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.

中文翻译：

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SU(2) 中的幂零 n 元组

我们描述了空间的连通分量$\text {Hom}(\Gamma ,SU(2))$离散幂零群的同态性$\伽马$. 由具有非阿贝尔图像的同态产生的连通分量结果是同胚的$\mathbb {RP}^{3}$. 我们给出明确的计算$\伽马$是一个有限生成的自由幂零团体。在论文的第二部分，我们研究过滤$B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$分类空间$BSU(2)$（由 Adem、Cohen 和 Torres-Giese 介绍），表明对于每个$q\geq 2$, 夹杂物在 2 是可逆的环上诱导具有系数的同调同构。大多数计算都是针对$SO(3)$和$U(2)$也是。