Computing $\widehat{\mathit{HF}}$ by factoring mapping classes

Abstract

Bordered Heegaard Floer homology is an invariant for $3$–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 $\widehat{HF}$ of a closed $3$–manifold, as well as the bordered Floer homology of any $3$–manifold with boundary.