Toroidal graph manifolds with small homology are not SU(2)-abelian

TL;DR

This paper proves that certain small homology toroidal graph manifolds admit non-abelian SU(2) representations, confirming a conjecture for this class using topological methods instead of gauge theory.

Contribution

It establishes the existence of irreducible SU(2) representations for small homology toroidal graph manifolds, providing a topological proof of a conjecture by Baldwin and Sivek.

Findings

Existence of irreducible SU(2) representations for |H_1(Y)| ≤ 5

Positive resolution of Baldwin and Sivek's conjecture for graph manifolds

Use of topological methods avoiding gauge theory

Abstract

We show that if $Y$ is a toroidal closed graph manifold rational homology $3$-sphere with $|H_1(Y;\mathbb{Z})| \le 5$, then there exists an irreducible representation $\fund{Y} \to SU(2)$, using topological methods and avoiding the use of gauge theory. This answers positively to a conjecture by Baldwin and Sivek in the case of graph manifolds.