Classifying space of a group

Remarks

A classifying space of $G$ is the same as an Eilenberg–MacLane space $K(G, 1)$. Any two classifying spaces of $G$ are homotopy equivalent. Equivalently, $X$ is path-connected, $\pi_1(X) \cong G$, and its universal cover is contractible; then $G$ acts freely on the universal cover and $X$ is the quotient.

Example: the circle $S^1$ is a classifying space of $\Z$. Its fundamental group is $\Z$, and its universal cover $\R$ is contractible, so $\pi_n(S^1) = 0$ for $n \ge 2$. In the same way, the torus is a classifying space of $\Z^2$, and the infinite real projective space $\R P^\infty$ is one of $\Z/2\Z$.

Non-example: the real projective plane $\R P^2$ has fundamental group $\Z/2\Z$, but its universal cover is the sphere $S^2$, so $\pi_2(\R P^2) \cong \pi_2(S^2) \cong \Z$. Clause 2 fails, so $\R P^2$ is not a classifying space of $\Z/2\Z$.