#### Volume 5, issue 2 (2005)

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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
 arXiv: math.AT/0406081
##### Abstract

We analyze in homological terms the homotopy fixed point spectrum of a $\mathbb{T}$–equivariant commutative $S$–algebra $R$. There is a homological homotopy fixed point spectral sequence with ${E}_{s,t}^{2}={H}_{gp}^{-s}\left(\mathbb{T};{H}_{t}\left(R;{\mathbb{F}}_{p}\right)\right)$, converging conditionally to the continuous homology ${H}_{s+t}^{c}\left({R}^{h\mathbb{T}};{\mathbb{F}}_{p}\right)$ of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations ${\beta }^{ϵ}{Q}^{i}$ 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 ${E}^{2r}$–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, 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 $R=THH\left(B\right)$ of many $S$–algebras, including $B=MU$, $BP$, $ku$, $ko$ and $tm\phantom{\rule{0.3em}{0ex}}f$. Similar results apply for all finite subgroups $C\subset \mathbb{T}$, 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
##### 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