## Algebraic & Geometric Topology

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

Daniel G Davis

#### 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 $H◃K$, then, in general, the $K∕H$–spectrum $XhH$ is not known to be a continuous $K∕H$–spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum $(XhH)hK∕H$. 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 $K∕H$–spectra, there is always an iterated homotopy fixed point spectrum, denoted $(XhH)hδK∕H$, and it is just $XhK$.

Additionally, we show that for any delta-discrete $G$–spectrum $Y$, there is an equivalence $YhδHhδK∕H≃YhδK$. 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 $K∕H$–action, there is always an equivalence $((Xδ)hδH)hδK∕H≃(Xδ)hδK.$ Therefore, delta-discrete $L$–spectra, by letting $L$ equal $H,K,$ and $K∕H$, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete $G$–spectra.

#### Article information

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

Dates
Accepted: 27 September 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715305

Digital Object Identifier
doi:10.2140/agt.2011.11.2775

Mathematical Reviews number (MathSciNet)
MR2846911

Zentralblatt MATH identifier
1230.55006

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715305

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