  We prove that if an integer homology three-sphere contains an embedded
incompressible torus, then its fundamental group admits irreducible
SU(2)-representations. Our methods use instanton Floer homology, and in
particular the surgery exact triangle, holonomy perturbations, and a
non-vanishing result due to Kronheimer-Mrowka, as well as results about
surgeries on cables due to Gordon.
