A higher chromatic analogue of the image of J
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 CP^\infty away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the role of CP^\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
Cite
@article{arxiv.1210.2472,
title = {A higher chromatic analogue of the image of J},
author = {Craig Westerland},
journal= {arXiv preprint arXiv:1210.2472},
year = {2017}
}
Comments
43 pages, comments welcome. Version 4: further clarified sign character, streamlined arguments with homotopy fixed point spectra, added a new section on the monochromatic J-homomorphism