Algebraic & Geometric Topology

Differentials in the homological homotopy fixed point spectral sequence

Robert R Bruner and John Rognes

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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 Es,t2=Hgps(T;Ht(R;Fp)), converging conditionally to the continuous homology Hs+tc(RhT;Fp) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations βϵQi 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 E2r–term of the spectral sequence there are 2r other classes in the E2r–term (obtained mostly by Dyer–Lashof operations on x) that are infinite cycles, ie 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=MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups CT, 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.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 2 (2005), 653-690.

Dates
Received: 2 June 2004
Revised: 3 June 2005
Accepted: 21 June 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796425

Digital Object Identifier
doi:10.2140/agt.2005.5.653

Mathematical Reviews number (MathSciNet)
MR2153113

Zentralblatt MATH identifier
1078.19003

Subjects
Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55S12: Dyer-Lashof operations 55T05: General
Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55P91: Equivariant homotopy theory [See also 19L47]

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

Citation

Bruner, Robert R; Rognes, John. Differentials in the homological homotopy fixed point spectral sequence. Algebr. Geom. Topol. 5 (2005), no. 2, 653--690. doi:10.2140/agt.2005.5.653. https://projecteuclid.org/euclid.agt/1513796425


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