NILPOTENT N-TUPLES IN SU(2)

We describe the connected components of the space Hom(Gamma, SU(2)) of homomorphisms for a discrete nilpotent group G. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to RP3. We give explicit calculations when G is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration BcomSU(2) = B(2, SU(2)) subset of center dot center dot center dot subset of B(q, SU(2)) subset of center dot center dot center dot of the classifying space BSU(2) (introduced by Adem, Cohen and Torres-Giese), showing that for every q >= 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.