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This article is available for purchase or by subscription. See below.
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 L1LK(2)S0 and LK(1)LK(2)S0 at the prime 2, where LK(n) is localization with respect to Morava K–theory and L1 localization with respect to 2–local K–theory. In L1LK(2)S0 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(𝔾2,E0), where 𝔾2 is the Morava stabilizer group and E0 = 𝕎[[u1]] is the ring of functions on the height 2 Lubin–Tate space. We show that the inclusion of the constants 𝕎 E0 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