The chromatic splitting conjecture at n = p = 2

Beaudry, Agnès

doi:10.2140/gt.2017.21.3213

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The chromatic splitting conjecture at $n=p=2$

Agnès Beaudry

Geometry & Topology 21 (2017) 3213–3230

DOI: 10.2140/gt.2017.21.3213

Abstract

We show that the strongest form of Hopkins’ chromatic splitting conjecture, as stated by Hovey, cannot hold at chromatic level $n = 2$ at the prime $p = 2$ . More precisely, for $V (0)$ , the mod $2$ Moore spectrum, we prove that $π_{k} L_{1} L_{K (2)} V (0)$ is not zero when $k$ is congruent to $- 3$ modulo $8$ . We explain how this contradicts the decomposition of $L_{1} L_{K (2)} S$ predicted by the chromatic splitting conjecture.

Keywords

K(2)-local, stable homotopy theory, Morava K-theory, chromatic assembly

Mathematical Subject Classification 2010

Primary: 55P60, 55Q45

References

Publication

Received: 3 April 2015

Revised: 7 December 2016

Accepted: 19 January 2017

Published: 31 August 2017

Proposed: Mark Behrens

Seconded: Stefan Schwede, Ralph Cohen

Authors

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

Volume 21, issue 6 (2017)