Algebraic & Geometric Topology

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

Daniel G Davis

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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.

Article information

Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2775-2814.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory [See also 19L47]

homotopy fixed point spectrum discrete $G$–spectrum iterated homotopy fixed point spectrum


Davis, Daniel G. Delta-discrete $G$–spectra and iterated homotopy fixed points. Algebr. Geom. Topol. 11 (2011), no. 5, 2775--2814. doi:10.2140/agt.2011.11.2775.

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