  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.
