Classifying map of a principal bundle

Fix a topological group $G$ and a chosen universal principal bundle $EG\to BG$ . For a topological principal $G$ -bundle $P\to M$ , a classifying map is a continuous map

c\colon M \longrightarrow BG

such that there is an isomorphism of principal $G$ -bundles

P \cong c^{*}(EG).

Existence and uniqueness

If the bundle is numerable, in particular if $M$ is paracompact Hausdorff as a standard smooth manifold is, then a classifying map exists and its homotopy class is uniquely determined by $P$ . Thus isomorphism classes of principal $G$ -bundles correspond to homotopy classes of maps $M\to BG$ .

See classification of principal bundles.

Examples

Trivial bundle. For $P=M\times G$ , a classifying map can be taken to be constant (and hence null-homotopic).
Hopf fibration. The principal $U(1)$ -bundle $S^3\to S^2$ is classified by a map $S^2\to BU(1)\simeq \mathbb{C}P^\infty$ representing a generator of $H^2(S^2;\mathbb{Z})$ .
Frame bundle viewpoint. The oriented orthonormal frame bundle of a Riemannian $n$ -manifold is a principal $SO(n)$ -bundle; its classifying map $M\to BSO(n)$ encodes characteristic classes such as the Stiefel–Whitney and Pontryagin classes.