On spaces with periodic cohomology

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 orient ed spherical fibration over X with total space homotopy equivalent to a fin ite dimensional complex. As applications we characterize discrete groups wh ich can act freely and properly on some R-n x S-m, show that every rank two p-group acts freely on a homotopy product of two spheres and construct exo tic free actions of many simple groups on such spaces.