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2016 Profinite and discrete $G\hskip-2pt$–spectra and iterated homotopy fixed points
Daniel Davis, Gereon Quick
Algebr. Geom. Topol. 16(4): 2257-2303 (2016). DOI: 10.2140/agt.2016.16.2257


For a profinite group G, let ()hG, ()hdG and ()hG 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 cG (a closed normal subgroup), we give various conditions for when the iterated homotopy fixed points (XhK)hGK exist and are XhG. For the Lubin–Tate spectrum En and G < cGn, the extended Morava stabilizer group, our results show that EnhK is a profinite GK–spectrum with (EnhK)hGK EnhG; we achieve this by an argument that possesses a certain technical simplicity enjoyed by neither the proof that (EnhK )hGK EnhG nor the Devinatz–Hopkins proof (which requires |GK| < ) of (EndhK)hdGK E ndhG, where EndhK is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the GK–homotopy fixed point spectral sequence for π((EnhK)hGK), with E2s,t = Hcs(GK;πt(EnhK)) (continuous cohomology), is isomorphic to both the strongly convergent Lyndon–Hochschild–Serre spectral sequence of Devinatz for π(EndhG) and the descent spectral sequence for π((EnhK )hGK ).


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Daniel Davis. Gereon Quick. "Profinite and discrete $G\hskip-2pt$–spectra and iterated homotopy fixed points." Algebr. Geom. Topol. 16 (4) 2257 - 2303, 2016.


Received: 29 July 2015; Revised: 13 October 2015; Accepted: 4 November 2015; Published: 2016
First available in Project Euclid: 28 November 2017

zbMATH: 1373.55010
MathSciNet: MR3546465
Digital Object Identifier: 10.2140/agt.2016.16.2257

Primary: 55P42
Secondary: 55S45 , 55T15 , 55T99

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

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 4 • 2016
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