Virtually Haken Conjecture

I like to read the Low-Dimensional Topology blog, despite the fact that I know almost nothing about the subject. (It’s possible I like to read it because I know nothing about the subject.)

Over the past year, several posts convey palpable excitement over a series of preprints that prove two conjectures: the virtually Haken conjecture and its generalization the virtually fibered conjecture. These were apparently the outstanding open conjectures after the proof of geometrization. This post in particular describes the techniques involved in the proof. To see how fast things changed over the past year, this post on the Wise conjecture (an important ingredient of the proof) makes it clear that from the perspective of March of this year it was very much an open question which way the result would most likely turn out.

I’d been meaning to learn more about the subject, just to have a better idea of what happened. (For example, I still don’t really understand what a Haken manifold is, even though I’ve read the definition. Fortuitously, Erica Klarreich has written a long
general-audience article that gives at least some of the flavor of what’s going on.

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