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15 June 2018 Integer homology 3-spheres admit irreducible representations in SL(2,C)
Raphael Zentner
Duke Math. J. 167(9): 1643-1712 (15 June 2018). DOI: 10.1215/00127094-2018-0004


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.


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Raphael Zentner. "Integer homology 3-spheres admit irreducible representations in SL(2,C)." Duke Math. J. 167 (9) 1643 - 1712, 15 June 2018.


Received: 5 December 2016; Revised: 11 January 2018; Published: 15 June 2018
First available in Project Euclid: 1 May 2018

zbMATH: 06904637
MathSciNet: MR3813594
Digital Object Identifier: 10.1215/00127094-2018-0004

Primary: 57M25
Secondary: 57R57

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

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 9 • 15 June 2018
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