Differentials in the homological homotopy fixed point spectral sequence

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 β ǫ 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 ∞ -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 = M U , 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.

Hopf algebra structure on topological Hochschild homology

“…The present paper advances the algebraic topological foundations for such systematic computations, especially by taking into account the Hopf algebra structure present in the topological Hochschild homology of commutative Salgebras. This program is continued in [BR05], which analyzes the differentials in the homological homotopy fixed point spectral sequence that approximates the cyclic fixed points of topological Hochschild homology, and in [L-N05], which identifies the action by Steenrod operations on the (continuous co-)homology of these fixed-and homotopy fixed point spectra.…”

“…The homotopy groups of an inverse limit are much better behaved. Nonetheless, it may be that future computations of the topological cyclic homology of S -algebras will follow a purely homological approach, see [BR05] and [L-N05].…”

BP-cohomology of projective Stiefel manifolds

“…We attempt to understand F S 1 tr (W n,k + , E) via the homotopy fixed point spectral sequence. For a spectrum Z with S 1 -action, we follow the exposition in [7] replacing homology with homotopy groups. We have a S 1 -equivariant filtration of ES 1 given by…”

Detecting periodic elements in higher topological Hochschild homology

“…Now we want to construct the first two columns of the homotopy fixed points spectral sequence for the group T n . We use the setup in [BR05]. Let the unit sphere S(C ∞ ) be our model for ES 1 with the S 1 -action given by the coordinatewise action.…”