Simple balanced three-manifolds, Heegaard Floer homology and the Andrews-Curtis conjecture

TL;DR

This paper uses Heegaard Floer theory to explore equivalence classes of simple balanced 3-manifolds, providing evidence related to the Andrews-Curtis conjecture by identifying non-trivial examples.

Contribution

It introduces a novel application of Heegaard Floer homology to distinguish simple balanced 3-manifolds from the trivial class, advancing understanding of the Andrews-Curtis conjecture.

Findings

Existence of simple balanced 3-manifolds outside the trivial class

Application of Heegaard Floer tools to manifold equivalence

Progress towards the Andrews-Curtis conjecture

Abstract

The first author introduced a notion of equivalence on a family of $3$-manifolds with boundary, called (simple) balanced $3$-manifolds in an earlier paper and discussed the analogy between the Andrews-Curtis equivalence for group presentations and the aforementioned notion of equivalence. Motivated by the Andrews-Curtis conjecture, we use tools from Heegaard Floer theory to prove that there are simple balanced $3$-manifolds which are not in the trivial equivalence class (i.e. the equivalence class of $S^2\times [-1,1]$).