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

  1. ^ Waldhausen, Friedhelm (1968). "On irreducible 3-manifolds which are sufficiently large". Annals of Mathematics. 87 (1): 56–88. doi:10.2307/1970594. JSTOR 1970594. MR 0224099.
  2. ^ 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. MR 3104553. S2CID 255586740.
  3. ^ 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. MR 2979855.
  4. ^ Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4. MR 2912704. S2CID 32593851.
  5. ^ Kahn, Jeremy; Markovic, Vladimir (2012). "Counting essential surfaces in a closed hyperbolic three-manifold". Geometry & Topology. 16 (1): 601–624. arXiv:1012.2828. doi:10.2140/gt.2012.16.601. MR 2916295.
  6. ^ Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
  7. ^ Bergeron, Nicolas; Wise, Daniel T. (2012). "A boundary criterion for cubulation". American Journal of Mathematics. 134 (3): 843–859. arXiv:0908.3609. doi:10.1353/ajm.2012.0020. MR 2931226. S2CID 14128842.
  8. ^ Przytycki, Piotr; Wise, Daniel (2017-10-19). "Mixed 3-manifolds are virtually special". Journal of the American Mathematical Society. 31 (2): 319–347. arXiv:1205.6742. doi:10.1090/jams/886. ISSN 0894-0347. S2CID 39611341.
  9. ^ "Piotr Przytycki and Daniel Wise receive 2022 Moore Prize". American Mathematical Society.

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