  We prove that the fundamental group of any integer homology 3-sphere
different from the 3-sphere admits irreducible representations of its
fundamental group in SL(2,C). For hyperbolic integer homology spheres this
comes with the definition, and for Seifert fibered integer homology spheres
this is well known. We prove that the splicing of any two non-trivial knots in
the 3-sphere admits an irreducible SU(2)-representation. By work of Boileau,
Rubinstein, and Wang, the general case follows. Using a result of Kuperberg, we
get the corollary that the problem of 3-sphere recognition is in the complexity
class coNP, provided the generalised Riemann hypothesis holds. To prove our
result, we establish a topological fact about the image of the
SU(2)-representation variety of a non-trivial knot complement into the
representation variety of its boundary torus, a pillowcase. For this, we use
holonomy perturbations of the Chern-Simons function in an exhaustive way - we
show that any area-preserving self-map of the pillowcase fixing the four
singular points, and which is isotopic to the identity, can be C^0-approximated
by maps which are realised geometrically through holonomy perturbations of the
flatness equation in a thickened torus. To conclude, we use a stretching
argument in instanton gauge theory, and a non-vanishing result of Kronheimer
and Mrowka for Donaldson's invariants of a 4-manifold which contains the
0-surgery of a knot as a splitting hypersurface.
