Twisted Morava K–theory and connective covers of Lie groups

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 (J. Topol. 8 (2015) 887–916). 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–Mac Lane spaces. This extends to the twisted case some of the results of Ravenel and Wilson (Amer. J. Math. 102 (1980) 691–748) and Kitchloo, Laures and Wilson (Adv. Math. 189 (2004) 192–236) for Morava $K$–theory. This also generalizes to all chromatic levels computations by Khorami (J. Topol. 4 (2011) 535–542) (and in part those of Douglas in Topology 45 (2006) 955–988) at chromatic level one, ie for the case of twisted $K$–theory. We establish that for natural twists in all cases, there are only two possibilities: either the twisted Morava homology vanishes, or 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.