Differentials in the homological homotopy fixed point spectral sequence

Abstract We analyze in homological terms the homotopy fixed point spec-trum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E2s,t = H −s gp (T;Ht(R;Fp)), converging conditionally to the continuous homology Hcs+t(R hT;Fp) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof op-erations βǫQi acting on this algebra spectral sequence, and that its dif-ferentials 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, 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 = MU, BP, ku, ko and tmf. Similar results apply for all finite sub-groups 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.