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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


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 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.

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
Forward citations
Received: 2 June 2004
Revised: 3 June 2005
Accepted: 21 June 2005
Published: 5 July 2005
Robert R Bruner
Department of Mathematics
Wayne State University
Detroit MI 48202
John Rognes
Department of Mathematics
University of Oslo
NO-0316 Oslo