A higher chromatic analogue of the image of J

作者：Westerland Craig^*

来源：Geometry and Topology, 2017, 21(2): 1033-1093.

DOI：10.2140/gt.2017.21.1033

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 infinity away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the role of CP infinity = 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.

• 出版日期2017