In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space for which all its homotopy groups are trivial) by a free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG.
For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg-Maclane space, or a K(G,1).
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—Ralph Waldo Emerson (1803–1882)