---
tag: ZC179DD2P4
kind: definition
name: Universal bundle
msc: [55R15, 55R35]
defines: 1HDH716Z24
uses: [8E37E74XVP, 9YA6S6HD9V, 5W95QFHXPR, R5WP8Z69Q8]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_bundle
---

## Definition

Let $G$ be a topological group and let $p \colon E \to B$ be a [[principal $G$-bundle|8E37E74XVP]].
For a space $X$, let $\operatorname{Prin}_G(X)$ be the set of isomorphism classes of principal $G$-bundles over $X$.
Let $[X, B]$ be the set of [[homotopy|9YA6S6HD9V]] classes of continuous maps $X \to B$.
The bundle $p$ is *universal* if, for every [[CW complex|5W95QFHXPR]] $X$, the map
$$[X, B] \to \operatorname{Prin}_G(X), \quad [f] \mapsto [f^* E],$$
is a bijection, where $f^* E$ is the pullback bundle.

## Remarks

The map is well defined because pullbacks along homotopic maps are isomorphic over CW complexes.
A universal bundle is written $EG \to BG$, and $BG$ is a [[classifying space|WSZ40H5E4T]] of $G$; it is unique up to homotopy equivalence.
A principal $G$-bundle over a CW complex whose total space is [[contractible|R5WP8Z69Q8]] is universal.
Such bundles exist for every topological group $G$, for instance by Milnor's join construction.

Example: $\Z/2\Z$ acts freely on the infinite sphere $S^\infty$ by $x \mapsto -x$, and $S^\infty \to \R P^\infty$ is a principal $\Z/2\Z$-bundle.
Since $S^\infty$ is contractible, this bundle is universal.

Non-example: the bundle $S^1 \to \R P^1$, $x \mapsto \set{x, -x}$, is a principal $\Z/2\Z$-bundle but not universal.
Take $X = \R P^2$: there are two principal $\Z/2\Z$-bundles over $\R P^2$, the trivial one and the double cover $S^2 \to \R P^2$.
But every map $\R P^2 \to \R P^1 \cong S^1$ is null-homotopic, because $H^1(\R P^2; \Z) = 0$, so $[\R P^2, \R P^1]$ has only one element and the map is not a bijection.