Bordered Heegaard Floer homology: Invariance and pairing

TL;DR

This paper develops a new invariance theory for 3-manifolds with boundary using bordered Heegaard Floer homology, introducing algebraic structures that facilitate decomposition and pairing of manifolds, and relates it to knot Floer homology.

Contribution

It constructs a bordered Heegaard Floer homology framework with algebraic invariants for manifolds with boundary, enabling decomposition and pairing results.

Findings

Defines differential graded algebras for boundary parametrizations

Establishes invariance of the constructed modules up to chain homotopy

Provides a new proof of the surgery exact triangle using bordered Floer homology

Abstract

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.