Generalized group characters and complex oriented cohomology theories

Abstract:

Let $BG$ be the classifying space of a finite group $G$. Given a multiplicative cohomology theory $E^{*}$, the assignment \[ G \longmapsto E^{*}(BG) \] is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories $E^{*}$, using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin’s Theorem is proved for all complex oriented $E^*$: the abelian subgroups of $G$ serve as a detecting family for $E^*(BG)$, modulo torsion dividing the order of $G$. When $E^*$ is a complete local ring, with residue field of characteristic $p$ and associated formal group of height $n$, we construct a character ring of class functions that computes $\frac {1}{p}E^*(BG)$. The domain of the characters is $G_{n,p}$, the set of $n$–tuples of elements in $G$ each of which has order a power of $p$. A formula for induction is also found. The ideas we use are related to the Lubin–Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, $E_n^*$–theory, etc. The $n$th Morava K–theory Euler characteristic for $BG$ is computed to be the number of $G$–orbits in $G_{n,p}$. For various groups $G$, including all symmetric groups, we prove that $K(n)^*(BG)$ is concentrated in even degrees. Our results about $E^*(BG)$ extend to theorems about $E^*(EG\times _G X)$, where $X$ is a finite $G$–CW complex.