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

Beaudry, Agnès; Goerss, Paul G; Henn, Hans-Werner

doi:10.2140/gt.2022.26.377

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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 $L_{1} L_{K (2)} S^{0}$ 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 $H^{*} (𝔾_{2}, E_{0})$ , where $𝔾_{2}$ is the Morava stabilizer group and $E_{0} = 𝕎 [[u_{1}]]$ is the ring of functions on the height $2$ Lubin–Tate space. We show that the inclusion of the constants $𝕎 \to E_{0}$ 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

References

Publication

Received: 8 October 2020

Revised: 18 February 2021

Accepted: 23 March 2021

Published: 5 April 2022

Proposed: Jesper Grodal

Seconded: Haynes R Miller, Stefan Schwede

Authors

Agnès Beaudry
Department of Mathematics University of Colorado Boulder Boulder, CO United States

Paul G Goerss
Department of Mathematics Northwestern University Evanston, IL United States

Hans-Werner Henn
Institut de Recherche Mathématique Avancée CNRS Université de Strasbourg Strasbourg France

Volume 26, issue 1 (2022)