Differentials in the homological homotopy fixed point spectral sequence

Bruner, Robert R; Rognes, John

doi:10.2140/agt.2005.5.653

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Differentials in the homological homotopy fixed point spectral sequence

Robert R Bruner and John Rognes

Algebraic & Geometric Topology 5 (2005) 653–690

DOI: 10.2140/agt.2005.5.653

arXiv: math.AT/0406081

Abstract

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 $E_{s, t}^{2} = H_{g p}^{- s} (T; H_{t} (R; F_{p}))$ , converging conditionally to the continuous homology $H_{s + t}^{c} (R^{h T}; F_{p})$ of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations $β^{ϵ} Q^{i}$ 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 $E^{2 r}$ –term of the spectral sequence there are $2 r$ other classes in the $E^{2 r}$ –term (obtained mostly by Dyer–Lashof operations on $x$ ) that are infinite cycles, ie survive to the $E^{\infty}$ –term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra $R = T H H (B)$ of many $S$ –algebras, including $B = M U$ , $B P$ , $k u$ , $k o$ and $t m f$ . Similar results apply for all finite subgroups $C \subset T$ , 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.

Keywords

homotopy fixed points, Tate spectrum, homotopy orbits, commutative $S$–algebra, Dyer–Lashof operations, differentials, topological Hochschild homology, topological cyclic homology, algebraic $K$–theory

Mathematical Subject Classification 2000

Primary: 19D55, 55S12, 55T05

Secondary: 55P43, 55P91

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Publication

Received: 2 June 2004

Revised: 3 June 2005

Accepted: 21 June 2005

Published: 5 July 2005

Authors

Robert R Bruner
Department of Mathematics Wayne State University Detroit MI 48202 USA

John Rognes
Department of Mathematics University of Oslo NO-0316 Oslo Norway

Volume 5, issue 2 (2005)