Fukaya categories and bordered Heegaard-Floer homology

TL;DR

This paper explores the relationship between Fukaya categories and bordered Heegaard-Floer homology, providing a symplectic topology perspective on 3-manifold invariants and connecting algebraic and geometric approaches.

Contribution

It offers a new interpretation of bordered Heegaard-Floer homology using Fukaya categories of symmetric products, bridging symplectic topology and low-dimensional topology.

Findings

Recasts bordered Heegaard-Floer homology in terms of Fukaya categories.

Establishes a connection between symplectic topology and 3-manifold invariants.

Provides a geometric perspective on algebraic structures in Heegaard-Floer theory.

Abstract

We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In particular we discuss the connection between the Fukaya category of the symmetric product and the bordered algebra introduced by Robert Lipshitz, Peter Ozsvath and Dylan Thurston, and recast bordered Heegaard-Floer homology in this language.