A higher chromatic analogue of the image of J

TL;DR

This paper extends Snaith's theorem to higher chromatic levels, providing new insights into the K(n)-local sphere, Picard groups, and supporting the chromatic redshift conjecture through advanced homotopy theory techniques.

Contribution

It introduces a higher chromatic analogue of Snaith's theorem, generalizes complex orientation to higher Eilenberg-MacLane spaces, and offers partial computations related to the K(n)-local sphere.

Findings

Partial computation of Picard-graded homotopy of the K(n)-local sphere.

Evidence supporting the chromatic redshift conjecture.

Development of a K(n)-local notion generalizing complex orientation.

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.