Generalized group characters and complex oriented cohomology theories
Michael J. Hopkins
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
mjh@math.mit.edu
Nicholas J. Kuhn
Department of Mathematics, University of Virginia, Charlottesville, VA 22903
njk4x@virginia.edu
Douglas C. Ravenel
Department of Mathematics, University of Rochester, Rochester, NY 14627
drav@math.rochester.edu
AMS classification numbers: Primary 55N22; Secondary 20C99, 55N91, 55R35
Though it seems a shame to mess with an undergraound cult classic, this July 1999 preprint is intended to replace earlier versions dating from 1989 and 1992. It is also intended to get those of you who regularly bug us about this off our case.
Cheers,
Nick
Let BG be the classifying space of a finite group G. Given a
multiplicative cohomology theory E^*, the assignment
G ---> 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 theories: 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 E^*(BG) tensored with the rationals. 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 nth 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.