Twisted Morava
–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
–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
–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
–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
–theory,
with mod 2 cohomology in place of integral cohomology.
Keywords
Morava $K$–theory, twisted Morava $K$–theory,
Atiyah–Hirzebruch spectral sequence, Whitehead tower,
connective covers of Lie groups, string group, fivebrane
group
