James Algebraic and Classical Topology

The Mathematical Works of J. H. C. Whitehead
1. Auflage 2014
ISBN: 978-1-4831-8477-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 364 Seiten, Web PDF

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.

Autoren/Hrsg.

James, I. M.

Inhaltsverzeichnis

1;Front Cover;1
2;Algebraic and Classical Topology;6
3;Copyright Page;7
4;Table of Contents;10
5;EDITORIAL PREFACE;8
6;ACKNOWLEDGMENT;9
7;PUBLICATIONS OF J. H. C. WHITEHEAD;12
8;CHAPTER 1. ON CERTAIN THEOREMS OF G. W. WHITEHEAD;18
8.1;1. There seems to be a sign wrong in each of the following Theorems;18
8.2;2. Theorem 2 in HSR;18
8.3;3. Theorem (3.2) in HP;19
8.4;4. Orientations;22
8.5;5. Operations on homotopy groups;24
8.6;6. Theorem (5.1) in HI;25
9;CHAPTER 2. NOTE ON THE WHITEHEAD PRODUCT;30
9.1;1. Introduction;30
9.2;2. A general theorem;30
9.3;3. Suspension properties;31
9.4;4. The cases Pa = 0 and Pa . 0;35
9.5;5. Appendix to (4.12);40
9.6;6. Appendix to (4.12), (4.15);41
9.7;REFERENCES;42
10;CHAPTER 3. NOTE ON FIBRE SPACES;44
10.1;1. Introduction;44
10.2;2. The main theorem;44
10.3;3. Further theorems;47
10.4;4. Note on cross-sections;49
10.5;5. Note on covering homotopies;51
10.6;REFERENCES;52
11;CHAPTER 4. THE HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES (I);54
11.1;1. Introduction;54
11.2;2. An expression for J;61
11.3;3. B as a cell-complex;62
11.4;4. Proof of (1.5);66
11.5;5. Proof of (1.6), (1.7);68
11.6;6. Proof of (1.23);71
11.7;7. Neighbourhood bundles;73
11.8;8. Examples;74
11.9;REFERENCES;76
12;CHAPTER 5. THE HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES (II);78
12.1;1. Introduction;78
12.2;2. Note on orientation;83
12.3;3. Some lemmas;85
12.4;4. Proof of (1.3);90
12.5;5. Proof of (1.4);92
12.6;6. Proof of (1.5), (1.6);93
12.7;7. Examples;94
12.8;8. Appendix;95
12.9;REFERENCES;96
13;CHAPTER 6. ON FIBRE SPACES IN WHICH THE FIBRE IS CONTRACT!BLE;98
13.1;REFERENCES;105
14;CHAPTER 7. OBSTRUCTIONS TO COMPRESSION;106
14.1;1. Introduction;106
14.2;2. Statement of results;107
14.3;3. The addition of maps;110
14.4;4. Proof of Theorem 2.1;112
14.5;5. Proof of (2.4), (2.5), (2.7);113
14.6;REFERENCES;115
15;CHAPTER 8. A FIRST APPROXIMATION TO HOMOTOPY THEORY;116
16;CHAPTER 9. THE THEORY OF CARRIERS AND S-THEORY;122
16.1;Introduction;122
16.2;I. THE THEORY OF CARRIERS;123
16.3;II. THE SUSPENSION CATEGORY;138
16.4;REFERENCES;152
17;CHAPTER 10. DUALITY IN HOMOTOPY THEORY;154
17.1;1. Introduction;154
17.2;2. Preliminaries;154
17.3;3. n-dvals;157
17.4;4. The basic duality;157
17.5;5. Functorial presentation;160
17.6;6. Weakly dual constructions;161
17.7;7. Applications;163
17.8;8. Polyhedral mapping cylinders;165
17.9;9. Proof of (4. 1), ..., (4. 5) ;166
17.10;References;178
18;CHAPTER 11. DUALITY IN TOPOLOGY;180
18.1;1. The principle of duality;180
18.2;2. Duality in group-theory;182
18.3;3. Homology and cohomolog;183
18.4;4. Duality in manifolds;186
18.5;5. Duality in homotopy theory;188
18.6;6. Note on coconnectedness;191
18.7;Bibliography;193
19;CHAPTER 12. DUALITY BETWEEN CW-LATTICES;196
