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Morava \(K\)-theories of classifying spaces and generalized characters for finite groups. (English) Zbl 0757.55006
Algebraic topology, Proc. Conf., S. Feliu de Guíxols/Spain 1990, Lect. Notes Math. 1509, 186-209 (1992).
[For the entire collection see Zbl 0741.00040.]
Let \(G\) be a finite group and let \(BG\) denote its classifying space. Ravenel has shown that \(K(n)^*(BG)\) has finite rank, where \(K(n)^*(- )\) denotes Morava \(K\)-theory for a prime \(p\). Hence one can define the Euler characteristic \(\chi_{n,p}(G)\) of \(K(n)^*(BG)\), i.e., the difference in ranks between the even and odd-dimensional components of \(K(n)^*(BG)\). Let \(G_{n,p}\) be the set of commuting \(n\)-tuples of elements of prime power order in \(G\). Generalizing a result of Kuhn, the authors prove the following Theorem: \(\chi_{n,p}(G)\) is the number of \(G\)-orbits in \(G_{n,p}\). Furthermore the authors state the Conjecture: \(K(n)^*(BG)\) is concentrated in even dimensions.
For each prime \(p\) and \(n>0\) there is a \(BP\)-module spectrum \(E(n)\) with coefficient ring \(E(n)_ *=v_ n^{-1}BP_ */(v_ k:k>n)\). The main result of the authors describes the connection between \(E(n)^*(BG)\) and the ring of \(\overline Q_ p\) valued conjugacy class functions on \(G_{n,p}\).