Periodic complexes and group actions. (English) Zbl 0992.55011

Adem, Alejandro; Smith, Jeff H.

Ann. Math. (2) 154, No. 2, 407-435 (2001).

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 \({\mathbb Z}/p \times {\mathbb Z}/p \times {\mathbb Z}/p\).

Reviewer: Martin D.Crossley (Swansea)

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