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Delta-discrete $G$–spectra and iterated homotopy fixed points

Daniel G Davis

Algebraic & Geometric Topology 11 (2011) 2775–2814

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 H K, 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 Y hδHhδKH Y hδK. 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 (X δ)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.

homotopy fixed point spectrum, discrete $G$–spectrum, iterated homotopy fixed point spectrum
Mathematical Subject Classification 2010
Primary: 55P42, 55P91
Received: 13 June 2010
Accepted: 27 September 2010
Published: 26 September 2011
Daniel G Davis
Department of Mathematics
University of Louisiana at Lafayette
Lafayette LA 70504