|
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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
In mathematics, a 3-manifold is P2-irreducible if it is irreducible and contains no 2-sided (real projective plane). ...
In mathematics, a 3-manifold is a 3-dimensional manifold. ...
In topology, a compact codimension one submanifold of a manifold is said to be 2-sided in when there is an embedding with for each and . This means, for example that a curve in a surface is 2-sided if it has a regular neighborhood which is a cartesian product...
In mathematics, an incompressible surface is a kind of two-dimensional surface inside of a 3-manifold. ...
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. In mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible 3-manifold with infinite fundamental group is virtually Haken, i. ...
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. Wolfgang Haken (born June 21, 1928) is a mathematician who specialized in topology, in particular 3-manifolds. ...
Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms. In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). ...
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. This article or section should be merged with Orientable manifold. ...
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.
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. William Thurston William Paul Thurston (born October 30, 1946) is an American mathematician. ...
Thurstons geometrization conjecture states that compact 3-manifolds can be decomposed into submanifolds that have geometric structures. ...
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. In mathematics, there are three definitions for atoroidal as applied to 3-manifolds: A 3-manifold is (geometrically) atoroidal if it does not contain an embedded, non_boundary parallel, incompressible torus. ...
In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. ...
In mathematics, Mostows rigidity theorem, sometimes called the strong rigidity theorem, essentially states that the geometry of a finite volume hyperbolic manifold (for dimension greater than two) is determined by the fundamental group and hence unique. ...
Examples of Haken manifolds Note that some families of examples are contained in others. In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ...
In mathematics, the knot complement of a tame knot K is the set-theoretic complement of the interior of the embedding of a solid torus into the 3-sphere. ...
A surface bundle over the circle gives a three manifold whose fundamental group can be consider as an extension of the fundamental group of the fiber (a surface) by the integers, also known as HNN-extension. ...
A Seifert fiber space is a 3-manifold together with a nice decomposition as a disjoint union of circles. ...
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.
See also |
Feeds |
Contact