Nilpotent $n$-tuples in $SU(2)$

TL;DR

This paper characterizes the connected components of homomorphism spaces from nilpotent groups to SU(2), revealing topological structures and homology properties of related classifying spaces, with explicit calculations for free nilpotent groups.

Contribution

It provides a detailed description of the topology of homomorphism spaces for nilpotent groups into SU(2) and analyzes the homology of a filtration of the classifying space BSU(2).

Findings

Connected components with non-abelian images are homeomorphic to RP^3.

Homology isomorphisms induced by inclusions in the filtration for coefficients where 2 is invertible.

Explicit calculations for free nilpotent groups and related groups like SO(3) and U(2).

Abstract

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.