Matveev Algorithmic Topology and Classification of 3-Manifolds

2003
ISBN: 978-3-662-05102-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 9, 478 Seiten, eBook

Reihe: Algorithms and Computation in Mathematics

Here is a thorough review of topics in 3-dimensional topology, derived from a decade of courses taught by the author. The author keeps the exposition to an elementary level by presenting the material mainly from the point of view of special polyhedra and special spines of 3-manifolds. The book culminates with the recognition procedure for Haken manifolds, and includes up-to-date results in computer enumeration of 3-mainfolds. The second edition adds new results, new proofs, and commentaries. Algorithmic Topology and Classification of 3-Manifolds serves as a standard reference for algorithmic 3-dimensional topology for both graduate students and researchers.

Zielgruppe

Research

Autoren/Hrsg.

Matveev, Sergei

Inhaltsverzeichnis

1. Simple and special polyhedra 1.1. Spines of 3-manifolds 1.2 Elementary moves on special spines 1.3 Special polyhedra which are not spines 2. Complexity theory of 3-manifolds 2.1. What is a complexity of a 3-manifold? 2.2 Properties of the complexity 2.3 Closed manifolds of small complexity 3. The Turaev-Viro invariants 3.1 The Turaev-Viro invariants 3.2 3-Manifolds having the same Turaev-Viro invariants 4. Haken theory of normal surfaces 4.1 Basic notion and Hakens scheme 4.2 Theory of normal curves 4.3 Normal surfaces in triangulated 3-manifolds 4.4 Examples of algorithms based on Hakens theory 4.5 Normal surfaces in handle decompositions 5. Algorithmic recognition of the 3-sphere 5.1 Thin position of links 5.2 Almost normal surfaces and the Rubinstein theorem 5.3 The algorithm 6. Classification of Haken 3-manifolds 6.1 Introduction 6.2 Theorem of Waldhausen 6.3 Simple skeletons and hierarchies 6.4 Jaco-Shalen-Johannson decomposition 6.5 The proof of the algorithmic classification theorem 7. Computer recognition of 3-manifolds 7.1 Simplifying moves on spines 7.2 Experimental results and conjectures 7.3 An efficient partial recognition algorithm 8. Computer calculation of the degree of maps between 3-manifolds 8.1 A conjecture of C. Legrand and H. Zieschang 8.2 Boundary cycles of Seifert 3-manifolds 8.3 Algorithm for calculating the degree 8.4 Computer implementation and results 9. Appendix 9.1 Tables of 3-manifolds up to complexity 6 9.2 Turaev-Viro invariants of manifolds up to complexity 6 9.3 Minimal spines of manifolds up to complexity 6 9.4 Sporadic 3-manifolds of complexity 7.