In
mathematics, a Haken manifold is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Briti ...
,
P²-irreducible 3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
that is sufficiently large, meaning that it contains a properly embedded
two-sided incompressible surface In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the surface of a suitcase is compressible, because ...
. 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
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''. That is, it has a finite cover (a covering ...
asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken. This conjecture was proven by
Ian Agol
Ian Agol (born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds.
Education and career
Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and ...
.
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 In mathematics, a normal surface is a surface inside a triangulated 3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just ...
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
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
Haken manifolds, as this simplifies the discussion; a
regular neighborhood
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrumen ...
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'', resulting in ''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
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolo ...
, 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, then 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
Friedhelm Waldhausen, Klaus Johannson, Geoffrey Hemion, Sergeĭ 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 Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurston ...
's
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
One form of Thurston's geometrization theo ...
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 groups. This result can be recovered from the combination of
Mostow rigidity Mostow may refer to: People
* George Mostow (1923–2017), American mathematician
** Mostow rigidity theorem Mostow may refer to: People
* George Mostow (1923–2017), American mathematician
** Mostow rigidity theorem
* Jonathan Mostow
Jonat ...
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 number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
*
Surface bundles over the circle
In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of map ...
, this is a special case of the example above.
*
Link complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
s
*Most
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
s have many incompressible tori
See also
*
Manifold decomposition
In topology, a branch of mathematics, a manifold ''M'' may be decomposed or split by writing ''M'' as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form ''M''.
Manifold ...
*
''P''2-irreducible manifold
References
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{{Manifolds
Differential geometry
Manifolds
3-manifolds