Suppose $X$ is some contractible space and $\Gamma$ a discrete topological group acting continuously on $X$. Is it possible for $X/\Gamma$ to have non-trivial higher homotopy groups $\pi_k$, $k\geq2$?
If the action is nice then $\Gamma \to X \to X/\Gamma$ should be a fibre bundle and the long exact sequence gives $\pi_k(X)=0\to\pi_k(X/\Gamma)\to\pi_{k-1}(\Gamma) = 0$ for $k\geq2$, since $\Gamma$ is discrete and $X$ contractible and so $\pi_k(X/\Gamma) = 0$ for $k\geq2$.