Universal principal bundle EG→BG

Let $G$ be a topological group. A universal principal $G$ -bundle is a numerable topological principal bundle $\pi:EG\to BG$ with contractible total space and the following universal property: for every space $X$ , pullback induces a bijection

[X,BG]\longrightarrow \{\text{numerable principal }G\text{-bundles over }X\}/\cong, \qquad [f]\longmapsto[f^*EG].

Here $[X,BG]$ means continuous maps modulo unbased homotopy, and $f^*EG=\{(x,e):f(x)=\pi(e)\}$ has the pullback topology and right action. Thus every numerable bundle is such a pullback, and two maps give isomorphic pullbacks exactly when they are homotopic. The base $BG$ is the orbit space $EG/G$ with its quotient topology.

Existence and scope

Universal bundles exist for topological groups, using the Milnor construction and the numerable classification theorem. Equivalently, a numerable principal bundle with contractible total space has the stated classification property. Models are unique up to equivariant homotopy equivalence, with bases unique up to homotopy equivalence.

On a paracompact Hausdorff base every locally trivial principal bundle is numerable, so the theorem classifies all such bundles. For a Lie group and a smooth manifold, continuous principal bundles admit compatible smooth structures unique up to smooth bundle isomorphism. The spaces $EG$ and $BG$ themselves need not be finite-dimensional manifolds.

Examples

Circle group. For $G=U(1)$ one model is $EU(1)=S^\infty$ with the free $U(1)$ -action by scalar multiplication, and $BU(1)=\mathbb{C}P^\infty$ .
A two-point group. For $G=\mathbb{Z}/2$ , take $EG=S^\infty$ with the antipodal action; then $BG=\mathbb{R}P^\infty$ .
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).