Profinite and discrete G–spectra and iterated homotopy fixed points

Davis, Daniel; Quick, Gereon

doi:10.2140/agt.2016.16.2257

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Profinite and discrete $G\hskip-2pt$–spectra and iterated homotopy fixed points

Daniel G Davis and Gereon Quick

Algebraic & Geometric Topology 16 (2016) 2257–2303

DOI: 10.2140/agt.2016.16.2257

Abstract

For a profinite group $G$ , let ${(-)}^{h G}$ , ${(-)}^{h_{d} G}$ and ${(-)}^{h^{'} G}$ denote continuous homotopy fixed points for profinite $G$ –spectra, discrete $G$ –spectra and continuous $G$ –spectra (coming from towers of discrete $G$ –spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K ◃_{c} G$ (a closed normal subgroup), we give various conditions for when the iterated homotopy fixed points ${(X^{h K})}^{h G ∕ K}$ exist and are $X^{h G}$ . For the Lubin–Tate spectrum $E_{n}$ and $G <_{c} G_{n}$ , the extended Morava stabilizer group, our results show that $E_{n}^{h K}$ is a profinite $G ∕ K$ –spectrum with ${(E_{n}^{h K})}^{h G ∕ K} ≃ E_{n}^{h G}$ ; we achieve this by an argument that possesses a certain technical simplicity enjoyed by neither the proof that ${(E_{n}^{h^{'} K})}^{h^{'} G ∕ K} ≃ E_{n}^{h^{'} G}$ nor the Devinatz–Hopkins proof (which requires $| G ∕ K | < \infty$ ) of ${(E_{n}^{d h K})}^{h_{d} G ∕ K} ≃ E_{n}^{d h G}$ , where $E_{n}^{d h K}$ is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the $G ∕ K$ –homotopy fixed point spectral sequence for $π_{*} ({(E_{n}^{h K})}^{h G ∕ K})$ , with $E_{2}^{s, t} = H_{c}^{s} (G ∕ K; π_{t} (E_{n}^{h K}))$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon–Hochschild–Serre spectral sequence of Devinatz for $π_{*} (E_{n}^{d h G})$ and the descent spectral sequence for $π_{*} ({(E_{n}^{h^{'} K})}^{h^{'} G ∕ K})$ .

Keywords

profinite $G$–spectrum, homotopy fixed point spectrum, iterated homotopy fixed point spectrum, Lubin–Tate spectrum, descent spectral sequence

Mathematical Subject Classification 2010

Primary: 55P42

Secondary: 55S45, 55T15, 55T99

References

Publication

Received: 29 July 2015

Revised: 13 October 2015

Accepted: 4 November 2015

Published: 12 September 2016

Authors

Daniel G Davis
Department of Mathematics University of Louisiana at Lafayette 1403 Johnston Street Maxim Doucet Hall, Room 217 Lafayette, LA 70504-3568 USA
http://www.ucs.louisiana.edu/~dxd0799/
Gereon Quick
Department of Mathematical Sciences Norwegian University of Science and Technology 7491 Trondheim Norway
http://www.math.ntnu.no/~gereonq/

Volume 16, issue 4 (2016)