A Universal Bundle for a Compact Lie Group

This chapter looks at a universal bundle for a compact Lie group. By Milnor's construction, every topological group has a universal bundle. Independently of Milnor's result, the chapter constructs a universal bundle for any compact Lie group G. This construction is based on the fact that every compact Lie group can be embedded as a subgroup of some orthogonal group O(k). The chapter first constructs a universal O(k)-bundle by finding a weakly contractible space on which O(k) acts freely. The infinite Stiefel variety V (k, ∞) is such a space. As a subgroup of O(k), the compact Lie group G will also act freely on V (k, ∞), thereby producing a universal G-bundle.