Differentials in the homological homotopy fixed point spectral sequence

Abstract

We analyze in homological terms the homotopy fixed point spectrum of a $\mathbb{T}$–equivariant commutative $S$–algebra $R$. There is a homological homotopy fixed point spectral sequence with $E_{s,t}^2=H_{gp}^{-s}\left(\mathbb{T};H_t\left(R;{\mathbb{F}}_p\right)\right)$, converging conditionally to the continuous homology $H_{s+t}^c\left(R^{h\mathbb{T}};{\mathbb{F}}_p\right)$ of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations ${\beta }^{ϵ}{Q}^i$ 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, ie survive to the $E^{\infty }$–term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra $R=THH\left(B\right)$ of many $S$–algebras, including $B=MU$, $BP$, $ku$, $ko$ and $tmf$. Similar results apply for all finite subgroups $C\subset \mathbb{T}$, 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.