Introduction

It was realized by Killing, Klein and others that the sphere and real projective space offered two different global models for geometry which were locally the same. In 1891, Killing formulated the problem of determining all such models. In 1926 Hopf revived the problem and also raised the more general question of studying manifolds covered by spheres, hence groups which act freely and properly on spheres.

If a group acts freely and properly on a sphere then the group is finite, since the sphere is a compact space, and it has periodic (Tate) cohomology [C-E, Ch. XVI §9]. The structure of finite groups with periodic cohomology is well known and is essentially based on the fact that if a group G has periodic cohomology then so does every subgroup of G, and if Cp is the cyclic group of order a prime p, then Cp × Cp does not have periodic cohomology. So if a finite group G has periodic cohomology then G does not contain subgroups of the form Cp × Cp for any prime p. It turns out that this is a sufficient condition for a finite group G to have periodic cohomology [Artin-Tate (unpublished), [C-E, Ch. XII] or [B, Ch. VI]].