Computing HF^ by factoring mapping classes

TL;DR

This paper provides explicit combinatorial descriptions of bimodules associated with arc slides in bordered Heegaard Floer homology, enabling computation of HF^ for closed and bounded 3-manifolds.

Contribution

It introduces explicit formulas for bimodules related to arc slides, advancing the computational framework of bordered Heegaard Floer homology.

Findings

Explicit bimodule descriptions for arc slides

Combinatorial formulas for HF^ of closed 3-manifolds

Extension to bordered Floer homology of manifolds with boundary

Abstract

Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these bimodules for arc slides explicitly, and then use them to give a combinatorial description of HF^ of a closed three-manifold, as well as the bordered Floer homology of any 3-manifold with boundary.