Affiliation: | Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ; Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903 ; Department of Mathematics, University of Rochester, Rochester, New York 14627 |
Abstract: | Let be the classifying space of a finite group . Given a multiplicative cohomology theory , the assignment is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories , 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 : the abelian subgroups of serve as a detecting family for , modulo torsion dividing the order of . When is a complete local ring, with residue field of characteristic and associated formal group of height , we construct a character ring of class functions that computes . The domain of the characters is , the set of -tuples of elements in each of which has order a power of . 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, -theory, etc. The th Morava K-theory Euler characteristic for is computed to be the number of -orbits in . For various groups , including all symmetric groups, we prove that is concentrated in even degrees. Our results about extend to theorems about , where is a finite -CW complex. |