Generalized Group Characters and Complex Oriented Cohomology Theories

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∗, particularly p–complete theories with an associated formal group of height n. We briefly remind our readers of the terms in this last sentence. A multiplicative cohomology theory E∗ is complex oriented if there exists a class x ∈ E2(CP∞) that restricts to a generator of the free rank one E∗ = E∗(pt)–module Ẽ(CP ). Such a class x is called a complex orientation of E. An orientation allows for the construction in E∗–theory of Chern classes for complex vector bundles. Furthermore, the behavior of these Chern classes under the tensor product of bundles is determined by an associated formal group law over the ring E∗. When localized at a prime p, such formal group laws are classified by ‘height’. Under the completeness hypotheses we will be assuming, a height n formal group law will force an element vn ∈ E2−2p n to be invertible, and thus we may informally refer to such theories as vn–periodic.