# On CW-complexes over groups with periodic cohomology

## TL;DR

This paper investigates CW-complexes over groups with periodic cohomology, establishing conditions under which D2 complexes are classified by Euler characteristic and solving Wall's D2 problem for certain non-abelian groups, with implications for Poincaré complexes.

## Contribution

It provides a classification criterion for D2 complexes over groups with 4-periodic cohomology and solves Wall's D2 problem for specific non-abelian groups, also showing existence of single 3-cell structures.

## Key findings

- D2 complexes over groups with 4-periodic cohomology are classified by Euler characteristic under certain conditions.
- Wall's D2 problem is solved for several families of non-abelian groups.
- Finite Poincaré 3-complexes over these groups admit a single 3-cell structure.

## Abstract

If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Wall's D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincar\'{e} $3$-complex $X$ with $\pi_1(X)=G$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $\mathbb{Z} G$ modules where $G$ has periodic cohomology.