Haken manifold
Encyclopedia
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.
Haken manifolds were introduced by . 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. gave an algorithm to determine if a 3-manifold was Haken.
Normal surface
s are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
. 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 any orientable compact manifold with a boundary component that is not a sphere has an infinite first homology group, which implies that it has a properly embedded 2-sided non-separating incompressible surface, and so is again a Haken manifold. Thus, we can pick another 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.
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.
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 hyperbolization theorem
for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds.
proved that atoroidal
, anannular, boundary-irreducible, Haken three-manifolds have finite mapping class group
s. This result can be recovered from the combination of Mostow rigidity with Thurston's geometrization theorem.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Haken manifold is a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, P²-irreducible
P²-irreducible
In mathematics, a 3-manifold is P2-irreducible if it is irreducible and contains no 2-sided \mathbb RP^2 ....
3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
that is sufficiently large, meaning that it contains a properly embedded two-sided
2-sided
In topology, a compact codimension one submanifold F of a manifold M is said to be 2-sided in M when there is an embeddingwith h=x for each x\in F andIn other words, if its normal bundle is trivial....
incompressible surface
Incompressible surface
In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
. 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
Virtually Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken...
asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
Haken manifolds were introduced by . 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. gave an algorithm to determine if a 3-manifold was Haken.
Normal surface
Normal surface
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 . A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices...
s are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
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-bundleI-bundle
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber....
. 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 any orientable compact manifold with a boundary component that is not a sphere has an infinite first homology group, which implies that it has a properly embedded 2-sided non-separating incompressible surface, and so is again a Haken manifold. Thus, we can pick another 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.
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
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
's hyperbolization theorem
Hyperbolization theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.-Statement:...
for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds.
proved that atoroidal
Atoroidal
In mathematics, there are three definitions for atoroidal as applied to 3-manifolds:*A 3-manifold is atoroidal if it does not contain an embedded, non-boundary parallel, incompressible torus....
, anannular, boundary-irreducible, Haken three-manifolds have finite mapping class group
Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
s. This result can be recovered from the combination of Mostow rigidity with Thurston's geometrization theorem.
Examples of Haken manifolds
Note that some families of examples are contained in others.- Compact, irreducible 3-manifolds with positive first Betti numberBetti numberIn algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
- Link complements
- Surface bundles over the circle
- Most Seifert fiber spaceSeifert fiber spaceA Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...
s have many incompressible tori