A higher chromatic analogue of the image of J

Westerland, Craig¹

¹ School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN 55455, United States

Geometry & topology, Tome 21 (2017) no. 2, pp. 1033-1093.

Résumé

We prove a higher chromatic analogue of Snaith’s theorem which identifies the $K$–theory spectrum as the localisation of the suspension spectrum of $ℂ{\mathbb{P}}^{\infty }$ away from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of $ℂ{\mathbb{P}}^{\infty }=K\left(\mathbb{Z},2\right)$. Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the $K\left(n\right)$–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 $K\left(n\right)$–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.

Détail

DOI : 10.2140/gt.2017.21.1033

Classification : 19L20, 55N15, 55P20, 55P42, 55Q51
Keywords: chromatic homotopy theory, Picard group, Snaith theorem, redshift conjecture

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