Periodic complexes and group actions

In this paper we show that the cohomology of a connected CW-complex is peri odic if and only if it is the base space of a spherical fibration with tota l space that is homotopically finite dimensional. As applications we charac terize those discrete groups that act freely and properly on R-n x S-m; we construct nonstandard free actions of rank-two simple groups on finite comp lexes Y similar or equal to S-n x 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 subgrou p isomorphic to (Z/p)(3).