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.

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

References in corpus (1)

• On Hopkins' Picard groups for the prime 3 and chromatic level 2

Cited by in corpus (5)

• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
• Chromatic Cyclotomic Extensions
• Coalgebraic Formal Curve Spectra and the Annular Tower
• Descent and cyclotomic redshift for chromatically localized algebraic K-theory
• Thom spectra, higher $THH$ and tensors in $\infty$-categories