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Equivariant homotopy theory for pro–spectra
Halvard Fausk
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Geometry and Topology 12 (2008)
103–176
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Abstract
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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.
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Keywords
equivariant homotopy, pro-spectra,
profinite groups
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Mathematical Subject Classification
Primary: 55P91
Secondary: 18G55
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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
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