Manifolds | 3-manifolds | Differential geometry

Haken manifold

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. 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 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. (Wikipedia).

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Related pages

P2-irreducible manifold | Compact space | Incompressible surface | I-bundle | Hyperbolization theorem | Atoroidal | Seifert fiber space | Mathematics | Manifold decomposition | 2-sided | Topological rigidity | 3-manifold | Virtually Haken conjecture | William Thurston | Normal surface | Mapping class group | Betti number