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A higher chromatic analogue of the image of $J$

Craig Westerland

Geometry & Topology 21 (2017) 1033–1093
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 away from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of = K(,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