On Cw-Complexes Over Groups With Periodic Cohomology

( )If G has 4-periodic cohomology, then finite 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 Poincare 3-complex X with G = pi(1)(X) admits a cell structure with a single 3-cell. The proof involves cancellation theorems for ZG modules where G has periodic cohomology.