Haken manifold
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In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a 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.
Haken manifolds are named after Wolfgang Haken, who pioneered the use of incompressible surfaces. He proved that Haken manifolds have a hierarchy.
To understand the concept of hierarchy, one must first understand regular neighborhoods. We will consider only the case of orientable Haken manifolds, as this simplifies the discussion; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a "thickened up" version of the surface. So the regular neighborhood is a 3-dimensional submanifold with boundary being two copies of the surface.
Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. (This is analogous (in one less dimension) to cutting a surface along a circle or arc.) The new manifold M' will have two new boundary components, each of which looks like S. It is a theorem that cutting a Haken manifold along an incompressible surface results in a Haken manifold. Thus, we can pick an incompressible surface in M' , and cut along that. If eventually this sequence of cutting results in a manifold whose pieces (or components) are just 3-balls, we call this sequence an hierarchy.
Summing up, Haken proved that given any Haken manifold, there is a (finite) sequence of cuttings (as above) that we can do, which results in pieces that are very simple, in fact are 3-balls. Note that not any sequence of cuttings will terminate this way, so we must be clever in how we pick each surface we cut along. In general, there will be infinitely many incompressible surfaces to pick from.
The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction. One proves the theorem for 3-balls. Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, that it is true for that Haken manifold. The key here is that the cutting takes place along a surface that was very "nice", i.e. incompressible. This makes proving the induction step feasible in many cases.
A pair of notable theorems have been proved along these lines. Friedhelm Waldhausen proved that Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism. So these three-manifolds are completely determined by their fundamental group. More recent is William Thurston's geometrization theorem for Haken manifolds.
Also worthy of note is Klaus Johannson's proof that atoroidal Haken three-manifolds have finite mapping class groups. In the hyperbolic case, this work is subsumed by the combination of Mostow rigidity with Thurston's theorem.