Decompositions of Manifolds (eBook)

Front Cover 1
Decompositions of Manifolds 4
Copyright Page 5
Contents 6
Preface 10
Acknowledgments 12
Introduction 14
Chapter I. Preliminaries 20
1. Elementary Properties of Upper Semicontinuous Decompositions 20
2. Upper Semicontinuous Decompositions 26
3. Proper Maps 28
4. Monotone Decompositions 30
Chapter II. The Shrinkability Criterion 35
5. Shrinkable Decompositions 36
6. Cellular Sets 48
7. Countable Decompositions and Shrinkability 54
8. Countable Decompositions of En 63
9. Some Cellular Decompositions of E3 74
10. Products of Decompositions with a Line 94
11. Squeezing a 2-Cell to an Arc 107
12. The Double Suspension of a Certain Homology Sphere 115
13. Applications of the Local Contractability of Manifold Homeomorphism Groups 120
Chapter III. Cell-Like Decompositions of Absolute Neighborhood Retracts 126
14. Absolute Retracts and Absolute Neighborhood Retracts 127
15. Cell-Like Sets 133
16. UV Properties and Decompositions 136
17. Cell-Like Decompositions and Dimension 142
18. The Cellularity Criterion and Decompositions 156
Chapter IV. The Cell-Like Approximation Theorem 161
19. Characterizing Shrinkable Decompositions of Manifolds—the Simple Test 162
20. Amalgamating Decompositions 164
21. The Concept of Embedding Dimension 171
22. Shrinking Special 0-Dimensional Decompositions 179
23. Shrinking Special (n – 3)-Dimensional Decompositions 184
24. The Disjoint Disks Property and the Cell-Like Approximation Theorem 191
25. Cell-Like Decompositions of 2-Manifolds–the Moore Theorem 200
Chapter V. Shrinkable Decompositions 203
26. Products of E2 and E1 with Decompositions 203
27. Products of E1 with Decompositions of E3 219
28. Spun Decompositions 225
29. Products of Generalized Manifolds 236
30. A Mismatch Property in Locally Spherical Decomposition Spaces 240
31. Sliced Decomposition of En+1 245
Chapter VI. Nonshrinkable Decompositions 252
32. Nonshrinkable Cellular Decompositions Obtained by Mixing 252
33. Nonshrinkable Null Sequence Cellular Decompositions Obtained by Amalgamating 254
34. Nested Defining Sequences for Decompositions 258
35. Cell-Like but Totally Noncellular Decompositions 264
36. Measures of Complexity in Decomposition Spaces 269
37. Defining Sequences for Decompositions 273
Chapter VII. Applications to Manifolds 277
38. Gropes and Closed n-Cell-Complements 278
39. Replacement Procedures for Improving Generalized Manifolds 287
40. Resolutions and Applications 297
41. Mapping Cylinder Neighborhoods 304
Appendix 313
References 316
Index 326