A higher chromatic analogue of the image of J

Westerland, Craig

doi:10.2140/gt.2017.21.1033

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A higher chromatic analogue of the image of $J$

Craig Westerland

Geometry & Topology 21 (2017) 1033–1093

DOI: 10.2140/gt.2017.21.1033

Abstract

We prove a higher chromatic analogue of Snaith’s theorem which identifies the $K$ –theory spectrum as the localisation of the suspension spectrum of $ℂ ℙ^{\infty}$ away from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of $ℂ ℙ^{\infty} = K (ℤ, 2)$ . Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the $K (n)$ –local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross–Hopkins dual of the sphere. Our main technical tool is a $K (n)$ –local notion generalising complex orientation to higher Eilenberg–MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.

Keywords

chromatic homotopy theory, Picard group, Snaith theorem, redshift conjecture

Mathematical Subject Classification 2010

Primary: 19L20, 55N15, 55P20, 55P42, 55Q51

References

Publication

Received: 2 September 2015

Accepted: 17 January 2016

Published: 17 March 2017

Proposed: Mark Behrens

Seconded: Ralph Cohen, Haynes Miller

Authors

Craig Westerland
School of Mathematics University of Minnesota 127 Vincent Hall 206 Church St SE Minneapolis, MN 55455 United States

Volume 21, issue 2 (2017)