Zentner, Raphael (2018) INTEGER HOMOLOGY 3-SPHERES ADMIT IRREDUCIBLE REPRESENTATIONS IN SL(2,C). DUKE MATHEMATICAL JOURNAL, 167 (9). pp. 1643-1712. ISSN 0012-7094, 1547-7398
Full text not available from this repository. (Request a copy)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; 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 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 C degrees-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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | INVARIANT; |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 17 Feb 2020 09:00 |
| Last Modified: | 17 Feb 2020 09:00 |
| URI: | https://pred.uni-regensburg.de/id/eprint/14394 |
Actions (login required)
 |
View Item |
