Foundational Essays on Topological Manifolds, Smoothing and Triangulations

Cover......Page 1
Title......Page 2
Copyright ©......Page 3
FOREWORD......Page 4
TABLE OF CONTENTS......Page 6
GUIDE......Page 8
Essay I:DEFORMATION OF SMOOTH AND PIECEWISE LINEAR MANIFOLD STRUCTURES......Page 10
Section headings in Essay I......Page 11
§ 1. INTRODUCTION......Page 12
§2. SOME DEFINITIONS, CONVENTIONS, AND NOTATIONS......Page 15
§3. HANDLES THAT CAN ALWAYS BE STRAIGHTENED......Page 24
§4. CONCORDANCE IMPLIES ISOTOPY......Page 33
§5. THE PRODUCT STRUCTURE THEOREM......Page 40
Appendix A. COLLARING THEOREMS......Page 49
Appendix B: SUBMERGING A PUNCTURED TORUS......Page 52
Appendix C. MAJORANT APPROXIMATION......Page 55
Essay II: DEFORMATION OF SLICED FAMILIES OF MANIFOLD STRUCTURES......Page 64
Section headings in Essay II......Page 65
§ O. INTRODUCTION......Page 66
§ 1. BUNDLE THEOREMS......Page 68
§2 . SLICED CONCORDANCE IMPLIES SLICED ISOTOPY......Page 82
Essay III: SOME BASIC THEOREMS ABOUT TOPOLOGICAL MANI FOLDS......Page 88
Section headings in Essay III......Page 89
§ O. INTRODUCTION......Page 90
§1. TRANSVERSALlTY......Page 92
§2. HANDLE DECOMPOSITIONS......Page 113
§ 3. MORSE FUNCTIONS.......Page 117
§4. THE SIMPLE HOMOTOPY TYPE OF A TOPOLOGICAL MANIFOLD......Page 126
§5. SIMPLE TYPES DECOMPOSITIONS AND DUALITY......Page 135
Appendix A: ON AVOIDING SURGERY BY USE OF STABLE STRUCTURES......Page 148
Appendix B: STRAIGHTENING POLYHEDRA IN CODIMENSION >=3.......Page 152
Appendix C: A TRANSVERSALITY LEMMA USEFUL FOR SURGERY.......Page 154
Essay IV: STABLE CLASSIFICATION OFSMOOTH AND PIECEWISE LINEAR MANIFOLD STRUCTURES......Page 162
Section headings in Essay IV......Page 163
§ 0. INTRODUCTION......Page 164
§ 1. RECOLLECTIONS CONCERNING MICROBUNDLES......Page 168
§2. ABSORBING THE BOUNDARY......Page 173
§3. MILNOR'S CRITERION FOR TRIANGULATING AND SMOOTHING......Page 175
§4. STATEMENTS ABOUT BIJECTIVITY OF THE SMOOTHING RULE AND THE PULL-BACK RULE.......Page 177
§5. THE MAIN LEMMA FOR BIJECTIVITY OF THE SMOOTHING RULE......Page 180
§6. BIJECTIVITY OF THE SMOOTHING RULE; ITS INVERSE......Page 182
§7. A PARALLEL CLASSIFICATION THEOREM......Page 185
§8. CLASSIFYING SPACES OBTAINED BY E. H. BROWN'S METHOD......Page 189
§9. CLASSIFYING REDUCTIONS......Page 196
CLASSIFICATION THEOREM 10.1.......Page 203
l0.4. Description of 0^-1......Page 204
10.7. TOP/CAT as an H-space......Page 205
10.9. The obstructions appear in Cech cohomology......Page 206
10.10. Interpreting the Postnikov decomposition of p: BCAT~ BTOP......Page 207
10.11. Classification relative to an open subset......Page 208
10.12 The Homotopy Groups of' TOP/CAT......Page 209
Appendix A. NORMAL BUNDLES......Page 212
Appendix B. EXAMPLES OF HOMOTOPY TORI......Page 217
Essay V: CLASSIFICATION OF SLICED FAMILIES OF MANIFOLD STRUCTURES......Page 224
Section headings in Essay V......Page 225
§0. INTRODUCTION......Page 226
§ 1 CLASSIFYING CAT STRUCTURES ON MANIFOLDS BY CAT STRUCTURES ON MICROBUNDLES......Page 229
§2 CAT STRUCTURES AND CLASSIFYING SPACES......Page 242
§3. THE RELATIVE CLASSIFICATION THEOREM......Page 249
§4. CLASSI FICATIONS FOR MANI FOLDS WITH BOUNDARY......Page 251
§5. THE HOMOTOPY GROUPS 1I"*(TOPm/CAT......Page 255
APPENDIX A: THE IMMERSION THEORETIC METHOD WITHOUT HANDLE DECOMPOSITIONS......Page 265
Appendix B: .CLASSIFICATION OF HOMOTOPY TORI: THE NECESSARY CALCULATIONS......Page 273
Appendix C: SOME TOPOLOGICAL SURGERY......Page 282
Stable homeomorphisms andthe annulus conjecture*......Page 300
REFERENCES......Page 307
ON THE TRIANGULATION OF MANIFOLDS ANDTHE HAUPTVERMUTUNG......Page 308
REFERENCES......Page 314
O. Introduction......Page 316
l. History.......Page 317
2. Failure of the Hauplvermutung and the triangulation conjecture......Page 318
3. The unrestricted triangulation conjectnre......Page 321
4. Structures on topological manifolds......Page 322
5. The product structure theorem.......Page 324
6. Simple homotopy theory [44] [46 A1.......Page 327
7. Handlebody theory......Page 328
9. Surgery.......Page 329
10. Cobordism theory : generalities......Page 330
11. Oriented cobordism.......Page 331
12. Unoriented cobordism......Page 333
13. Spin cobordism.......Page 334
14. The periodicity of Casson and Sullivan......Page 336
15. Hauptvermutung and triangulation for nonnal invariants; Sullivan's thesis(......Page 337
16. Manifolds homotopy equivalent to real projective space pn......Page 340
REFERENCES......Page 342
Note on involutions......Page 347
COMBINED BIBLIOGRAPHY......Page 348
Index......Page 362
Back Cover......Page 365