#### Volume 21, issue 6 (2017)

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

### Agnès Beaudry

Geometry & Topology 21 (2017) 3213–3230
##### 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 $V\left(0\right)$, the mod $2$ Moore spectrum, we prove that ${\pi }_{k}{L}_{1}{L}_{K\left(2\right)}V\left(0\right)$ is not zero when $k$ is congruent to $-3$ modulo $8$. We explain how this contradicts the decomposition of ${L}_{1}{L}_{K\left(2\right)}S$ 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
##### Publication
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