Classification of Principal Bundles
Any topological group G admits a classifying space BG: the quotient of some weakly contractible space EG, i.e. a topological space for which all its homotopy groups are trivial by a free action of G. The classifying space has the property that any G principal bundle over a paracompact manifold B is isomorphic to a pullback of the principal bundle . In fact, more is true, as the set of isomorphism classes of principal G bundles over the base B identifies with the set of homotopy classes of maps B → BG.
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