Homology lens spaces and $\mathrm{S}\mathrm{L}(2,\mathbb{C})$

We prove that if $Y$ is a closed, oriented 3-manifold with first homology $H_1(Y;\mathbb{Z})$ of order less than $5$, then there is an irreducible representation ${\pi }_1(Y)\to \mathrm{S}\mathrm{L}(2,\mathbb{C})$ unless $Y$ is homeomorphic to $S^3$, a lens space, or ${\mathbb{R}\mathbb{P}}^3\mathrm{\#}{\mathbb{R}\mathbb{P}}^3$. By previous work it suffices to consider the case $H_1(Y;\mathbb{Z})\cong \mathbb{Z}\mathrm{/}4\mathbb{Z}$, which we accomplish using holonomy perturbation techniques in instanton Floer homology.