zbMATH — the first resource for mathematics

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.

MSC:
 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
Full Text: