mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 British ...

P²-irreducible In mathematics, a P2-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided \mathbb RP^2 (real projective plane). An orientable manifold is P2-irreducible if and only if it is irreducible.. Every non-orientable P2-irredu ...

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 In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F of a manifold M is said to be 2-sided in M when there is an embedding ::h\colon F\times 1,1to M with h(x,0)=x for each x\in F and ::h(F\times 1,1\ ...

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 s ...

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 that intersects each tetrahedron so that each component of intersection is a ''triangle'' or a ''quad'' (see figure). A triangle cuts off a vertex of the tetrahedron wh ...

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 topolog ...

, 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 Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory. Career Wald ...

, 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 ...

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 theor ...

for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds. proved that

atoroidal In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non- boundary parallel, incompressible t ...

, anannular, boundary-irreducible, Haken three-manifolds have finite

mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...

s. 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 * Jonathan Mostow Jonathan Mostow (born November 28, 1961) is an American film director, screenwriter, and producer. He has directed f ...

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 mappi ...

, 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

References

* * * * * * * {{Manifolds Differential geometry Manifolds 3-manifolds

Haken hierarchy

Applications

Examples of Haken manifolds

See also

References