Heegaard Floer Homology and Balanced Presentations of Groups

TL;DR

This paper introduces a new Heegaard Floer homology invariant for groups with balanced presentations, showing its independence from certain choices and invariance under specific transformations, advancing the understanding of group invariants.

Contribution

It defines a Heegaard Floer homology for groups with balanced presentations and proves its invariance under stable Andrews-Curtis transformations, providing a novel group invariant.

Findings

$ ext{HF}_P(G)$ is independent of auxiliary choices.

$ ext{HF}_P(G)$ is invariant under stable Andrews-Curtis transformations.

The invariance relies on two unproven claims.

Abstract

Let $G$ be a group with a finite balanced presentation $P$. We associate a Heegaard Floer homology group $\widehat{HF}_P(G)$ with the pair $(G,P)$ based on some extra choices and technical assumptions. We show that $\widehat{HF}_P(G)$ is independent from these choices and also is invariant under stable Andrews-Curtis transformations on $P$, based on two claims which are not settled in this paper.