Equivariant homotopy theory for pro–spectra

Fausk, Halvard

doi:10.2140/gt.2008.12.103

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Equivariant homotopy theory for pro–spectra

Halvard Fausk

Geometry & Topology 12 (2008) 103–176

DOI: 10.2140/gt.2008.12.103

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

Mathematical Subject Classification 2000

Primary: 55P91

Secondary: 18G55

References

Publication

Received: 20 December 2006

Revised: 16 April 2007

Accepted: 23 July 2007

Published: 8 February 2008

Proposed: Haynes Miller

Seconded: Tom Goodwillie, Paul Goerss

Authors

Halvard Fausk
Department of Mathematics University of Oslo 1053 Blindern, 0316 Oslo Norway

Volume 12, issue 1 (2008)