Chromatic splitting for the K.2/–local sphere at p D 2

Agnès Beaudry; Paul G. Goerss; Hans Werner Henn

doi:10.2140/gt.2022.26.377

Chromatic splitting for the K.2/–local sphere at p D 2

Agnès Beaudry, Paul G. Goerss, Hans Werner Henn

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We calculate the homotopy type of L₁L_K.2/S⁰ and L_K.1/L_K.2/S⁰ at the prime 2, where L_K.n/ is localization with respect to Morava K–theory and L₁ localization with respect to 2–local K–theory. In L₁L_K.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 D W ŒŒu₁ 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.

Original language	English (US)
Pages (from-to)	377-476
Number of pages	100
Journal	Geometry and Topology
Volume	26
Issue number	1
DOIs	https://doi.org/10.2140/gt.2022.26.377
State	Published - 2022

ASJC Scopus subject areas

Geometry and Topology

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title = "Chromatic splitting for the K.2/–local sphere at p D 2",

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 D W {\OE}{\OE}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.",

author = "Agn{\`e}s Beaudry and Goerss, {Paul G.} and Henn, {Hans Werner}",

note = "Funding Information: This material is based upon work supported by ANR-16-CE40-0003 ChroK and by the National Science Foundation under grants DMS-1308916 and DMS-1612020/1725563. Publisher Copyright: {\textcopyright} 2022, Mathematical Sciences Publishers. All rights reserved.",

year = "2022",

doi = "10.2140/gt.2022.26.377",

language = "English (US)",

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pages = "377--476",

journal = "Geometry and Topology",

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publisher = "University of Warwick",

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T1 - Chromatic splitting for the K.2/–local sphere at p D 2

AU - Beaudry, Agnès

AU - Goerss, Paul G.

AU - Henn, Hans Werner

N1 - Funding Information: This material is based upon work supported by ANR-16-CE40-0003 ChroK and by the National Science Foundation under grants DMS-1308916 and DMS-1612020/1725563. Publisher Copyright: © 2022, Mathematical Sciences Publishers. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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 D W ŒŒu1 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.

AB - 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 D W ŒŒu1 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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Chromatic splitting for the K.2/–local sphere at p D 2

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