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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 ... [Link]