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

TL;DR

This paper computes the homotopy types of certain localized spheres at the prime 2, confirming parts of the Chromatic Splitting Conjecture and revealing additional summands through analysis of group cohomology.

Contribution

It provides explicit calculations of the homotopy type of localized spheres at p=2, confirming the Chromatic Splitting Conjecture predictions and identifying extra summands, using group cohomology techniques.

Findings

Confirmed all predicted summands by the Chromatic Splitting Conjecture

Discovered additional summands beyond predictions

Simplified group cohomology calculations via inclusion of constants

Abstract

We calculate the homotopy type of $L_1L_{K(2)}S^0$ 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 $L_1L_{K(2)}S^0$ 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 $H^\ast_c(\mathbb{G}_2,E_0)$ 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.