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) (��;H_t(R;��_p)), converging conditionally to the continuous homology H^c_(s+t)(R^(h��);��_p) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations β^ϵQ^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, 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⊂��, 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.
Algebraic Geometry | Geometry and Topology
Bruner, R. R. & Rognes, J. Differentials in the homological homotopy fixed point spectral sequence. Algebraic & Geometric Topology 5 (2005) 653–690. https://doi.org/10.2140/agt.2005.5.653