19.1;1. Introduction;196
19.2;2. Extension and compression;196
19.3;3. External duality for pairs;197
19.4;4. The category CJ;197
19.5;5. The algebraic duality;199
19.6;6. The geometric duality;200
19.7;7. External duality;201
19.8;8. Dual attachments;202
19.9;9. The dual sequences;203
19.10;10. Combinatorial duals;204
19.11;11. Dual exact couples;205
19.12;REFERENCES;206
20;CHAPTER 13. DUALITY IN RELATIVE HOMOTOPY THEORY;208
20.1;1. Introduction;208
20.2;2. Preliminaries;209
20.3;3. Complete lattices;212
20.4;4. Dual lattices;215
20.5;5. n-duals;221
20.6;6. The duality;223
20.7;7. Weak duality;224
20.8;8. Dual attachments;226
20.9;9. External duality;228
20.10;10. Combinatorial n-duals;230
20.11;11. Dual exact couples;234
20.12;12. Mapping lattices;237
20.13;13. Proofs of (6.2), …, (6.7);239
20.14;14. Proof of (8.3);240
20.15;15. H-isomorphic complexes;241
20.16;REFERENCES;242
21;CHAPTER 14. HOMOLOGY WITH ZERO COEFFICIENTS;244
21.1;Introduction;244
21.2;1. The first example;244
21.3;2. On co-cochains;244
21.4;3. A process of construction;245
21.5;4. A further example;246
21.6;REFERENCES;247
22;CHAPTER 15. NOTE ON THE CONDITION n-colc;248
22.1;REFERENCES;249
23;CHAPTER 16. ON INVOLUTIONS OF SPHERES;250
23.1;REFERENCES;251
24;CHAPTER 17. ON 2-SPHERES IN 3-MANIFOLDS;254
24.1;1. Let M be a connected;254
24.2;2. Proof of (1.2);255
24.3;3. Consequences of (1.1);256
24.4;REFERENCES;259
25;CHAPTER 18. ON FINITE COCYCLES AND THE SPHERE THEOREM;260
25.1;1. Introduction;260
25.2;2. The functors Hnf;261
25.3;3. 77-simple elements;261
25.4;4. The case of a simplicial covering;262
25.5;5 The set S(II, G);265
25.6;6. Proof of (1.2);267
25.7;7. Proof of (1.4);268
25.8;REFERENCES;270
26;CHAPTER 19. A PROOF AND EXTENSION OF DEHN'S LEMMA;272
26.1;1. Introduction;272
26.2;2. Preliminaries;272
26.3;3. Proof of Dehn's lemma;273
26.4;4. Proof of (1.1);274
26.5;REFERENCES;276
27;CHAPTER 20. THE IMMERSION OF AN OPEN 3-MANIFOLD IN EUCLIDEAN 3-SPACE;278
27.1;1. Introduction;278
27.2;2. Proof of (1.1);279
27.3;3. Proof of (2.1);281
27.4;4. Proof of (2.4);283
27.5;5. The thickening theorems;285
27.6;REFERENCES;287
28;CHAPTER 21. MANIFOLDS WITH TRANSVERSE FIELDS IN EUCLIDEAN SPACE;288
28.1;1. Introduction;288
28.2;2. Regular Lipschitz maps;298
28.3;3. The metric a for Gk, n;301
28.4;4. A lemma on local homeomorphisms;303
28.5;5. Function spaces;305
28.6;6. Proof of (1.3);306
28.7;7. Proof of (1.5);308
28.8;8. Proof of (1.7);309
28.9;9. Approximation theorems;312
28.10;10. Proof of (1.10);319
28.11;11. Addendum to (1.3), (1.10);326
28.12;12. Proof of (1.11);326
28.13;13. Cr-complexes;327
28.14;14. Transversals to a complex;331
28.15;15. Proof of (1.12);332
28.16;16. The spaces F(s, t, K);332
28.17;17. Proof of (1.13);337
28.18;18. Direct limits of spaces;337
28.19;19. The spaces R8, G8,n;341
28.20;20. Theorems (1.3), …, (1.12) for k = 8;342
28.21;21. Proof of (1.17);344
28.22;REFERENCES;345
29;CHAPTER 22. IMBEDDING OF MANIFOLDS IN EUCLIDEAN SPACE;348
29.1;1. The main theorems;348
29.2;2. Definitions and lemmas;353
29.3;REFERENCES;358
30;CONTENTS OF VOLUMES I TO IV;360