Universal 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 of $G$; it is unique up to homotopy equivalence. A principal $G$-bundle over a CW complex whose total space is contractible 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.