Duke Mathematical Journal

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

Raphael Zentner

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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.

Article information

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

Received: 5 December 2016
Revised: 11 January 2018
First available in Project Euclid: 1 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

instanton gauge theory holonomy perturbations knots representation varieties $\operatorname{SU}(2)$ $\operatorname{SL}(2,C)$ 3-sphere recognition 3-manifold groups


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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