## Duke Mathematical Journal

### Integer homology $3$-spheres admit irreducible representations in $\operatorname{SL}(2,\mathbb{C})$

Raphael Zentner

#### 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 $\operatorname{SL}(2,\mathbb{C})$. 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 $\operatorname{SU}(2)$-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 $\operatorname{SU}(2)$-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 $C^{0}$-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.

#### Article information

Source
Duke Math. J., Volume 167, Number 9 (2018), 1643-1712.

Dates
Revised: 11 January 2018
First available in Project Euclid: 1 May 2018

https://projecteuclid.org/euclid.dmj/1525140014

Digital Object Identifier
doi:10.1215/00127094-2018-0004

Mathematical Reviews number (MathSciNet)
MR3813594

Zentralblatt MATH identifier
06904637

#### Citation

Zentner, Raphael. Integer homology $3$ -spheres admit irreducible representations in $\operatorname{SL}(2,\mathbb{C})$. Duke Math. J. 167 (2018), no. 9, 1643--1712. doi:10.1215/00127094-2018-0004. https://projecteuclid.org/euclid.dmj/1525140014

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