We prove that the fundamental group of any integer homology -sphere different from the -sphere admits irreducible representations of its fundamental group in . 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 admits an irreducible -representation. Using a result of Kuperberg, we get the corollary that the problem of -sphere recognition is in the complexity class , provided the generalized Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the -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 -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 -manifold which contains the -surgery of a knot as a splitting hypersurface.
"Integer homology -spheres admit irreducible representations in ." Duke Math. J. 167 (9) 1643 - 1712, 15 June 2018. https://doi.org/10.1215/00127094-2018-0004