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 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 $\mathbb R^n\times \mathbb S^m$, 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.