Representing 3-Manifolds by Filling Dehn Surfaces

Nonfiction, Science & Nature, Mathematics, Combinatorics, Topology

• Author: Rubén Vigara, Álvaro Lozano-Rojo
• ISBN: 9789814725507
• Publisher: World Scientific Publishing Company
• Publication: March 11, 2016
• Imprint: WSPC
• Language: English

This book provides an introduction to the beautiful and deep subject of filling Dehn surfaces in the study of topological 3-manifolds. This book presents, for the first time in English and with all the details, the results from the PhD thesis of the first author, together with some more recent results in the subject. It also presents some key ideas on how these techniques could be used on other subjects.

Representing 3-Manifolds by Filling Dehn Surfaces is mostly self-contained requiring only basic knowledge on topology and homotopy theory. The complete and detailed proofs are illustrated with a set of more than 600 spectacular pictures, in the tradition of low-dimensional topology books. It is a basic reference for researchers in the area, but it can also be used as an advanced textbook for graduate students or even for adventurous undergraduates in mathematics. The book uses topological and combinatorial tools developed throughout the twentieth century making the volume a trip along the history of low-dimensional topology.

Contents:

• Preliminaries:

  • Sets
  • Manifolds
  • Curves
  • Transversality
  • Regular deformations
  • Complexes

• Filling Dehn Surfaces:

  • Dehn Surfaces in 3-manifolds
  • Filling Dehn Surfaces
  • Notation
  • Surgery on Dehn Surfaces. Montesinos Theorem

• Johansson Diagrams:

  • Diagrams Associated to Dehn Surfaces
  • Abstract Diagrams on Surfaces
  • The Johansson Theorem
  • Filling Diagrams

• Fundamental Group of a Dehn Sphere:

  • Coverings of Dehn Spheres
  • The Diagram Group
  • Coverings and Representations
  • Applications
  • The Fundamental Group of a Dehn g-torus

• Filling Homotopies:

  • Filling Homotopies
  • Bad Haken Moves
  • "Not so Bad" Haken Moves
  • Diagram Moves
  • Duplication
  • Amendola's Moves

• Proof of Theorem 5.8:

  • Pushing Disks
  • Shellings. Smooth Triangulations
  • Complex f-moves
  • Inflating Triangulations
  • Filling Pairs
  • Simultaneous Growings
  • Proof of Theorem 5.8

• The Triple Point Spectrum:

  • The Shima's Spheres
  • Some Examples of Filling Dehn Surfaces
  • The Number of Triple Points as a Measure of Complexity: Montestinos Complexity
  • The Triple Point Spectrum
  • Surface-complexity

• Knots, Knots and Some Open Questions:

  • 2-Knots: Lifting Filling Dehn Surfaces
  • 1-Knots
  • Open Problems

Readership: Graduate students and researchers interested in low-dimensional topology.
Key Features:

• It provides deep results in a new subject of mathematical research. Moreover, it introduces new mathematical tools and techniques useful in different areas of low-dimensional topology
• The book uses topological and combinatorial tools developed all along the twentieth century making the volume a trip along the history of low-dimensional topology
• A spectacular set of pictures, in the better tradition of low-dimensional topology books, which give deep insight of the techniques and constructions done in the book