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 $SL(2,\mathbb{C})$. 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 $S^3$ admits an irreducible $SU\left(2\right)$-representation. Using a result of Kuperberg, we get the corollary that the problem of $3$-sphere recognition is in the complexity class $\mathsf{coNP}$, provided the generalized Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the $SU\left(2\right)$-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 $C^0$-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.