#### Volume 12, issue 1 (2008)

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Equivariant homotopy theory for pro–spectra

### Halvard Fausk

Geometry & Topology 12 (2008) 103–176
##### Abstract

We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The $G$–homotopy theory is “pieced together” from the $G∕U$–homotopy theories for suitable quotient groups $G∕U$ of $G$; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In the model category of equivariant spectra Postnikov towers are studied from a general perspective. We introduce pro–$G$–spectra and construct various model structures on them. A key property of the model structures is that pro–spectra are weakly equivalent to their Postnikov towers. We discuss two versions of a model structure with “underlying weak equivalences”. One of the versions only makes sense for pro–spectra. In the end we use the theory to study homotopy fixed points of pro–$G$–spectra.

##### Keywords
equivariant homotopy, pro-spectra, profinite groups
Primary: 55P91
Secondary: 18G55