The $D(2)$ property for $D_8$

Abstract

Wall’s $D\left(2\right)$ 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 give a complete solution for the case where it is $D_8$ the dihedral group of order 8.