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.

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.

Haken manifolds are named after Wolfgang Haken, who pioneered the use of incompressible surfaces. He proved that Haken manifolds have a hierarchy. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one, but it was left to Jaco and Oertel, almost 20 years later, to show there was 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.

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[edit] Haken Hierarchy

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, i.e. a trivial I-bundle. So the regular neighborhood is a 3-dimensional submanifold with boundary containing 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.) 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 a hierarchy.

[edit] Applications

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.

Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or not. His outline was filled in by substantive efforts by Waldhausen, Johannson, Hemion, Matveev, et al. Since there is an algorithm to check if a 3-manifold is Haken (cf. Jaco-Oertel), the basic problem of recognition of 3-manifolds can be considered to be solved for Haken manifolds.

Friedhelm Waldhausen proved that closed Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism (for the case of boundary, a condition on peripheral structure is needed). So these three-manifolds are completely determined by their fundamental group. In addition, Waldhausen proved that the fundamental groups of Haken manifolds have solvable word problem; this is also true for virtually Haken manifolds.

The hierarchy played a crucial role in William Thurston's geometrization theorem for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds.

Also worthy of note is Klaus Johannson's proof that atoroidal, anannular, boundary-irreducible, Haken three-manifolds have finite mapping class groups. This result can be recovered from the combination of Mostow rigidity with Thurston's geometrization theorem.

[edit] Examples of Haken manifolds

Note that some families of examples are contained in others.

[edit] References

  • Wolfgang Haken, Theorie der Normalflächen. Acta Math. 105 1961 245--375.
  • Wolfgang Haken, Some results on surfaces in $3$-manifolds. 1968 Studies in Modern Topology pp. 39--98 Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.)
  • Wolfgang Haken, Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I. Math. Z. 80 1962 89--120.
  • William Jaco and Ulrich Oertel, An algorithm to decide if a $3$-manifold is a Haken manifold. Topology 23 (1984), no. 2, 195--209.
  • Klaus Johannson, On the mapping class group of simple $3$-manifolds. Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 48--66, Lecture Notes in Math., 722, Springer, Berlin, 1979.
  • Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large. Ann. of Math. (2) 87 1968 56--88.

[edit] See also