# Periodic complexes and group actions

### Abstract

In this paper we show that the cohomology of a connected CW-complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on $\mathbb{R}^n \times \mathbb{S}^m$; we construct nonstandard free actions of rank-two simple groups on finite complexes $Y\simeq \mathbb{S}^n \times \mathbb{S}^m$; and we prove that a finite $p$-group $P$ acts freely on such a complex if and only if it does not contain a subgroup isomorphic to $(\mathbb{Z}/p)^3$.