Exotic presentations of quaternion groups and Wall's D2 problem

TL;DR

This paper investigates the D2 problem related to quaternion groups, demonstrating that a specific complex is not a counterexample and introducing new presentations with distinct homotopy types.

Contribution

It shows a known quaternion group complex is homotopy equivalent to a known presentation, and introduces new presentations with unique homotopy types.

Findings

The Cohen-Dyer complex is homotopy equivalent to a Mannan-Popiel presentation.

An infinite family of quaternion group presentations with novel homotopy types.

The results clarify the structure of potential counterexamples to the D2 problem.

Abstract

The D2 problem of C. T. C. Wall asks whether every finite cohomologically 2-dimensional CW-complex is homotopy equivalent to a finite 2-complex. Several potential counterexamples have been proposed, the longest standing of which is a CW-complex constructed by Cohen and Dyer whose fundamental group is a quaternion group of order 32. We show that this CW-complex is homotopy equivalent to the presentation 2-complex of a presentation constructed by Mannan-Popiel, thus showing it is not a counterexample to the D2 problem. We next introduce an infinite family of presentations for a quaternion group of order $4n$ and prove that they achieve homotopy types which are not achieved by the presentations of Mannan-Popiel.