#### Algebraic and Geometric Topology 5 (2005),
paper no. 27, pages 653-690.

## Differentials in the homological homotopy fixed point spectral sequence

### Robert R. Bruner, John Rognes

**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^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 beta^epsilon 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^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(B) of many S-algebras, including B = MU, BP, ku, ko and
tmf. Similar results apply for all finite subgroups C of 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

**AMS subject classification**.
Primary: 19D55, 55S12, 55T05 .
Secondary: 55P43, 55P91 .

**E-print:** `arXiv:math.AT/0406081`

**DOI:** 10.2140/agt.2005.5.653

Submitted: 2 June 2004.
(Revised: 3 June 2005.)
Accepted: 21 June 2005.
Published: 5 July 2005.

Notes on file formats
Robert R. Bruner, John Rognes

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

and

Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway

Email: rrb@math.wayne.edu, rognes@math.uio.no

AGT home page

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