Complex cobordism, classifying spaces of finite groups, and generalized characters.

Abstract

In this paper, we explore various methods for calculating the Brown-Peterson cohomology of a classifying space of a finite group. The first method uses the Morava structure theorem and calculations of Morava's extraordinary K-theories, denoted K(n). A particular group is shown to have non-trivial Morava K-theory for each odd prime p and each $n\ge 2.$ Using this data, the Brown-Peterson cohomology is shown not to be Landweber flat. The second method extends Hopkins, Kuhn, and Ravenel's theory for E(n)-cohomology and generalized characters. Using observations about the Hopf algebroid $E(n)\sb\* E(n),$ an injective map from rationalized Brown-Peterson cohomology into a space of generalized characters is constructed.