Classifying space BG

Let $G$ be a Lie group (more generally, a reasonable topological group).

Definition (universal bundle and classifying space)

A classifying space $BG$ for $G$ is a space equipped with a principal $G$-bundle

such that:

• $EG$ is contractible, and
• $G$ acts freely on $EG$ with quotient $BG=EG/G$.

The bundle $EG\to BG$ is called the universal principal $G$-bundle.

Classification theorem

If $B$ is a paracompact topological space (in particular, if $B$ is a paracompact smooth manifold), then isomorphism classes of 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

1. 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 $[S^2,BU(1)]\cong \mathbb Z$.

2. 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.

3. Bundles over the circle.
   Specializing the classification theorem to $B=S^1$ shows that principal $G$-bundles over the circle are classified by $\pi_0(G)$, matching the explicit clutching description by gluing.