Algebraic & Geometric Topology 5 (2005) 653–690

DOI: 10.2140/agt.2005.5.653

arXiv: math.AT/0406081

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 E²_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 β^εQⁱ 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^∞–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.

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

Primary: 19D55, 55S12, 55T05

Secondary: 55P43, 55P91

References

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Publication

Received: 2 June 2004
Revised: 3 June 2005
Accepted: 21 June 2005
Published: 5 July 2005

Authors