Classification of principal G-bundles by homotopy classes of maps into BG

Let $G$ be a Lie group and let $M$ be a paracompact smooth manifold. Write $\mathrm{Prin}_G(M)$ for the set of isomorphism classes of principal G-bundles over $M$ (isomorphisms are principal bundle isomorphisms covering $\mathrm{id}_M$).

Let $EG \to BG$ be the universal principal G-bundle over the classifying space $BG$. For any continuous map $f \colon M \to BG$, there is a pullback principal $G$-bundle

constructed via the usual pullback construction (and guaranteed to be a principal bundle by pullback functoriality for principal bundles ).

Theorem (classification by maps to BG)

The assignment

is a well-defined bijection from the set of homotopy classes of maps $M\to BG$ to isomorphism classes of principal $G$-bundles over $M$.

Equivalently:

1. (Existence) For every principal $G$-bundle $P\to M$ there exists a (continuous) classifying map $f_P\colon M\to BG$ such that $P \cong f_P^*(EG)$ as principal $G$-bundles.
2. (Uniqueness) If $f,g\colon M\to BG$ are homotopic, then $f^*(EG)\cong g^*(EG)$; conversely, if $f^*(EG)\cong g^*(EG)$ then $f$ and $g$ are homotopic.

Examples

1. Contractible base. If $M$ is contractible, then $[M,BG]$ has one element, so every principal $G$-bundle over $M$ is isomorphic to the trivial principal bundle .
2. Circle and disconnected structure group. Since $[S^1,BG]\cong \pi_1(BG)\cong \pi_0(G)$, principal $G$-bundles over $S^1$ are classified by connected components of $G$. For instance, with $G=O(1)\cong \{\pm 1\}$ there are two classes; the nontrivial one corresponds (via the usual passage to associated line bundles) to the Möbius bundle.
3. Hopf bundle as a pullback. For $G=U(1)$ and $M=S^2$, one has $[S^2,BU(1)]\cong \mathbb{Z}$. Under this identification, the Hopf fibration represents a generator (so it is a pullback of $EU(1)\to BU(1)$ along a degree-one classifying map).