $X$ of a virtually overtwisted contact structure on $L(p,q)$ has another symplectic structure that fills the universally tight contact structure on $L(p,q)$. Moreover, we show that the Stein filling of $L(p,q)$ with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.

紧的不可半充切触三维流形的无限族

An infinite family of tight, not semi-fillable contact three-manifolds

证明了Honda在曲面上某些圆丛上发现的无穷族虚过扭紧切触结构不允许辛半填充。该论点使用Mrowka，Ozsvath和Yu的结果上的投影不变的解决方案的塞伯格-威滕方程的圆柱和非平凡的Kronheimer Mrowka投影不变量的辛填充。

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 …

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…

切触3-流形的一族局部辛填充障碍

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

我们推广了熟悉的概念overtwedness和Giroux挠在3维接触流形，定义了一个无限层次的局部填充障碍称为平面扭转，其整数值的顺序$k \ge 0$可以被解释为衡量一个层次的“紧密度”的接触流形。我们特别指出，任何具有平面挠率的切触流形都不允许在闭辛4-流形中嵌入切触型，并且在嵌入切触同调中具有零切触不变量，并且我们给出了对任何$k \ge 2$具有平面k-挠率但不具有Giroux挠率的切触流形的例子.我们还表明，补充的装订支持打开书从来没有平面扭转。背景中的统一思想是根据打开的书籍沿其装订的接触纤维和来分解接触流形。作为这些结果的技术基础，我们建立了存在性，唯一性和紧性定理的某些类J-全纯曲线在炸毁总结开卷，这些也意味着代数障碍的平面性和嵌入的部分平面域。结果进一步应用于后续论文弱辛填充（arXiv：1003.3923，联合K。Niederkrueger），非精确辛配边（arXiv：1008.2456）和辛域论（arXiv：1009.3262，与J. Latschev联合）。

弱辛填充与全纯曲线

Weak Symplectic Fillings and Holomorphic Curves

证明了切触3-流形的弱辛填充的几个结果，包括：（1）任何平面切触流形的弱填充都可以变形为Stein填充的爆破。(2)具有完全分离平面挠率的切触流形不是弱可填充的-这给出了许多没有吉鲁挠率的切触流形没有弱填充的新例子。(3)弱可填充性在沿着辛前拉格朗日环面的接触流形的拼接下被保留-这给出了许多没有吉鲁挠的接触流形的新例子，这些接触流形是弱可填充的，但不是强可填充的。我们建立了障碍弱填充通过两个平行的方法使用全纯曲线。在第一种方法中，我们推广了原来的Gromov-Eliashberg“主教盘”的论点，研究通过一个主教家庭的全纯环的Giroux扭转与边界上的“锚定overtwisted环”的特殊情况。第二种方法使用穿孔的全纯曲线，是基于观察到，每一个弱填充可以变形的领附近，从而导致一个稳定的哈密顿结构的边界上。这也使得有可能应用辛域论的技术，我们证明在一个测试案例中，通过显示弱和强填充之间的区别转化为接触的同源性之间的区别扭曲和untwisted系数。