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

TL;DR

This paper analyzes the slice spectral sequence of a 32-periodic $C_{4}$-spectrum related to real cobordism, providing detailed differentials and extensions, and connects to the Kervaire invariant problem.

Contribution

It offers a detailed description of the slice spectral sequence for a specific $C_{4}$-spectrum, extending previous work on real $K$-theory and contributing to the understanding of Kervaire invariant classes.

Findings

Complete spectral sequence with differentials and extensions provided.

Connected spectral sequence analysis to the Kervaire invariant problem.

Extended understanding of $C_{4}$-spectra related to real cobordism.

Abstract

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 $j\geq 7$.