The Milnor Construction: Homotopy Classification of Principal Bundles

For a given topological group G, we consider all the principal bundles. At first, it looks like finding all principal G-bundles over a space might be a great task, but there is a special construction of a principal G-bundle due to Milnor. It has the property that all other numerable principal G-bundles over all possible spaces are induced from this particular bundle. Thus, it is called the universal principal Gbundle, and its base space is called the classifying space B(G) = BG = BG of the group G.