2004/06/30 by Robert R. Bruner, Robert R Bruner, John Rognes · 1 citation
Computer Science · Mathematics · #Commutative property #Fixed point #Hochschild homology #Homological algebra #Homology (biology) #Homotopy #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #Spectral sequence #Topological and Geometric Data Analysis #Whitehead theorem #math.AT #msc:19D55 #msc:55P43 #msc:55P91 #msc:55S12 #msc:55T05
paper · pdf · doi:10.2140/agt.2005.5.653
published as Algebr. Geom. Topol. 5 (2005) 653-690 · Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-27.abs.html
openalex publication_date 2005/07/05 · arxiv created 2005/07/14 · arxiv updated 2014/09/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S -algebra R. There is a homological homotopy fixed point spectral sequence with E 2 s,t = H -s gp (T; H t (R; F p )), converging conditionally to the continuous homology H c s+t (R hT ; F p ) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations Q i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E 2r -term of the spectral sequence there are 2r other classes in the E 2r -term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E -term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH (B) of many S -algebras, including B = M U , BP , ku, ko and tmf . Similar results apply for all finite subgroups C T, and for the Tate-and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K -theory of commutative S -algebras.