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 Ion 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).
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. There were a whole list of related conjectures, and people had some beliefs about which conjectures are stronger. The works of Dani Wise (Done, in part, while he was visiting us at the Hebrew University!) proved various unexpected connections between these conjectures, and gave hope that the picture will be completely clarified.
Another crucial work is the proof by Vladimir Markovic and Jeremy Kahn of the surface subgroup conjecture, that every complete hyperbolic -manifold contains a closed -injective surface. Equivalently, contains a closed surface subgroup. (Lewis Bowen had, some years ago, an incomplete proof with many good influential ideas.) Markovic and Kahn proved (as discussed on Danny Calegary’s blog in 2009) the conjecture by showing that every closed hyperbolic 3-manifold contains many immersed surfaces S which are very nearly totally geodesic, and whose fundamental group therefore injects into that of M.
So this is a great event.
One additional thing that appeals to me is that cubical complexes (not only in three dimension!) are crucial, and so is the theory of CAT spaces described by combinatorial properties of cubical complexes.
More from Thurston’s paper:
A nice moment in mathematics
To our readers: After you practiced the meaning of virtually -P, you need to move to the term “residually-X” . And then you are almost ready for LERF (= locally extended residually finite)!