REPRESENTING 3-MANIFOLDS BY FILLING DEHN SURFACES

by Ruben Vigara Benito, Alvaro Lozano-rojo

Overview

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.

Product Details

ISBN-13:	9789814725507
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/11/2016
Series:	SERIES ON KNOTS AND EVERYTHING , #58
Sold by:	Barnes & Noble
Format:	eBook
Pages:	300
File size:	14 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface vii

1 Preliminaries 1

1.1 Sets 1

1.2 Manifolds 1

1.3 Curves 2

1.4 Transversality 3

1.5 Begular deformations 4

1.6 Complexes 6

2 Filling Dehn surfaces 9

2.1 Dehn surfaces in 3-manifolds 9

2.2 Filling Dehn surfaces 11

2.3 Notation 13

2.4 Surgery on Dehn surfaces. Montesinos Theorem 15

2.4.1 Type 0 arcs 16

2.4.2 Type 1 arcs 17

2.4.3 Type 2 arcs 18

2.4.4 Surgeries 19

2.4.5 Spiral piping 21

3 Johansson diagrams 25

3.1 Diagrams associated to Dehn surfaces 25

3.2 Abstract diagrams on surfaces 26

3.3 The Johansson Theorem 30

3.4 Filling diagrams 41

4 Fundamental group of a Dehn sphere 51

4.1 Coverings of Dehn spheres 51

4.2 The diagram group 53

4.3 Coverings and representations 54

4.4 Applications 58

4.5 The fundamental group of a Dehn g-torus 60

5 Filling homotopies 63

5.1 Filling homotopies 63

5.2 Bad Haken moves 68

5.3 "Not so bad" Haken moves 70

5.4 Diagram moves 72

5.5 Duplication 77

5.6 Amendola's moves 82

6 Proof of Theorem 5.8 85

6.1 Pushing disks 85

6.2 Shellings. Smooth triangulations 103

6.2.1 Shellings 103

6.2.2 Smooth triangulations 106

6.2.3 Shellings of 2-disks 108

6.3 Complex f-moves 111

6.3.1 Finger move 3/2 111

6.3.2 Singular saddles 113

6.3.3 Pushing disks along a 2-cells 115

6.3.4 Pushing disks along 3-cells 118

6.3.5 Inflating double points 122

6.3.6 Passing through spiral pipings 125

6.4 Inflating triangulations 130

6.4.1 Inflating T 130

6.4.2 Naming the regions of Σ_T 134

6.4.3 Σ_T fills M 137

6.4.4 Inflating filling immersions 139

6.5 Filling pairs 144

6.6 Simultaneous growings 145

6.7 Proof of Theorem 5.8 148

7 The triple point spectrum 153

7.1 The Shima's spheres 153

7.2 Some examples of filling Dehn surfaces 158

7.2.1 A filling Dehn sphere in S² × S¹ 158

7.2.2 A filling Dehn torus in S² × S¹ 158

7.2.3 A filling Dehn sphere in L(3, 1) 160

7.3 The number of triple points as a measure of complexity: Montesinos complexity 162

7.4 The triple point spectrum 169

7.5 Surface-complexity 171

8 Knots, knots and some open questions 173

8.1 2-Knots: lifting filling Dehn surfaces 173

8.2 1-Knots 175

8.3 Open problems 177

8.3.1 Filling Dehn surfaces and filling Dehn spheres 177

8.3.2 Filling homotopies. Moves 177

8.3.3 Montesinos complexity. Triple point spectrum 178

8.3.4 Knots 180

Appendix A Proof of Key Lemma 2 183

Appendix B Proof of Lemma 6.46 237

Appendix C Proof of Proposition 6.57 251

Bibliography 267