The Virtually Haken Conjecture
A Haken 3-manifold is a compact 3-dimensional manifold M which is irreducible (in a certain strong sense) but contains an incompressible surface S. (An embedded surface S is incompressible if the embedding indices an injection of its fundamental group to the fundamental group of M. A 3-manifold is virtually
finite Haken if it has a finite cover which is Haken. (This is a typical way how the word “virtually” occurs in algebra and topology.)
The virtually Haken conjecture states that every compact, orientable, irreducible 3-dimensional manifold with infinite fundamental group is virtually Haken. The big news is that Ian Agol has just announced the proof of the virtually Haken conjecture!
Danny Calegary have just wrote three detailed posts about it over his blog “Geometry and the Imagination”: Agol’s Virtual Haken Theorem (part 1), Agol’s Virtual Haken Theorem (part 2): Agol-Groves-Manning strike back, and Agol’s Virtual Haken Theorem (part 3): return of the hierarchies. (Everything I say is taken from there.)
To quote Danny:
I think it is no overstatement to say that this marks the end of an era in 3-manifold topology, since the proof ties up just about every loose end left over on the list of problems in 3-manifold topology from Thurston’s famous Bulletin article (with the exception of problem 23 — to show that volumes of closed hyperbolic 3-manifolds are not rationally related — which is very close to some famous open problems in number theory).
Here are also few relevant posts from the blog: Low dimensional topology. A post about Wise conjecture (that Agol proved) with references and links; An earlier post on Wise’s work; A post VHC post; Update (August ’14): Here is a post by Tim Gowers on Agol’s lecture at ICM2014. The videotaped lecture can be found here. Ian’s ICM paper can be found here. Dani Wises’s ICM talk is here.
Problems 16-18 in Thurston’s Bulletin paper.
A whole array of conjectures and a whole array of results: Wise, Haglund-Wise, Bergeron-Wise, Sageev, Kahn-Markovic, …
Perleman’s geometrization theorem reduces the conjecture to the case of hyperbolic manifolds. Continue reading