irreducible 3-manifold that contains an orientable, incompressible surface.

A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken. This conjecture was proven by Ian Agol.

Haken manifolds were introduced by . proved that Haken manifolds have a hierarchy, where they can

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