Topological Embeddings (eBook)

Front Cover 1
Topologicul Embeddings 4
Copyright Page 5
Contents 8
Preface 12
Acknowledgments 14
Chapter 1. Main Problem and Preliminary Notions 16
1.1. Introduction 16
1.2. Main Problem 16
1.3. Topological Manifolds 17
1.4. The Category of Polyhedra and Piecewise Linear Maps 19
1.5. Method of Attacking the Main Problem 23
1.6. Piecewise Linear Manifolds and Piecewise Linear Tools 24
1.7. Local Flatness, (Pinched) Collars, and (Pinched) Bicollars 48
1.8. Cellular Sets and Applications 59
Chapter 2. Wild Embeddings, Knotted Embeddings, and Related Topics 66
2.1. Introductory Definitions 66
2.2. The Group of a Knot and Knotted Codimension Two Spheres 66
2.3. Local Homotopy Groups, Wild Codimension Two Cells and Spheres, and Tame Nonlocally Flat Codimension Two Cells and Spheres 71
2.4. Wild 1-Cells, 1-Spheres, 2-Cells, and 2-Spheres in S3 75
2.5. En Modulo an Arc Crossed with E1 Is En+1 89
2.6. Everywhere Wild Cells and Spheres in En?3 of All Codimensions 99
2.7. Some Wild Polyhedra in Low Codimensions 106
Chapter 3. Flattening, Unknotting, and Taming Special Embeddings 113
3.1. Introduction 113
3.2. Almost Polyhedral Arcs Are Flat 114
3.3. An (n - 1)-Sphere in Sn?4 which Is Locally Flat Modulo a Point Is Flat 115
3.4. Flattening Cells, Half-Strings, and Strings 120
3.5. PL Unknotting Infinite Polyhedra in the Trivial Range 125
3.6. ?(x)-Taming Locally Tame Embeddings of Infinite Polyhedra in The Trivial Range 133
3.7. ?-Taming Polyhedra in the Trivial Range Which Lie in Hyperplanes 147
3.8. ?(x) Taming Embeddings Which are Locally Tame Modulo Nice Subsets 149
3.9. Nonlocally Flat Points of a Codimension One Submanifold 151
Chapter 4. Engulfing and Applications 163
4.1. Introduction 163
4.2. Stallings’ Engulfing 164
4.3. The Generalized Poincaré Theorem 168
4.4. The Hauptvermutung for Open Cells 169
4.5. Flattening Topological Sphere Pairs and Cell Pairs 173
4.6. Zeeman Engulfing 181
4.7. The Penrose, Whitehead, Zeeman Embedding Theorem and Irwin’s Embedding Theorem 185
4.8. The Cellularity Criterion in High Dimensions 193
4.9. Locally Nice Codimension One Spheres in Sn?5 Are Weakly Flat 198
4.10. Radial Engulfing 200
4.11. The PL Approximation of Stable Homeomorphisms of En 208
4.12. Topological Engulfing 215
4.13. Topological H-Cobordisms and the Topological Poincaré Theorem 221
4.14. Infinite Engulfing 229
Chapter 5. Taming and PL Approximating Embeddings 235
5.1. Introduction 235
5.2. ?ernavski?’s Straightening Technique Applied to Cell Pairs and to Singular Points of Topological Embeddings 236
5.3. Taming Embeddings of PL Manifolds around the Boundary in All Codimensions 256
5.4. PL Approximating Topological Embeddings 264
5.5. ?-Taming Allowable Embeddings of PL Manifolds 274
5.6. Local Contractibility of the Homeomorphism Group of a Manifold and Codimension Zero Taming 285
Appendix: Some Topics for Further Study 309
Bibliography 311
Author Index 324
Subject Index 327