Twisted Morava \(K\)-theory and connective covers of Lie groups (Q2667195)

Twisted Morava \(K\)-theory was introduced by Craig Westerland and the first author [\textit{H. Sati} and \textit{C. Westerland}, J. Topol. 8, No. 4, 887--916 (2015; Zbl 1330.55007)] together with enough computational techniques like: universal coefficient theorem (Theorem 15), Atiyah-Hirzebruch spectral sequence (Theorem 16), etc. The present authors use these techniques to compute the twisted Morava \(K\)-theory of all connective covers of the stable orthogonal group and stable unitary group, their classifying spaces, spheres and also Eilenberg-Mac Lane spaces. Morava \(K\)-theory \(K(n)\) is an ``extension'' of \(K\)-theory: it is a complex oriented cohomology theory, defined for every chromatic level (the height of the corresponding formal group law) integer \(n\) and prime \(p\). Motivated by string theory, a twisted form of Morava \(K\)-theory and \(E\)-theory should describe an extension of the untwisted setting on anomalies at chromatic level two. The coefficients of the theory \(K(n)_* = \mathbb Z/p[v_n,v_n^{-1}]\) form a graded field, which implies that the twisted Morava \(K\)-theory of Sati and Westerland has the useful computational Künneth isomorphism \(K(n)_*(X \times Y) \cong K(n)_*(Y) \otimes_{K(n)_*} K(n)_*(X).\) The main results of the paper are: 1. A general vanishing theorem for twisted Morava \(K\)-theory: the composition of the map in twisted Morava \(K\)-homology with the Bockstein map and the twisted Morava \(K\)-homology of the base of a principal \(K(\mathbb Z, n+1)\)-bundle \(\xi: E \to B\) vanishes \(K(n)_*(B,\xi) = 0\), if the induced map on Morava \(K\)-homology is a map of Hopf algebras. 2. The untwisting theorem: the twisted Morava \(K\)-homology of all groups in the Whitehead tower of the orthogonal and unitary groups and their classifying spaces is isomorphic to the underlying untwisted Morava \(K\)-homology.