Algebraic and Classical Topology (eBook)

The Mathematical Works of J. H. C. Whitehead

I. M. James (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
364 Seiten
Elsevier Science (Verlag)
978-1-4831-8477-7 (ISBN)

Infos

Algebraic and Classical Topology contains all the published mathematical work of J. H. C. Whitehead, written between 1952 and 1960. This volume is composed of 21 chapters, which represent two groups of papers. The first group, written between 1952 and 1957, is principally concerned with fiber spaces and the Spanier-Whitehead S-theory. In the second group, written between 1957 and 1960, Whitehead returns to classical topology after a long interval, and participates in the renewed assault on the problems which fascinated him most. This book will prove useful to mathematicians.

Inhalt

Front Cover 1
Algebraic and Classical Topology 6
Copyright Page 7
Table of Contents 10
EDITORIAL PREFACE 8
ACKNOWLEDGMENT 9
PUBLICATIONS OF J. H. C. WHITEHEAD 12
CHAPTER 1. ON CERTAIN THEOREMS OF G. W. WHITEHEAD 18
1. There seems to be a sign wrong in each of the following Theorems 18
2. Theorem 2 in HSR 18
3. Theorem (3.2) in HP 19
4. Orientations 22
5. Operations on homotopy groups 24
6. Theorem (5.1) in HI 25
CHAPTER 2. NOTE ON THE WHITEHEAD PRODUCT 30
1. Introduction 30
2. A general theorem 30
3. Suspension properties 31
4. The cases Pa = 0 and Pa . 0 35
5. Appendix to (4.12) 40
6. Appendix to (4.12), (4.15) 41
REFERENCES 42
CHAPTER 3. NOTE ON FIBRE SPACES 44
1. Introduction 44
2. The main theorem 44
3. Further theorems 47
4. Note on cross-sections 49
5. Note on covering homotopies 51
REFERENCES 52
CHAPTER 4. THE HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES (I) 54
1. Introduction 54
2. An expression for J 61
3. B as a cell-complex 62
4. Proof of (1.5) 66
5. Proof of (1.6), (1.7) 68
6. Proof of (1.23) 71
7. Neighbourhood bundles 73
8. Examples 74
REFERENCES 76
CHAPTER 5. THE HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES (II) 78
1. Introduction 78
2. Note on orientation 83
3. Some lemmas 85
4. Proof of (1.3) 90
5. Proof of (1.4) 92
6. Proof of (1.5), (1.6) 93
7. Examples 94
8. Appendix 95
REFERENCES 96
CHAPTER 6. ON FIBRE SPACES IN WHICH THE FIBRE IS CONTRACT!BLE 98
REFERENCES 105
CHAPTER 7. OBSTRUCTIONS TO COMPRESSION 106
1. Introduction 106
2. Statement of results 107
3. The addition of maps 110
4. Proof of Theorem 2.1 112
5. Proof of (2.4), (2.5), (2.7) 113
REFERENCES 115
CHAPTER 8. A FIRST APPROXIMATION TO HOMOTOPY THEORY 116
CHAPTER 9. THE THEORY OF CARRIERS AND S-THEORY 122
Introduction 122
I. THE THEORY OF CARRIERS 123
II. THE SUSPENSION CATEGORY 138
REFERENCES 152
CHAPTER 10. DUALITY IN HOMOTOPY THEORY 154
1. Introduction 154
2. Preliminaries 154
3. n-dvals 157
4. The basic duality 157
5. Functorial presentation 160
6. Weakly dual constructions 161
7. Applications 163
8. Polyhedral mapping cylinders 165
9. Proof of (4. 1), ..., (4. 5) 166
References 178
CHAPTER 11. DUALITY IN TOPOLOGY 180
1. The principle of duality 180
