This paper contains results on the Tate cohomology of finite groups, with applications to the homotopy theory of 4-manifolds. Let \(\pi\) be a finite group with periodic cohomology of period 4. The authors give isomorphisms between the Tate cohomology of \(\pi\) for integral coefficients and for certain coefficients involving Whitehead’s \(\Gamma\) functor. Let \(X\) be a compact 4-manifold with fundamental group \(\pi\) of the type described. The authors give results on the homotopy groups of \(X\) and, using a generalised spectral sequence, on the group of self-homotopy equivalences.