Virtually Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.
After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds.
The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968,[1] although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list.
A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica.[2] The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fundamental group on certain auxiliary spaces (CAT(0) cube complexes)[3] It used as an essential ingredient the freshly-obtained solution to the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.[4][5] Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of Wise[6] and a criterion of Nicolas Bergeron and Wise for the cubulation of groups.[7]
In 2018 related results were obtained by Piotr Przytycki and Daniel Wise proving that mixed 3-manifolds are also virtually special, that is they can be cubulated into a cube complex with a finite cover where all the hyperplanes are embedded which by the previous mentioned work can be made virtually Haken.[8][9]
See also
Notes
- ↑ Waldhausen, Friedhelm (1968). "On irreducible 3-manifolds which are sufficiently large". Annals of Mathematics 87 (1): 56–88. doi:10.2307/1970594. https://pub.uni-bielefeld.de/record/1782185.
- ↑ Agol, Ian (2013). With an appendix by Ian Agol, Daniel Groves, and Jason Manning. "The virtual Haken Conjecture". Doc. Math. 18: 1045–1087. doi:10.4171/dm/421. https://www.math.uni-bielefeld.de/documenta/vol-18/33.html.
- ↑ Haglund, Frédéric; Wise, Daniel (2012). "A combination theorem for special cube complexes". Annals of Mathematics 176 (3): 1427–1482. doi:10.4007/annals.2012.176.3.2.
- ↑ Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics 175 (3): 1127–1190. doi:10.4007/annals.2012.175.3.4.
- ↑ Kahn, Jeremy; Markovic, Vladimir (2012). "Counting essential surfaces in a closed hyperbolic three-manifold". Geometry & Topology 16 (1): 601–624. doi:10.2140/gt.2012.16.601.
- ↑ Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
- ↑ Bergeron, Nicolas; Wise, Daniel T. (2012). "A boundary criterion for cubulation". American Journal of Mathematics 134 (3): 843–859. doi:10.1353/ajm.2012.0020.
- ↑ Przytycki, Piotr; Wise, Daniel (2017-10-19). "Mixed 3-manifolds are virtually special" (in en). Journal of the American Mathematical Society 31 (2): 319–347. doi:10.1090/jams/886. ISSN 0894-0347. https://www.ams.org/jams/2018-31-02/S0894-0347-2017-00886-5/.
- ↑ "Piotr Przytycki and Daniel Wise receive 2022 Moore Prize" (in en). https://www.ams.org/news?news_id=6854.
References
- Dunfield, Nathan; Thurston, William (2003), "The virtual Haken conjecture: experiments and examples", Geometry and Topology 7: 399–441, doi:10.2140/gt.2003.7.399.
- Kirby, Robion (1978), "Problems in low dimensional manifold theory.", Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) 7: pp. 273–312, ISBN 9780821867891, https://books.google.com/books?id=Ve5uy_dx35cC&dq=kirby%20problem%20list&pg=PA273.
External links
- Klarreich, Erica (2012-10-02). "Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back" (in en). https://www.quantamagazine.org/from-hyperbolic-geometry-to-cube-complexes-and-back-20121002/.
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