The topological spherical space form problem—II existence of free actions

Abstract

RECENT advances in calculation of projective class groups and of surgery obstruction groups lead us to hope that it will shortly be possible to give a fairly complete account of the classification of free actions of finite groups on spheres. In the present paper, we determine which groups can so act, thus solving a problem of several years’ standing. Further, we show that these actions can be taken to be smooth actions on smooth homotopy spheres. Previously known results can be summarised as follows, where we say the finite group 7~ satisfies the “pq-condition” (p, q primes not necessarily distinct) if all subgroups if v of order pq are cyclic. 0.1. (Cartan and Eilenberg[3]). If rr acts freely on S”-‘, it has periodic cohomology with minimum period dividing n. Moreover, P has periodic cohomology if and only if it satisfies all p2-conditions. And the p* condition is equivalent to the Sylow p-subgroup zrr, of r being cyclic or perhaps (if p = 2) generalised quaternionic. 0.2. (Wolf [19]). If 7~ acts freely and orthogonally on a sphere, it satisfies all pq-conditions. Conversely, if r is soluble and satisfies all pq-conditions, free orthogonal actions exist. However, for rr non-soluble, the only non-cyclic composition factor allowed is the simple group of order 60. 0.3. (Milnor [9], see also Lee [8]). If 7~ acts freely on any sphere, it satisfies all 2p-conditions. 0.4. (Petrie [I 11). Any extension of a cyclic group of odd order m by a cyclic group of odd prime order q prime to m can act freely on S*“-‘. Petrie’s result shows that pq-conditions are not all necessary for free topological actions. it is therefore not so surprising that