Twisted Morava K-theory and connective covers of Lie groups

TL;DR

This paper computes twisted Morava K-theory for various classifying spaces and connective covers of Lie groups, revealing that twists either vanish or replicate untwisted homology, extending prior chromatic level results.

Contribution

It extends computational techniques to all connective covers of classical groups and classifying spaces, generalizing previous Morava K-theory results to the twisted setting across all chromatic levels.

Findings

Twisted Morava K-theory either vanishes or equals untwisted homology for natural twists.

Computations include all connective covers of classical groups and their classifying spaces.

Results extend to a variant with mod 2 cohomology.

Abstract

Twisted Morava K-theory, along with computational techniques, including a universal coefficient theorem and an Atiyah-Hirzebruch spectral sequence, was introduced by Craig Westerland and the first author. We employ these techniques to compute twisted Morava K-theory of all connective covers of the stable orthogonal group and stable unitary group, and their classifying spaces, as well as spheres and Eilenberg-MacLane spaces. This extends to the twisted case some of the results of Ravenel and Wilson and of Kitchloo, Laures, and Wilson for Morava K-theory. This also generalizes to all chromatic levels computations by Khorami (and in part those of Douglas) at chromatic level one, i.e. for the case of twisted K-theory. We establish that for natural twists in all cases, there are only two possibilities: either that the twisted Morava homology vanishes, or that it is isomorphic to untwisted homology. We also provide a variant on the twist of Morava K-theory, with mod 2 cohomology in place of integral cohomology.