Classifying space of a group

Summary

A classifying space of a discrete group $G$ is a path-connected space $BG$ whose fundamental group is $G$ and whose higher homotopy groups all vanish. It is the Eilenberg–MacLane space $K(G, 1)$, and among CW complexes it is unique up to homotopy equivalence. Through $BG$, group theory and topology meet: the homology of the group is the homology of the space $BG$. For topological groups, the same symbol denotes a space that classifies principal $G$-bundles, see principal bundle.