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

By M. Hill, M. Hopkins, D. Ravenel. 2015

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We describe the slice spectral sequence of a 32-periodic $C_{4}$-spectrum $K_{[2]}$ 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 $\underline{\pi }_{*}K_{[2]}$, 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 $K_{[2]}$ is 256-periodic and detects the Kervaire invariant classes $\theta_{j}$ in the stable homotopy groups of spheres. 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 $j\geq 7$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3e0cbc3ea2d425f3cc10538a915d1ee6b1a28b51