Fukaya categories of the torus and Dehn surgery

TL;DR

This paper explores the Fukaya category of the punctured 2-torus, linking it to Heegaard Floer theory for 3-manifolds with boundary, and provides insights into Dehn surgery invariants and A-infinity structures.

Contribution

It introduces the first nontrivial case of extending Heegaard Floer theory to 3-manifolds with boundary using quilted Floer cohomology and analyzes the governing parameters of A-infinity structures.

Findings

Exactness of the Dehn surgery triangle in Heegaard Floer homology

A-infinity structures governed by two parameters (m^6, m^8)

Nonzero m^6 parameter in the Fukaya category

Abstract

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 of the punctured 2-torus are governed by just two parameters (m^6,m^8), extracted from the Hochschild cohomology of A. For the Fukaya category itself, m^6 is nonzero.