## Periodic complexes and group actions.(English)Zbl 0992.55011

A space is said to have periodic cohomology if there is a cohomology class $$\alpha$$, of positive degree, such that taking cup products with $$\alpha$$ gives an isomorphism in sufficiently high degrees. The main result of this paper is a homotopy-theoretic characterization of CW-complexes which have periodic cohomology. The authors enumerate many interesting applications of this. For example, a discrete group has periodic cohomology if, and only if, the group acts freely and properly on a finite dimensional complex that is homotopy equivalent to a sphere. And a finite $$p$$-group acts freely on a finite complex homotopy equivalent to a product of two spheres if, and only if, the group does not contain a subgroup isomorphic to $${\mathbb Z}/p \times {\mathbb Z}/p \times {\mathbb Z}/p$$.

### MSC:

 55P99 Homotopy theory 20J06 Cohomology of groups 57S25 Groups acting on specific manifolds 55N25 Homology with local coefficients, equivariant cohomology 55R25 Sphere bundles and vector bundles in algebraic topology 57Q91 Equivariant PL-topology 57Q05 General topology of complexes

### Keywords:

periodic cohomology; group actions
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