nLab
universal principal bundle

Idea

For $G$ a topological group there is a notion of $G$-principal bundles $P\to X$ over any topological space $X$. Under continuous maps $f:X\to Y$ there is a notion of pullback of principal bundles $f^{*}:G\mathrm{Bund}(Y)\to G\mathrm{Bund}(X)$.

A universal $G$-principal bundle is a $G$-principal bundle, which is usually written $EG\to BG$, such that for every CW-complex $X$ the map

$$[X,BG]\to G\mathrm{Bund}(X){/}_{\sim }$$[X, B G] \to G Bund(X)/_\sim

from homotopy classes of continuous functions $X\to BG$ given by $[f]\mapsto f^{*}EG$, is an isomorphism.

In this case one calls $BG$ a classifying space for $G$-principal bundles.

The universal principal bundle is characterized, up to equivalence, by its total space $EG$ being contractible.

More generally, we can ask for a universal bundle for numerable bundles, that is principal bundles which admit a trivialisation over a numerable open cover. Such a bundle exists, and classifies numerable bundles over all topological spaces, not just paracompact spaces or CW-complexes.

Related concepts

• classifying space

• universal principal infinity-bundle.

References

Among the earliest references that consider the notion of universal bundles is

• Shiing-shen Chern, Sun, … (1949)

A review is for instance in

• Stephen Mitchell, Universal principal bundles and classifying spaces (pdf)