Floer homology beyond borders

TL;DR

Bordered Floer homology extends Heegaard Floer theory to 3-manifolds with boundary, offering both conceptual insights and computational tools, with recent developments enhancing its applications.

Contribution

This survey summarizes recent advances in bordered Floer homology, highlighting new theoretical developments and practical computational methods.

Findings

Enhanced computational techniques for 3-manifolds with boundary

New applications in low-dimensional topology

Deeper understanding of Floer homology invariants

Abstract

Bordered Floer homology is an invariant for 3-manifolds with boundary, defined by the authors in 2008. It extends the Heegaard Floer homology of closed 3-manifolds, defined in earlier work of Zolt\'an Szab\'o and the second author. In addition to its conceptual interest, bordered Floer homology also provides powerful computational tools. This survey outlines the theory, focusing on recent developments and applications.