Universal principal bundle EG→BG

Let $G$ be a Lie group . A universal principal bundle for $G$ consists of a free $G$-space $EG$ that is contractible, together with the quotient space

and the quotient map $\pi\colon EG \to BG$.

Definition (Universal principal G-bundle)

The map $\pi\colon EG\to BG$ is called a universal principal $G$-bundle if:

1. The right $G$-action on $EG$ is free and $\pi$ exhibits $EG$ as a principal G-bundle over $BG$.
2. The total space $EG$ is contractible.
3. (Universal property for paracompact bases) For every paracompact space $X$ (in particular, any smooth manifold ), every principal $G$-bundle $P\to X$ is isomorphic to a pullback $f^{*}(EG)\to X$ for some map $f\colon X\to BG$. Such an $f$ is a classifying map , and its homotopy class in [X,BG] is determined uniquely by $P$.

The pair $(EG, BG)$ is unique up to $G$-equivariant homotopy equivalence (and $BG$ up to homotopy equivalence).

Examples

1. Circle group. For $G=U(1)$ one model is $EU(1)=S^\infty$ with the free $U(1)$-action by scalar multiplication, and $BU(1)=\mathbb{C}P^\infty$.
2. A two-point group. For $G=\mathbb{Z}/2$, take $EG=S^\infty$ with the antipodal action; then $BG=\mathbb{R}P^\infty$.
3. Classifying bundles over manifolds. For any principal $G$-bundle $P\to M$ over a smooth manifold $M$, choosing $EG\to BG$ produces a classifying map $M\to BG$ whose pullback recovers $P$ (up to isomorphism).