Classifying space

Definition

Let $G$ be a topological group. A classifying space of $G$, denoted $BG$, is defined as the quotient of a weakly contractible space (a space all whose homotopy groups are trivial) by a free action of $G$.

In particular when $G$ is a discrete group, a classifying space of $G$, denoted $BG$, is a path-connected space whose fundamental group is $G$ and for which the higher homotopy groups are trivial.

Another way of saying this is that the classifying space of a discrete group is a path-connected space with the given discrete group as fundamental group, and whose universal cover is a weakly contractible space (often, a contractible space).

Classifying spaces for discrete groups are special cases of Eilenberg-Maclane spaces.

A topological space occurs as a classifying space for a discrete group iff it is aspherical.