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Morava \(K\)-theory of classifying spaces: Some calculations. (English) Zbl 0896.55005
The author develops some powerful techniques for calculating Morava K-theories of classifying spaces of finite and compact Lie groups. As a highlight, he shows that \(K(2)^{odd} \neq 0\), where \(P\) is the \(3\)-Sylow subgroup of \(GL_4(\mathbb Z /3)\). This disproves a conjecture of M. J. Hopkins, N. J. Kuhn and D. C. Ravenel [Lect. Notes Math. 1509, 186-209 (1992; Zbl 0757.55006)], that for every finite group \(G\) and positive integer \(n\), \(K(n)^*(BG)\) is additively generated as a \(K(n)^*\)-module by transferred Euler classes of complex representations of subgroups of \(G\), and so must be concentrated in even dimensions. The author also calculates Morava K-theories of semidirect products of cyclic groups with elementary abelian groups and studies complex-oriented cohomology of \(BO(k)\). There are two main ingredients to the author’s calculations. The first ingredient is the Hochschild-Serre spectral sequence associated with an extension of the group \(\mathbb Z /p\). The second is Hopkins-Kuhn-Ravenel character theory.

55N22 Bordism and cobordism theories and formal group laws in algebraic topology
20J06 Cohomology of groups
55N15 Topological \(K\)-theory
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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