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Algorithmic Topology and Classification of 3-Manifolds

Sergei Matveev

Resumen/Descripción – provisto por la editorial

No disponible.

Palabras clave – provistas por la editorial

Topology; Programming Techniques; Differential Geometry; Algorithms; Symbolic and Algebraic Manipulation

Disponibilidad
Institución detectada Año de publicación Navegá Descargá Solicitá
No detectada 2007 SpringerLink

Información

Tipo de recurso:

libros

ISBN impreso

978-3-540-45898-2

ISBN electrónico

978-3-540-45899-9

Editor responsable

Springer Nature

País de edición

Reino Unido

Fecha de publicación

Información sobre derechos de publicación

© Springer-Verlag Berlin Heidelberg 2007

Cobertura temática

Tabla de contenidos

Simple and Special Polyhedra

We wish to study the geometry and topology of 3-manifolds. To this end we will need the central notion of spine of a 3-manifold. Indeed, we will be able to refine the notion of spine to get a class of spines that give us a natural presentation of 3-manifolds.

Palabras clave: Simplicial Complex; Triple Line; Elementary Move; Bubble Move; Regular Neighborhood.

Pp. 1-57

Complexity Theory of 3-Manifolds

Denote by M the set of all compact 3-manifolds. We wish to study it systematically and comprehensively. The crucial question is the choice of filtration in M . It would be desirable to have a finite number of 3-manifolds in each term of the filtration, all of them being in some sense simpler than those in the subsequent terms. A useful tool here would be a measure of “complexity” of a 3-manifold. Given such a measure, we might hope to enumerate all “simple” manifolds before moving on to more complicated ones. There are several well-known candidates for such a complexity function. For example, take the Heegaard genus g ( M ), defined to be the minimal genus over all Heegaard decompositions of M . Other examples include the minimal number of simplices in a triangulation of M and the minimal crossing number in a surgery presentation for M .

Palabras clave: Boundary Curve; Complexity Theory; Lens Space; Solid Torus; Regular Neighborhood.

Pp. 59-106

Haken Theory of Normal Surfaces

Normal surfaces were introduced by Kneser in 1929 [66]. The theory of normal surfaces was further developed by W. Haken in the early 1960s [38]. Its fundamental importance to the algorithmic topology cannot be overestimated. Most of the work on 3-manifolds since then is based on or related to it.

Palabras clave: Normal Surface; Admissible Solution; Klein Bottle; Match System; Linear Homogeneous Equation.

Pp. 107-145

Applications of the Theory of Normal Surfaces

The theory of normal surfaces is used extensively in algorithmic topology. Algorithms based on it most often follow the General Scheme described below. Suppose that we wish to solve a problem about a given 3-manifold M .

Palabras clave: Klein Bottle; Double Curve; Seifert Surface; Incompressible Surface; Simple Spine.

Pp. 147-190

Algorithmic Recognition of S3

As we have mentioned earlier, recognizing irreducibility of 3-manifolds requires the existence of a recognition algorithm for the sphere S 3. Another motivation for constructing such an algorithm is the following. To the late 1970s topologists elaborated methods for proving an algorithmic classification theorem for Haken manifolds (though a complete proof appeared only in 1997, see Chap. 6). These methods play a crucial role in solving many other problems about Haken manifolds, but they do not work for manifolds which are not sufficiently large. What can one do with them?

Palabras clave: Algorithmic Recognition; Homology Sphere; Special Spine; Simple Spine; Elementary Disc.

Pp. 191-211

Classification of Haken 3-Manifolds

Palabras clave: Klein Bottle; Solid Torus; Boundary Circle; Regular Neighborhood; Incompressible Surface.

Pp. 213-325

3-Manifold Recognizer

In preceding chapters we described a number of important algorithms, which make heavy use of the Haken method of normal surfaces. As a rule, algorithms based on that method have exponential complexity and hence are impractical. In particular, although the recognition problem for Haken manifolds has an algorithmic solution, there is no chance of it being be realized by a computer program, at least in the foreseeable future. On the other hand, quite often experienced topologists recognize 3-manifolds rather quickly. They use other algorithms which, whether based on rigorous mathematics or intuition, are much more efficient. However, this gain does not come for free. The price is that one has to allow an algorithm to be only partial (i.e., not an algorithm at all in the formal meaning of this term). The problem of finding an efficient partial algorithm for answering a particular class of geometric questions is in itself a well-stated mathematical problem. Trying to solve it, we inevitably discover new structural properties of geometric objects.

Palabras clave: Hyperbolic Manifold; Solid Torus; Regular Neighborhood; Relative Spine; Label Molecule.

Pp. 327-381

The Turaev–Viro Invariants

Palabras clave: Euler Characteristic; Mapping Class Group; Spine Move; Lens Space; Klein Bottle.

Pp. 383-420