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Generalized group characters and complex oriented cohomology theories. (English) Zbl 1007.55004
This is a paper of long gestation and wide scope. It examines the functor $$G\mapsto E^*(BG)$$ which takes a finite group $$G$$ into the $$E$$-cohomology of its classifying space where $$E^*(-)$$ is a complex oriented cohomology theory. Complex oriented cohomology theories include ordinary integral cohomology, complex K-theory, complex cobordism, the variety of theories related to Brown-Peterson cohomology, and the Morava K-theories, $$K(n)^*(-)$$.
For a finite group $$G$$, $${\mathcal A}(G)$$ is the category whose objects are the Abelian subgroups of $$G$$. A morphism from $$B$$ to $$A$$ is a $$G$$-map from $$G/B$$ to $$G/A$$. With $$G$$ a finite group of order $$|G|$$ and $$E$$ as above, ${1\over|G|} E^*(BG)\to \lim_{A\in{\mathcal A}(G)} {1\over|G|} E^*(BA)$ is an isomorphism. This result is an analogue of Artin’s theorem; the splitting principle is employed in the proof.
A second result states that for a finite group $$G$$, the Morava K-theory Euler characteristic $$\dim K(n)^{\text{even}}(BG)- \dim K(n)^{\text{odd}}(BG)$$ is the number of $$G$$-orbits in a $$G$$-set $$G_{n,p}$$. The prime $$p$$ is the one associated to the Morava K-theory. $$G_{n,p}$$ is the set of $$n$$-tuples of commuting order-$$p$$ elements of $$G$$. $$G$$ acts on $$G_{n,p}$$ by conjugation.
If we take as our example of complex-oriented $$E^*$$ ordinary cohomology, $${1\over|G|} E^*(BG)$$ is not very interesting. But decades ago, Landweber observed that $$MU^*(BZ/2)$$ is torsion-free – vastly different to its homology analogue. If $$E^*$$ is a $$p$$-complete integral life of $$K(n)^*$$, then $$E^*(BG)$$ will be torsion-free if and only if $$K(n)^*(BG)$$ is concentrated in even degrees. This leads the authors to seek conditions to ensure this even concentration (e.g. if the group $$G$$ is “good”).
The authors interpret their results in terms of generalized characters and invariant rings. The reader is referred to the paper’s introduction for the necessary definitions and precise statements. The authors’ approach can be applied to compute the kernel of $$\pi^0_s(BG)\to MU^0(BG)$$ up to finite index.

MSC:
 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 55N91 Equivariant homology and cohomology in algebraic topology 55N34 Elliptic cohomology
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References:
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