Please refer to Low Dimensional Topology blog.

**virtually Haken conjecture** states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken.

**Virtual Haken 猜想 **设\(M\) 为紧致的可定向的不可约的基本群无限的\(3\)-流形, 则\(M\)有一个\(Haken\) 流形的有限覆盖.

这个猜想通常归于Friedhelm Waldhausen \(1968\)年的一篇文章, 尽管他没有正式的陈述过这个猜想.

\(3\)-流形的有限覆盖一直是\(3\)-流形拓扑理论中重要但进展缓慢的课题.

Ian Agol (UC Berkeley) announced a proof of the Wise’s Conjecture on March 12, 2012 when he was speaking in a seminar lecture at the Institut Henri Poincaré. In particular, this implies the Virtually Haken Conjecture. His proof is based on joint work with Daniel Groves (UI Chicago) and Jason Manning (SUNY Buffalo). It makes heavy use of the work of Dani Wise (McGill) on the Virtually Fibred Conjecture, as well as the proof of the Surface Subgroup Conjecture by Jeremy Kahn (Brown) and Vlad Markovic (Caltech). 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 , pdf .