Dynamical invariants of mapping torus categories

Fukaya categories of the torus and Dehn surgery

This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded as modules over the Fukaya category of the punctured 2-torus. We extract a short proof of exactness of the Dehn surgery triangle in Heegaard Floer homology. We show that A-infinity structures on the graded algebra A formed by the cohomology of two basic objects in the Fukaya category …

Fukaya categories of two-tori revisited

We construct an A-infinity structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal A-infinity products are obtained as derivatives of those of transversal A-infinity products.

A∞-范畴中的单位性与映射空间

Unitalities and mapping spaces in $A_\infty$-categories

本文在任意基环上证明严格单位与一般单位的A∞-范畴无穷范畴等价，并刻画了dg-范畴与A∞-范畴中内部hom和映射空间的多种模型，推广了Toën与Faonte的工作。

A note on mapping class group actions on derived categories

Let $X_n$ be a cycle of $n$ projective lines, and $\bT_n$ a symplectic torus with $n$ punctures. Using the theory of spherical twists introduced by Seidel and Thomas (2001), I will define an action of the pure mapping class group of $\bT_n$ on $D^b(Coh(X_n))$. The motivation comes from homological mirror symmetry for degenerate elliptic curves, which was studied by the author with Treumann and Zaslow in arXiv:1103.2462 .

Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_ϕ$ associated to a Weinstein domain $M$, and an exact, compactly supported symplectomorphism $ϕ$. $T_ϕ$ is another Weinstein domain and its contact boundary is independent of $ϕ$. In this paper, we distinguish $ϕ$ from $T_{1_M}$, under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and whose symplectic co…

Fukaya categories of surfaces, spherical objects, and mapping class groups

We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least two whose Chern character represents a non-zero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank one local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrap…