The slice spectral sequence for the $C_{4}$ analog of real $K$-theory

M. A. Hill, M. J. Hopkins, D. C. Ravenel

Published 2015-02-26Version 1

We describe the slice spectral sequence of a 32-periodic $C_{4}$-spectrum $K_{\bf H}$ related to the $C_{4}$ norm ${N_{C_{2}}^{C_{4}}MU_{\bf R}}$ of the real cobordism spectrum $MU_{\bf R}$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor $\underline{\pi }_{*}K_{\bf H}$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real $K$-theory spectrum $K_{\bf R}$ was first analyzed by Dugger. The $C_{8}$ analog of $K_{\bf H}$ is 256-periodic and detects the Kervaire invariant classes $\theta_{j}$. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that $\theta_{j}$ does not exist for $j\geq 7$.