Symplectic S^1 x N^3, subgroup separability, and vanishing Thurston norm

关于纤维化$3$-流形中链环的辛Khovanov同调

Towards a symplectic Khovanov homology for links in fibered $3$-manifolds

本文针对纤维化闭3-流形中的横向链环定义辛Khovanov型同调，并猜想相关曲面范畴的组合dga描述，该dga是带边Heegaard Floer同调中链代数的高维类似物。

Obstructions for exact submanifolds with symplectic applications

Suppose $X^{N}$ is a closed oriented manifold, $α\in H^*(X;\mathbb{R})$ is a cohomology class, and $Z \in H_{N-k}(X)$ is an integral homology class. We ask the following question: is there an oriented embedded submanifold $Y^{N-k} \subset X$ with homology class $Z$ such that $α|_Y = 0 \in H^*(Y;\mathbb{R})$? In this article, we provide a family of computable obstructions to the existence of such 'exact' submanifolds in a given homology class which arise from studying formal deformations of the de Rham complex. In the final section, we apply these obstructions to prove that the following symple…

三分4-流形上的辛形式

Symplectic forms on trisected 4-manifolds

本文研究闭光滑4-流形三分结构可构造辛结构的显式判据，结合此前辛流形存在相容三分的结论，给出4-流形容许辛结构的新刻画，并引出三维几何拓扑相关的若干研究问题。

高维共叶结构的可填充性障碍

Fillability obstructions for high-dimensional confoliations

本文研究高维共叶结构的辛可填充性，推广了Massot-Niederkrüger-Wendl的边界Legendrian开书，并应用于过扭接触流形乘积的不可填充性及Bourgeois接触结构的弱辛填充不存在性。

On the asphericity of a symplectic M3 x S1

An interesting question in symplectic topology, which was posed by C. H. Taubes, concerns the topology of closed (i.e. compact and without boundary) connected oriented three dimensional manifolds whose product with a circle admits a symplectic structure. The only known examples of such manifolds are those which fiber over a circle. Taubes asked whether these examples are the only examples of such manifolds. An affirmative answer to Taubes' question would have the following consequence: any such manifold either is diffeomorphic to the product of a two-sphere with a circle or is irreducible and …

Higher-Order Polynomial Invariants of 3-Manifolds Giving Lower Bounds for the Thurston Norm

We define an infinite sequence of new invariants, delta_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the delta_n give obstructions to a 3-manifold fibering over S^1 and to a 3-manifold being Seifert fibered. Moreover, we s…

Norms on the cohomology of a 3-manifold and SW theory

The aim of this paper is to discuss some applications of the relation between Seiberg-Witten theory and two natural norms defined on the first cohomology group of a closed 3-manifold N - the Alexander and Thurston norms. We start by giving a "new" proof of McMullen's inequality between these norms, and then use these norms to study two problems related to symplectic 4-manifolds of the form S^1xN. First we prove that - as long as N is irreducible - the unit balls of these norms are related in a way similar to the case of fibered 3-manifolds, supporting the conjecture that N is fibered. Second, …

Twisted Alexander polynomials detect fibered 3-manifolds

A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S^1 x N^3 admits a symplectic structure, then N fibers over S^1. In fact we will completely determine the…

关于可填充切触结构的Euler类为一猜想

On the Euler class one conjecture for fillable contact structures

本文证明每个定向闭双曲3-流形存在有限覆盖，其边界上某偶格点不是任何弱辛可填充切触结构或横截定向紧叶状结构的实Euler类，从而给出Thurston Euler类为一猜想的新反例。

Teichmüller空间上的水平度量

The horocyclic metric on Teichm{ü}ller spaces

本文从共形视角研究Teichmüller空间上的水平度量，将Thurston的理论扩展到任意亏格的Riemann面，并通过复Legendre变换建立Weil-Petersson symplectic形式的对偶关系。

Representations of 3-manifold groups in PGL(n,C) and their restriction to the boundary

We study here the space of representations of a fundamental group of a 3-manifold into PGL(n,C). Thurston, Neumann and Zagier initiated a strategy (in the case of PGL(2,C)) consisting in: triangulate the manifold, assign shapes to each pieces and then try to glue back. This leads to the "gluing equations" and the Neumann-Zagier symplectic space. Building on the works of Dimofte-Gabella-Goncharov and Bergeron-Falbel-Guilloux, we complete the picture in the case of PGL(n,C). We recover a situation very similar to the case of PGL(2,C). This allows for example to obtain a combinatorial proof of a …

Monotonicity of Degrees of Generalized Alexander Polynomials of Groups and 3-Manifolds

We investigate the behavior of the higher-order degrees, db_n, of a finitely presented group G. These db_n are functions from H^1(G;Z) to Z whose values are the degrees certain higher-order Alexander polynomials. We show that if def(G) is at least 1 or G is the fundamental group of a compact, orientable 3-manifold then db_n is a monotonically increasing function of n for n at least 1. This is false for general groups. As a consequence, we show that if a 4 manifold of the form X times S^1 admits a symplectic structure then X ``looks algebraically like'' a 3-manifold that fibers over S^1, suppor…

关于近复4维流形上的Kodaira型猜想

A Kodaira type conjecture on almost complex 4 manifolds

本文在近复流形上定义了精细Dolbeault上同调，证明了特定维数相等条件蕴含辛结构且等价于广义$\partial\bar\partial$-引理，从而给出了紧近复4维流形成为几乎Kähler流形的充分条件。