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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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

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 ).

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
Received: 29 July 2015
Revised: 13 October 2015
Accepted: 4 November 2015
Published: 12 September 2016
Daniel G Davis
Department of Mathematics
University of Louisiana at Lafayette
1403 Johnston Street
Maxim Doucet Hall, Room 217
Lafayette, LA 70504-3568
Gereon Quick
Department of Mathematical Sciences
Norwegian University of Science and Technology
7491 Trondheim