Classifying space BG

Let $G$ be a topological group. A classifying space $BG$ for $G$ is the base of a universal principal bundle $EG\to BG$ : the bundle is numerable, $EG$ is contractible, and pullback classifies numerable principal $G$ -bundles by continuous maps into $BG$ up to homotopy. In particular, $BG=EG/G$ with its quotient topology. The defining bundle data and universal property are those of the linked universal-bundle definition.

Classification theorem

If $B$ is a paracompact topological space that is Hausdorff (in particular, if $B$ is a paracompact smooth manifold), then isomorphism classes of topological principal G-bundles over $B$ are in natural bijection with homotopy classes of maps $[B,BG]$ .

Concretely:

given a map $f:B\to BG$ , the pullback bundle $f^*(EG)\to B$ is a principal $G$ -bundle;
every principal $G$ -bundle over $B$ is isomorphic to such a pullback for some $f$ ;
two maps yield isomorphic bundles if and only if they are homotopic.

Good covers (see good covers) are often used to build and compare classifying maps via transition functions.

Examples

Circle bundles. For $G=U(1)$ , one has $BU(1)\simeq \mathbb{CP}^\infty$ . The Hopf bundle over $S^2$ corresponds to a generator of $[S^2,BU(1)]\cong \mathbb Z$ .

The Möbius twist via a discrete group. For $G=\mathbb Z/2$ , one has $B(\mathbb Z/2)\simeq \mathbb{RP}^\infty$ . The nontrivial element of $[S^1,B(\mathbb Z/2)]\cong \mathbb Z/2$ classifies the principal $\mathbb Z/2$ -bundle whose associated line bundle is the Möbius bundle.

Bundles over the circle. Specializing the classification theorem to $B=S^1$ shows that principal $G$ -bundles over the circle are classified by conjugacy classes in the component group $\pi_0(G)$ . In particular, they are all trivial when $G$ is connected. This matches the explicit clutching description by gluing.