2. Duality in group-theory 182
3. Homology and cohomolog 183
4. Duality in manifolds 186
5. Duality in homotopy theory 188
6. Note on coconnectedness 191
Bibliography 193
CHAPTER 12. DUALITY BETWEEN CW-LATTICES 196
1. Introduction 196
2. Extension and compression 196
3. External duality for pairs 197
4. The category CJ 197
5. The algebraic duality 199
6. The geometric duality 200
7. External duality 201
8. Dual attachments 202
9. The dual sequences 203
10. Combinatorial duals 204
11. Dual exact couples 205
REFERENCES 206
CHAPTER 13. DUALITY IN RELATIVE HOMOTOPY THEORY 208
1. Introduction 208
2. Preliminaries 209
3. Complete lattices 212
4. Dual lattices 215
5. n-duals 221
6. The duality 223
7. Weak duality 224
8. Dual attachments 226
9. External duality 228
10. Combinatorial n-duals 230
11. Dual exact couples 234
12. Mapping lattices 237
13. Proofs of (6.2), …, (6.7) 239
14. Proof of (8.3) 240
15. H-isomorphic complexes 241
REFERENCES 242
CHAPTER 14. HOMOLOGY WITH ZERO COEFFICIENTS 244
Introduction 244
1. The first example 244
2. On co-cochains 244
3. A process of construction 245
4. A further example 246
REFERENCES 247
CHAPTER 15. NOTE ON THE CONDITION n-colc 248
REFERENCES 249
CHAPTER 16. ON INVOLUTIONS OF SPHERES 250
REFERENCES 251
CHAPTER 17. ON 2-SPHERES IN 3-MANIFOLDS 254
1. Let M be a connected 254
2. Proof of (1.2) 255
3. Consequences of (1.1) 256
REFERENCES 259
CHAPTER 18. ON FINITE COCYCLES AND THE SPHERE THEOREM 260
1. Introduction 260
2. The functors Hnf 261
3. 77-simple elements 261
4. The case of a simplicial covering 262
5 The set S(II, G) 265
6. Proof of (1.2) 267
7. Proof of (1.4) 268
REFERENCES 270
CHAPTER 19. A PROOF AND EXTENSION OF DEHN'S LEMMA 272
1. Introduction 272
2. Preliminaries 272
3. Proof of Dehn's lemma 273
4. Proof of (1.1) 274
REFERENCES 276
CHAPTER 20. THE IMMERSION OF AN OPEN 3-MANIFOLD IN EUCLIDEAN 3-SPACE 278
1. Introduction 278
2. Proof of (1.1) 279
3. Proof of (2.1) 281
4. Proof of (2.4) 283
5. The thickening theorems 285
REFERENCES 287
CHAPTER 21. MANIFOLDS WITH TRANSVERSE FIELDS IN EUCLIDEAN SPACE 288
1. Introduction 288
2. Regular Lipschitz maps 298
3. The metric a for Gk, n 301
4. A lemma on local homeomorphisms 303
5. Function spaces 305
6. Proof of (1.3) 306
7. Proof of (1.5) 308
8. Proof of (1.7) 309
9. Approximation theorems 312
10. Proof of (1.10) 319
11. Addendum to (1.3), (1.10) 326
12. Proof of (1.11) 326
13. Cr-complexes 327
14. Transversals to a complex 331
15. Proof of (1.12) 332
16. The spaces F(s, t, K) 332
17. Proof of (1.13) 337
18. Direct limits of spaces 337
19. The spaces R8, G8,n 341
20. Theorems (1.3), …, (1.12) for k = 8 342
21. Proof of (1.17) 344
REFERENCES 345
CHAPTER 22. IMBEDDING OF MANIFOLDS IN EUCLIDEAN SPACE 348
1. The main theorems 348
2. Definitions and lemmas 353
REFERENCES 358
CONTENTS OF VOLUMES I TO IV 360

Produkt-Details

• Erscheint lt. Verlag: 9.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4831-8477-3 / 1483184773
• ISBN-13: 978-1-4831-8477-7 / 9781483184777