The D(2) property for $D_8$

TL;DR

This paper investigates Wall's D(2) problem for certain dihedral groups, providing partial and complete solutions for groups of order 2^n, specifically focusing on D_8.

Contribution

It offers a partial solution for dihedral groups of order 2^n and fully resolves the case for D_8, advancing understanding of the D(2) property in geometric topology.

Findings

Partial solution for dihedral groups of order 2^n

Complete solution for D_8 case

Advances in understanding the D(2) property for specific groups

Abstract

Wall's D(2) problem asks if a cohomologically 2-dimensional geometric 3-complex is necessarily homotopy equivalent to a geometric 2-complex. We solve part of the problem when the fundamental group is dihedral of order $2^n$, and offer a complete solution for the case where it is $D_8$ the dihedral group of order 8.