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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 $q\geq 2$
, 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.
