## Algebraic & Geometric Topology

### Differentials in the homological homotopy fixed point spectral sequence

#### 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 $Es,t2=Hgp−s(T;Ht(R;Fp))$, 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 $C⊂T$, 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.

#### Article information

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

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

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

#### 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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