Classifying space

Summary

The classifying space $BG$ of a topological group $G$ is the base of a universal bundle $EG \to BG$. Principal $G$-bundles over a CW complex $X$ correspond to homotopy classes of maps $X \to BG$. Cohomology classes of $BG$ give the characteristic classes of such bundles. For a discrete group $G$, the space $BG$ is the classifying space of the group in the sense of an Eilenberg–MacLane space $K(G, 1)$.