Homotopy Theory of Classifying Spaces of Compact Lie Groups

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomology then they are homotopic. It has also been shown that if a space X has the same mod p cohomology, cup product, and Steenrod operations as a classifying space BG then (at least if p is odd and G is a classical group) X is actually homotopy equivalent to BG after mod p completion. Similar methods have also been used to obtain new results on Steenrod’s problem of constructing spaces with a given polynomial cohomology ring. The aim of this paper is to describe these results and the methods used to prove them.