Embeddings in manifolds

Contents......Page 8
Acknowledgments......Page 12
Introduction: The Main Problem......Page 14
0.1. More definitions and notation......Page 19
Exercises......Page 20
0.2. The Seifert-van Kampen Theorem......Page 21
0.3. The ultimate duality theorem......Page 24
Exercises......Page 26
0.5. Higher homotopy groups......Page 27
0.6. Absolute neighborhood retracts......Page 30
Exercises......Page 31
0.7. Dimension theory......Page 32
0.8. The Hurewicz Isomorphism Theorem and its localization......Page 34
0.10. Acyclic complexes and contractible manifolds......Page 35
0.11. The 2-dimensional PL Sch¨onflies Theorem......Page 37
1.1. Knotted and flat piecewise linear embeddings......Page 41
Exercises......Page 46
1.2. Tame and locally flat topological embeddings......Page 47
1.3. Local co-connectedness properties......Page 48
Exercises......Page 52
1.4. Suspending and spinning......Page 53
Exercises......Page 57
2.1. Antoine’s necklace and Alexander’s horned sphere......Page 59
Exercises......Page 70
2.2. Function spaces......Page 71
2.3. Shrinkable decompositions and the Bing shrinking criterion......Page 73
2.4. Cellular sets and the Generalized Sch¨onflies Theorem......Page 77
Exercises......Page 87
Exercises......Page 88
2.6. The product of R^1 with an arc decomposition......Page 89
2.7. Everywhere wild cells and spheres......Page 95
2.8. Miscellaneous examples of wild embeddings......Page 97
2.9. Embeddings that are piecewise linear modulo one point......Page 113
Exercises......Page 120
3 Engulfing, Cellularity, and Embedding Dimension......Page 121
3.1. Engulfing without control......Page 122
3.2. Application: The cellularity criterion......Page 130
3.3. Engulfing with control......Page 138
3.4. Application: Embedding dimension......Page 147
3.5. Embeddings of Menger continua......Page 154
3.6. Embedding dimension and Hausdorff dimension......Page 160
4 Trivial-range Embeddings......Page 169
4.1. Unknotting PL embeddings of polyhedra......Page 170
Exercise......Page 175
4.2. Spaces of embeddings and taming of polyhedra......Page 176
4.3. Taming 1-LCC embeddings of polyhedra......Page 183
4.4. Unknotting 1-LCC embeddings of compacta......Page 185
4.5. Chart-by-chart analysis of topological manifolds......Page 186
4.6. Detecting 1-LCC embeddings......Page 187
4.7. More wild embeddings......Page 193
4.8. Even more wild embeddings......Page 196
Exercises......Page 204
5 Codimension-three Embeddings......Page 205
5.1. Constructing PL embeddings of polyhedra......Page 206
Exercises......Page 213
5.2. Constructing PL embeddings of manifolds......Page 214
Exercise......Page 219
5.3. Unknotting PL embeddings of manifolds......Page 220
5.4. Unknotting PL embeddings of polyhedra......Page 235
Exercise......Page 250
5.5. 1-LCC approximation of embeddings of compacta......Page 251
5.6. PL approximation of embeddings of manifolds......Page 267
5.7. Taming 1-LCC embeddings of polyhedra......Page 285
5.8. PL approximation of embeddings of polyhedra......Page 287
Exercises......Page 296
6 Codimension-two Embeddings......Page 297
6.1. Piecewise linear knotting and algebraic unknotting......Page 298
6.2. Topological flattening and algebraic knotting......Page 307
6.3. Local flatness and local homotopy properties......Page 317
Exercises......Page 321
6.4. The homology of an infinite cyclic cover......Page 322
Exercises......Page 334
6.5. Properties of the Alexander polynomial......Page 335
by PL embeddings......Page 349
6.7. A homotopy equivalence that is not homotopic to an embedding......Page 358
6.8. Disk bundle neighborhoods and taming......Page 370
7 Codimension-one Embeddings......Page 373
7.1. Codimension-one separation properties......Page 374
7.2. The 1-LCC characterization of local flatness for collared embeddings......Page 377
Exercises......Page 381
7.3. Unknotting close 1-LCC embeddings of manifolds......Page 382
7.4. The Cell-like Approximation Theorem......Page 390
7.5. Determining n-cells by embeddings of M_N^(n-1) in S^n......Page 395
7.6. The 1-LCC characterization of local flatness......Page 401
7.7. Locally flat approximations......Page 406
7.8. Kirby-Siebenmann obstruction theory......Page 426
7.9. Detecting 1-LCC embeddings......Page 427
7.10. Sewings of crumpled n-cubes......Page 434
7.11. Wild examples and mapping cylinder neighborhoods......Page 441
Exercises......Page 448
8.1. Manifold characterizations......Page 449
8.3. Ends of manifolds......Page 451
8.4. Ends of maps......Page 456
8.5. Quinn’s obstruction and the topological characterization of manifolds......Page 459
8.6. Exotic homology manifolds......Page 461
Exercise......Page 462
8.8. Approximating stable homeomorphisms of R^n by PL homeomorphisms......Page 463
8.9. Rigidity: Homotopy equivalence implies homeomorphism......Page 466
8.10. Simplicial triangulations......Page 467
Bibliography......Page 469
Abbreviations......Page 485
Index......Page 487