Representing 3-manifolds By Filling Dehn Surfaces

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:SetsManifoldsCurvesTransversalityRegular deformationsComplexesFilling Dehn Surfaces:Dehn Surfaces in 3-manifoldsFilling Dehn SurfacesNotationSurgery on Dehn Surfaces. Montesinos TheoremJohansson Diagrams:Diagrams Associated to Dehn SurfacesAbstract Diagrams on SurfacesThe Johansson TheoremFilling DiagramsFundamental Group of a Dehn Sphere:Coverings of Dehn SpheresThe Diagram GroupCoverings and RepresentationsApplicationsThe Fundamental Group of a Dehn g-torusFilling Homotopies:Filling HomotopiesBad Haken Moves"Not so Bad" Haken MovesDiagram MovesDuplicationAmendola's MovesProof of Theorem 5.8:Pushing DisksShellings. Smooth TriangulationsComplex f-movesInflating TriangulationsFilling PairsSimultaneous GrowingsProof of Theorem 5.8The Triple Point Spectrum:The Shima's SpheresSome Examples of Filling Dehn SurfacesThe Number of Triple Points as a Measure of Complexity: Montestinos ComplexityThe Triple Point SpectrumSurface-complexityKnots, Knots and Some Open Questions:2-Knots: Lifting Filling Dehn Surfaces1-KnotsOpen Problems
Readership: Graduate students and researchers interested in low-dimensional topology.
3-Manifolds;Immersed Surfaces;Dehn Surfaces;Johansson Diagrams;Set of Moves;Regular HomotopiesKey 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 topologyThe book uses topological and combinatorial tools developed all along the twentieth century making the volume a trip along the history of low-dimensional topologyA 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