Integer homology 3-spheres admit irreducible representations in SL(2,C)

Abstract

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; for Seifert-fibered integer homology spheres, this is well known. We prove that the splicing of any two nontrivial knots in S3 admits an irreducible SU(2)-representation. Using a result of Kuperberg, we get the corollary that the problem of 3-sphere recognition is in the complexity class coNP, provided the generalized Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the SU(2)-representation variety of a nontrivial knot complement into the representation variety of its boundary torus, a pillowcase, using holonomy perturbations of the Chern–Simons function in an exhaustive way—showing that any area-preserving self-map of the pillowcase fixing the four singular points, and which is isotopic to the identity, can be C0-approximated by maps realized geometrically through holonomy perturbations of the flatness equation in a thickened torus. We conclude with a stretching argument in instanton gauge theory and a nonvanishing 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.