On spaces with periodic cohomology

Abstract

Abstract. We define a generalized notion of cohomological periodicity for a connected CW-complex X, and show that it is equivalent to the existence of an oriented spherical fibration over X with total space homotopy equivalent to a finite dimensional complex. As applications we characterize discrete groups which can act freely and properly on some R n × S m , show that every rank two p-group acts freely on a homotopy product of two spheres and construct exotic free actions of many simple groups on such spaces.

A quotient of the set [BG,BU(n)] for a finite group G of small rank

“…They look at the Grothendieck group of Vect(BG) and show that it is isomorphic to p||G| R(G p ) G , where R(G p ) is the complex representation ring of G p restricted to thee elements that are stable under the action of G. The present work is intended to be a step toward classifying finite groups that act freely on a finite CW-complex that is homotopy equivalent to the product of two spheres. The classification of such groups began with Adem and Smith [1,2]. Using Theorem 1.3 and this author's thesis [18], a finite group can be shown to act freely on a finite CW-complex that is homotopy equivalent to the product of two spheres by demonstrating the appropriate element of p||G| Char G n (G p ).…”