On CW-complexes over groups with periodic cohomology

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.