Haken manifold
a compact, P²-irreducible 3-manifold that contains a properly embedded two-sided incompressible surface
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, 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 Wolfgang Haken (1961). Haken (1962) proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. William Jaco and Ulrich Oertel (1984) gave an algorithm to determine if a 3-manifold was Haken.
Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
The public source identifies “Haken manifold” as a compact, P²-irreducible 3-manifold that contains a properly embedded two-sided incompressible surface. This brief keeps that definition visible, then builds a research path around Haken, manifold and compact.
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