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2011 Delta-discrete $G$–spectra and iterated homotopy fixed points
Daniel G Davis
Algebr. Geom. Topol. 11(5): 2775-2814 (2011). DOI: 10.2140/agt.2011.11.2775

Abstract

Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G–spectrum. If H and K are closed subgroups of G, with HK, then, in general, the KH–spectrum XhH is not known to be a continuous KH–spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (XhH)hKH. To address this situation, we define homotopy fixed points for delta-discrete G–spectra and show that the setting of delta-discrete G–spectra gives a good framework within which to work. In particular, we show that by using delta-discrete KH–spectra, there is always an iterated homotopy fixed point spectrum, denoted (XhH)hδKH, and it is just XhK.

Additionally, we show that for any delta-discrete G–spectrum Y, there is an equivalence YhδHhδKHYhδK. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G–spectrum Xδ that is equivalent to X and, though XhH is not even known in general to have a KH–action, there is always an equivalence ((Xδ)hδH)hδKH(Xδ)hδK. Therefore, delta-discrete L–spectra, by letting L equal H,K, and KH, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G–spectra.

Citation

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Daniel G Davis. "Delta-discrete $G$–spectra and iterated homotopy fixed points." Algebr. Geom. Topol. 11 (5) 2775 - 2814, 2011. https://doi.org/10.2140/agt.2011.11.2775

Information

Received: 13 June 2010; Accepted: 27 September 2010; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1230.55006
MathSciNet: MR2846911
Digital Object Identifier: 10.2140/agt.2011.11.2775

Subjects:
Primary: 55P42, 55P91

Rights: Copyright © 2011 Mathematical Sciences Publishers

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Vol.11 • No. 5 • 2011
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