Irreducible SU(2)-representations of toroidal integral homology 3-spheres

Somewhat an analogue of the L-space conjecture for Heegaard Floer homology is the following question: Does instanton Floer homology detect the 3-sphere among integral homology 3-spheres? The generators of the underlying instanton chain complex is given by irreducible SU(2)-representations of the fundamental group. Hence a weaker question than the above is whether all integral homology 3-spheres admit irreducible SU(2)-representations. It is conjectured that this holds, since it has been proved for large classes by various authors. We show that toroidal integral homology 3-spheres have irreducible SU(2)-representations.This is joint work with Tye Lidman and Juanita Pinzon Caicedo.