Embeddings in Manifolds (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 106)

A topological embedding is a homeomorphism of one space onto a subspace of another. The book analyzes how and when objects like polyhedra or manifolds embed in a given higher-dimensional manifold. The main problem is to determine when two topological embeddings of the same object are equivalent in the sense of differing only by a homeomorphism of the ambient manifold. Knot theory is the special case of spheres smoothly embedded in spheres; in this book, much more general spaces and much more general embeddings are considered. A key aspect of the main problem is taming: when is a topological embedding of a polyhedron equivalent to a piecewise linear embedding? A central theme of the book is the fundamental role played by local homotopy properties of the complement in answering this taming question. The book begins with a fresh description of the various classic examples of wild embeddings (i.e., embeddings inequivalent to piecewise linear embeddings). Engulfing, the fundamental tool of the subject, is developed next. After that, the study of embeddings is organized by codimension (the difference between the ambient dimension and the dimension of the embedded space). In all codimensions greater than two, topological embeddings of compacta are approximated by nicer embeddings, nice embeddings of polyhedra are tamed, topological embeddings of polyhedra are approximated by piecewise linear embeddings, and piecewise linear embeddings are locally unknotted. Complete details of the codimension-three proofs, including the requisite piecewise linear tools, are provided. The treatment of codimension-two embeddings includes a self-contained, elementary exposition of the algebraic invariants needed to construct counterexamples to the approximation and existence of embeddings. The treatment of codimension-one embeddings includes the locally flat approximation theorem for manifolds as well as the characterization of local flatness in terms of local homotopy properties. 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