Chromatic splitting for the $K(2)$-local sphere at $p=2$

Abstract

We calculate the homotopy type of $L_1{L}_{K(2)}{S}^0$ and $L_{K(1)}{L}_{K(2)}{S}^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$-theory and $L_1$ localization with respect to $2$-local $K$ theory. In $L_1{L}_{K(2)}{S}^0$ we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology $H_c^{\ast }({\mathbb{G}}_2,E_0)$ where ${\mathbb{G}}_2$ is the Morava stabilizer group and $E_0=\mathbb{W}[[u_1]]$ is the ring of functions on the height $2$ Lubin-Tate space. We show that the inclusion of the constants $\mathbb{W}\to E_0$ induces an isomorphism on group cohomology, a radical simplification.

Clarifications, details, and explanations have been added throughout the paper. There are a few new minor results (e.g. Corollary 9.1.9), but there have been no significant changes to the main results and proofs

References in corpus (2)

• Topological resolutions in K(2)-local homotopy theory at the prime 2
• Finite subgroups of extended Morava stabilizer groups

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