universal bundle

Let $G$ be a topological group. A universal bundle for $G$ is a principal bundle $p:EG\to BG$ such that for any principal bundle $\pi :E\to B$, with $B$ a CW-complex, there is a map $\phi :B\to BG$, unique up to homotopy, such that the pullback bundle ${\phi }^{*}(p)$ is equivalent to $\pi$, that is such that there is a bundle map ${\phi }^{\prime }$.

with ${\phi }^{\prime }(B)=\phi$, such that any bundle map of any bundle over $B$ extending $\phi$ factors uniquely through ${\phi }^{\prime }$.

As is obvious from the universal property, the universal bundle for a group $G$ is unique up to unique homotopy equivalence.

The base space $BG$ is often called a classifying space of $G$, since homotopy classes of maps to it from a given space classify $G$-bundles over that space.

There is a useful criterion for universality: a bundle is universal if and only if all the homotopy groups of $EG$, its total space, are trivial. This allows us to construct the universal bundle any subgroup from that of a larger group. Assume $H\le G$ and that $p:EG\to BG$ is a universal bundle for $G$. Then $H$ also acts freely on $EG$ which is contractable so $p_H:EH=EB\to BH=EB/H$ must be a universal bundle for $H$.

In 1956, John Milnor gave a general construction of the universal bundle for any topological group $G$ (see Annals of Mathematics, Second Series, Volume 63 Issue 2 and Issue 3 for details). His construction uses the infinite join of the group $G$ with itself to define the total space of the universal bundle.