Some General Theorems on the Cohomology of Classifying Spaces of Compact Lie Groups

Abstract

This paper is divided into two parts. The first part proves a number of general theorems on the cohomology of the classifying spaces of compact Lie groups. These theorems are proved by transfer methods, relying heavily on the double coset theorem [F$_{1}$]. Several of these results are well known while others are quite new. For the most part the proofs of the theorems are independent of each other and are quite short. Nevertheless they are true in great generality. Several are proven for arbitrary compact Lie groups and arbitrary cohomology theories. Perhaps the most interesting of the new results relates the cohomology of the classifying space of an arbitrary compact Lie group with that of the normalizer of a maximal torus. The second part of the paper generalizes many theorems to certain equivariant cohomology theories. Some of these theorems appear in [F$_{2}$].