Universal bundle

Summary

A universal bundle for a topological group $G$ is a principal $G$-bundle $EG \to BG$ with a classifying property. Every principal $G$-bundle over a CW complex is its pullback along a map, unique up to homotopy. Its base $BG$ is the classifying space of $G$. A principal bundle with contractible total space has this property under mild conditions, which gives a practical way to recognize universal bundles. Universal bundles turn the classification of bundles into the computation of homotopy classes of maps.