Differentials in the homological homotopy fixed point spectral sequence

Bruner, Robert R¹ ; Rognes, John²

¹Department of Mathematics, Wayne State University, Detroit MI 48202, USA
²Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway

Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 653-690

Résumé

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.

Détail

DOI : 10.2140/agt.2005.5.653

Keywords: homotopy fixed points, Tate spectrum, homotopy orbits, commutative $S$–algebra, Dyer–Lashof operations, differentials, topological Hochschild homology, topological cyclic homology, algebraic $K$–theory

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