The slice spectral sequence for the $C_4$ analog of real $K$-theory

Abstract

We describe the slice spectral sequence of a 32-periodic $C_4$-spectrum $K_{[2]}$ related to the $C_4$ norm $N_{C_2}^{C_4}M{U}_{\mathbf{R}}$ of the real cobordism spectrum $M{U}_{\mathbf{R}}$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor ${\underline{\pi }}_{\ast }{K}_{[2]}$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real $K$-theory spectrum $K_{\mathbf{R}}$ 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 $j\ge 7$.

82 pages, 4 tables and 17 figures, some using color

References in corpus (1)

• On the non-existence of elements of Kervaire invariant one