Morse theory for group presentations

TL;DR

This paper introduces a new combinatorial approach using refined discrete Morse theory to analyze group presentations and CW-complexes, providing tools to verify the Andrews--Curtis conjecture for certain cases.

Contribution

It develops a novel combinatorial method based on refined discrete Morse theory to study 3-deformations of CW-complexes and group presentations, with explicit homotopy equivalences.

Findings

Some potential counterexamples to the Andrews--Curtis conjecture satisfy the conjecture.

The method provides explicit descriptions of attaching maps in Morse complexes.

Bounds on the dimension of complexes involved in deformations are established.

Abstract

We introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplified Morse CW-complex, with an explicit description of the attaching maps and bounds on the dimension of the complexes involved in the deformation. We apply this technique to show that some known potential counterexamples to the Andrews--Curtis conjecture do satisfy the conjecture.