#### Volume 26, issue 1 (2022)

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

### Agnès Beaudry, Paul G Goerss and Hans-Werner Henn

Geometry & Topology 26 (2022) 377–476
DOI: 10.2140/gt.2022.26.377
##### Abstract

We calculate the homotopy type of ${L}_{1}{L}_{K\left(2\right)}{S}^{0}$ and ${L}_{K\left(1\right)}{L}_{K\left(2\right)}{S}^{0}$ at the prime 2, where ${L}_{K\left(n\right)}$ is localization with respect to Morava $K$–theory and ${L}_{1}$ localization with respect to $2$–local $K$–theory. In ${L}_{1}{L}_{K\left(2\right)}{S}^{0}$ 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 ${H}^{\ast }\left({\mathbb{𝔾}}_{2},{E}_{0}\right)$, where ${\mathbb{𝔾}}_{2}$ is the Morava stabilizer group and ${E}_{0}=\mathbb{𝕎}\left[\left[{u}_{1}\right]\right]$ 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.

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##### Keywords
chromatic splitting conjecture, chromatic homotopy theory, Morava K–theory localization of the sphere
##### Mathematical Subject Classification
Primary: 55P42, 55P60, 55Q51