The slice spectral sequence for the C4$C_{4}$ analog of real K-theory

Abstract

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