The decategorification of bordered Heegaard Floer homology

TL;DR

This paper explores the Grothendieck group of modules over the algebra associated with bordered Heegaard Floer homology, defining an invariant that relates to the homology of 3-manifolds and categorifies the Alexander polynomial formula for satellites.

Contribution

It introduces a new invariant in the Grothendieck group framework for bordered Floer homology modules, linking it to classical topological invariants and gluing properties.

Findings

Invariant recovers the kernel of the boundary homology inclusion when finite

Invariant is zero otherwise

Categorifies the Alexander polynomial formula for satellites

Abstract

Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism \phi from F to \bdy Y, a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invariant lying in this group for every bordered 3-manifold. We prove that this invariant recovers the kernel of the inclusion of H_1(\bdy Y; Z) into H_1(Y; Z) if H_1(Y, \bdy Y; Z) is finite, and is 0 otherwise. We also study the properties of this invariant corresponding to gluing. As one application, we show that the pairing theorem for bordered Floer homology categorifies the classical Alexander polynomial formula for satellites.