Fukaya categories of the torus and Dehn surgery

Fukaya categories in Koszul duality theory

In this paper, we define $A_{\infty}$-Koszul duals for directed $A_{\infty}$-categories in terms of twists in their $A_{\infty}$-derived categories. Then, we compute a concrete formula of $A_{\infty}$-Koszul duals for path algebras with directed $A_n$-type Gabriel quivers. To compute an $A_\infty$-Koszul dual of such an algebra $A$, we construct a directed subcategory of a Fukaya category which are $A_\infty$-derived equivalent to the category of $A$-modules and compute Dehn twists as twists. The formula unveils all the ext groups of simple modules of the parh algebras and their higher composi…

3-流形上可填充接触结构的不存在性与存在性

Nonexistence and existence of fillable contact structures on 3-manifolds

本文构造了无穷多个无辛可填充接触结构的双曲闭3-流形，并证明沿特定纽结和链环的大系数Dehn手术承认Stein可填充接触结构，支持高手术猜想。

Berge duals and universally tight contact structures

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.

Fukaya categories of plumbings and multiplicative preprojective algebras

Given an arbitrary graph $Γ$ and non-negative integers $g_v$ for each vertex $v$ of $Γ$, let $X_Γ$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*Σ_v$ according to this graph, where $Σ_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_Γ$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that $g_v=0$ for all $v$ and $Γ$ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw ar…

Dehn twists exact sequences through Lagrangian cobordism

In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism theory to the immersed category. As a result, we manifest Seidel's exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodward's family Dehn twist sequence (including the codimension-1 case missing in the literature) as consequences of our surgery/cobordism constructions. Moreover, we obtain an expression of the a…

Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds

We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twi…

3 - 环面的接触手术数

Contact Surgery Numbers of the 3-torus

研究3 - 环面接触结构的接触手术数，证明沿Legendrian博罗梅安环手术得过度扭曲结构且有无限多个非接触同胚结构，给出三分量链环相关障碍，还用特定方法证明\((\mathbb{T}^3, \xi_1)\)斯坦填充的总辛同调不为零。

Contact structures and geometric topology

This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to fillings (Eliashberg, Etnyre), and detection of non-loose Legendrian knots with the help of contact surgery.

Spherical twists and Lagrangian spherical manifolds

We study Dehn twists along Lagrangian submanifolds that are finite quotients of spheres. We decribe the induced auto-equivalences to the derived Fukaya category and explain its relation to twists along spherical functors.

Weinstein流形与扇形的缠绕Fukaya范畴的几何生成

Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

该研究通过几何方法证明2n维Weinstein流形/扇形的缠绕Fukaya范畴由Liouville场n阶临界点不稳定流形生成，证得Seidel关于开闭映射为同构的猜想，并扩展了证明设定。

Monopoles and lens space surgeries

Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a non-trivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a non-vanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations.

浸入弗洛尔上同调在拉格朗日手术下的不变性

Invariance of immersed Floer cohomology under Lagrangian surgery

该研究证明了浸入拉格朗日膜的弗洛尔上同调在特定条件下对拉格朗日手术具有不变性，并推广了相关势不变性结果。

A Holomorphic 0-Surgery Model for Open Books with Application to Cylindrical Contact Homology

We give a simple model in the complex plane of the 0-surgery along a fibered knot of a closed 3-manifold M to yield a mapping torus M'. This model allows explicit relations between pseudoholomorphic curves in the symplectizations of M and M'. As an application we use it to compute the cylindrical contact homology of open books resulting from a positive Dehn twist on a torus with boundary.

Fukaya categories of Lagrangian cobordisms and duality

We introduce a new type of duality structure for $A_\infty$-categories called a relative weak Calabi-Yau pairing which generalizes Kontsevich and Soibelman's notion of a weak (proper) Calabi-Yau structure. We prove the existence of a relative weak Calabi-Yau pairing on Biran and Cornea's Fukaya category of Lagrangian cobordisms $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$. Here $M$ is a symplectic manifold which is closed or tame at infinity. This duality structure on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ extends the relative Poincaré duality satisfied by Floer complexes for…

Koszul duality patterns in Floer theory

We study symplectic invariants of the open symplectic manifolds $X_Γ$ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree $Γ$. For any tree $Γ$, we calculate (DG-)algebra models of the Fukaya category $\mathcal{F}(X_Γ)$ of closed exact Lagrangians in $X_Γ$ and the wrapped Fukaya category $\mathcal{W}(X_Γ)$. When $Γ$ is a Dynkin tree of type $A_n$ or $D_n$ (and conjecturally also for $E_6,E_7,E_8$), we prove that these models for the Fukaya category $\mathcal{F}(X_Γ)$ and $\mathcal{W}(X_Γ)$ are related by (derived) Koszul duality. As an application, we give explicit…