  We describe the connected components of the space $\text{Hom}(\Gamma,SU(2))$
of homomorphisms for a discrete nilpotent group $\Gamma$. The connected
components arising from homomorphisms with non-abelian image turn out to be
homeomorphic to $\mathbb{RP}^3$. We give explicit calculations when $\Gamma$ is
a finitely generated free nilpotent group. In the second part of the paper we
study the filtration $B_{\text{com}}SU(2) = B(2,SU(2))\subset\cdots \subset
B(q,SU(2))\subset\cdots$ of the classifying space $BSU(2)$ (introduced by Adem,
Cohen and Torres-Giese), showing that for every $q\geq2$, the inclusions induce
a homology isomorphism with coefficients over a ring in which 2 is invertible.
Most of the computations are done for $SO(3)$ and $U(2)$ as well.
