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

Abstract

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 $\ddot{I}{€}_1(Y)\to \mathrm{SL}(2,\mathbb{C})$ unless $Y$ is homeomorphic to $S^3$, a lens space, or ${\mathbb{RP}}^3\#{\mathbb{RP}}^3$. By previous work it suffices to consider the case $H_1(Y;\mathbb{Z})\cong \mathbb{Z}/4\mathbb{Z}$, which we accomplish using holonomy perturbation techniques in instanton Floer homology.

31 pages, 4 figures