Classifying map of a principal bundle

Fix a Lie group $G$ and a chosen universal principal bundle $\pi\colon EG\to BG$.

Definition (Classifying map)

Let $M$ be a smooth manifold and let $P\to M$ be a principal G-bundle . A classifying map for $P$ is a continuous map

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

If $M$ is paracompact (in particular, if $M$ is a smooth manifold), then:

• a classifying map exists, and
• its homotopy class in [M,BG] is uniquely determined by $P$.

Equivalently, isomorphism classes of principal $G$-bundles correspond to homotopy classes of classifying maps (see the classification theorem ).

Examples

1. Trivial bundle. For $P=M\times G$, a classifying map can be taken to be constant (and hence null-homotopic).
2. 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 the generator of $H^2(S^2;\mathbb{Z})$.
3. 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.