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

TL;DR

This paper proves that the fundamental group of any integer homology 3-sphere, other than the 3-sphere itself, admits irreducible representations in SL(2,C), with implications for 3-sphere recognition complexity.

Contribution

It establishes the existence of irreducible SL(2,C) representations for all non-trivial integer homology 3-spheres, extending known cases and linking to complexity class results.

Findings

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

Splicing of non-trivial knots admits irreducible SU(2)-representations.

3-sphere recognition problem is in coNP assuming GRH.

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, 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.