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 was subsequently outlined in three lectures March 26 and 28th at the Workshop on Immersed Surfaces in 3-Manifolds at the Institut Henri Poincaré. A preprint of the claimed proof has been posted on the ArXiv.[2] The proof built on results of Kahn and Markovic[3] in their proof of the Surface subgroup conjecture and results of Wise in proving the Malnormal Special Quotient Theorem[4] and results of Bergeron and Wise for the cubulation of groups.[5]
See also
- virtually fibered conjecture
- virtually positive Betti number conjecture
Notes
- ^ Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87 1968 56–88.[1],[2]
- ^ Ian Agol, Daniel Groves, and Jason Manning, A boundary criterion for cubulation. http://arxiv.org/abs/1204.2810
- ^ Kahn and Markovic, Immersing almost geodesic surfaces in a closed hyperbolic manifold http://arxiv.org/abs/0910.5501, Counting essential surfaces in a closed hyperbolic 3-manifold, http://arxiv.org/abs/1012.2828
- ^ Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
- ^ Nicolas Bergeron and Daniel T. Wise, A boundary criterion for cubulation, http://arxiv.org/abs/0908.3609
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. 7, pp. 273–312 Unknown parameter
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