James Homotopy Theory

The Mathematical Works of J. H. C. Whitehead
1. Auflage 2014
ISBN: 978-1-4831-8476-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 468 Seiten, Web PDF

Homotopy Theory contains all the published mathematical work of J. H. C. Whitehead, written between 1947 and 1955. This volume considers the study of simple homotopy types, particularly the realization of problem for homotopy types. It describes Whitehead's version of homotopy theory in terms of CW-complexes. This book is composed of 21 chapters and begins with an overview of a theorem to Borsuk and the homotopy type of ANR. The subsequent chapters deal with four-dimensional polyhedral, the homotopy type of a special kind of polyhedron, and the combinatorial homotopy I and II. These topics are followed by reviews of other homotopy types, such as group extensions with homotopy operators, cohomology systems, secondary boundary operator, algebraic homotopy, and the G-dual of a semi-exact couple. The last chapters examine the connected complex homotopy types and the second non-vanishing homotopy groups. This book will be of great value to mathematicians.

Autoren/Hrsg.

James, I. M.

Inhaltsverzeichnis

1;Front Cover;1
2;Homotopy Theory;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. NOTE ON A THEOREM DUE TO BORSUK;18
8.1;1. Introduction;18
8.2;2. Another theorem;19
8.3;3. Proof of Theorem;21
8.4;4. Proof of Theorem;22
8.5;5. Note on identification spaces;23
8.6;REFERENCES;25
9;CHAPTER 2. ON THE HOMOTOPY TYPE OF ANR'S;26
9.1;1. Introduction;26
9.2;2. A lemma on mapping cylinders;29
9.3;3. Proof of Theorem;31
9.4;4. Proof of Theorem;33
9.5;5. Lemmas on homology;34
9.6;6. Proof of Theorem;35
9.7;7. Proof of Theorem;35
9.8;8. Two lemmas;36
9.9;REFERENCES;37
10;CHAPTER 3. ON SIMPLY CONNECTED, 4-DIMENSIONAL POLYHEDRA (ABSTRACT);40
10.1;REFERENCES;43
11;CHAPTER 4. ON SIMPLY CONNECTED, 4-DIMENSIONAL POLYHEDRA;44
11.1;1. Introduction;44
11.2;2. The co-homology groups;46
11.3;3. The co-homology ring;54
11.4;4. The Pontrjagin squares;56
11.5;5. Simple 4-dimensional co-homology rings;59
11.6;6. Theorem 3 implies Theorem;61
11.7;7. Cell complexes;62
11.8;8. Homology and co-homology in a cell complex;65
11.9;9. A lemma on extensions of maps;68
11.10;10. Reduced complexes;69
11.11;11. p3 (K3), where K3 is reduced;71
11.12;12. Homotopy and co-homology in a reduced complex;73
11.13;13. Proof of Theorem;77
11.14;14. Proof of Theorem;79
11.15;15. Proof of Lemma;84
11.16;16. Note on cell complexes;85
11.17;REFERENCES;88
12;CHAPTER 5. THE HOMOTOPY TYPE OF A SPECIAL KIND OF POLYHEDRON;90
12.1;1. Introduction;90
12.2;2. A2n- co-homology systems;90
12.3;3. A lemma on realizability;92
12.4;4. Reduced complexes;93
12.5;5. The group;93
12.6;6. Proof of Theorems;97
12.7;7. The case;98
12.8;8. Note on homotopy groups;99
12.9;References;100
13;CHAPTER 6. COMBINATORIAL HOMOTOPY;102
13.1;1. Introduction;102
13.2;2. n-types;105
13.3;3. Jm-complexes;107
13.4;4. Cell complexes;110
13.5;5. CW-complexes;112
13.6;6. Proof of Theorems;122
13.7;7. Note on n-homotopy;123
13.8;8. A process of identification;124
13.9;9. Countable complexes;128
13.10;10. Proof of Lemma;131
13.11;REFERENCES;134
14;CHAPTER 7. COMBINATORIAL HOMOTOPY;136
14.1;1. Introduction;136
14.2;2. Crossed modules;136
14.3;3. A lemma on crossed homomorphisms;140
14.4;4. Homotopy systems;142
14.5;5. The homotopy system of a complex;144
14.6;6. Realizability;148
14.7;7. Homotopy and equivalence;150
14.8;8. Chain groups;151
14.9;9. Chain mappings and homomorphisms;154
14.10;10. Chain homotopy;157
14.11;11. Chain equivalence: Proof of Theorem;160
14.12;12. Covering complexes;162
14.13;13. Chain mappings: Proof of Theorems 5 and 6;166
14.14;14. Proof of Theorem;171
14.15;15. Examples;172
14.16;16. Note on;176
14.17;17. The groups;177
14.18;REFERENCES;179
15;CHAPTER 8. SIMPLE HOMOTOPY TYPES;180
15.1;1. Introduction;180
15.2;2. The group;181
15.3;3. Chain systems;189
15.4;4. Deformation retracts;192
15.5;5. Simple equivalence;194
15.6;6. Null-equivalent systems;198
15.7;7. Conjugate systems;204
15.8;8. Mapping cylinders;205
15.9;9. The groupoid;210
15.10;10. Homotopy types of complexes;213
15.11;11. Combinatorial invariance;216
15.12;12. Lens spaces;218
15.13;13. Formal deformations;220
15.14;14. Lemmas on formal deformations;222
15.15;15. Proof of Theorem;226
15.16;16. n-types;228
15.17;17. Homotopy systems;230
15.18;18. Infinite complexes;235
15.19;REFERENCES;236
16;CHAPTER 9. ON THE REALIZABILITY OF HOMOTOPY GROUPS;238
16.1;REFERENCES;240
17;CHAPTER 10. ON GROUP EXTENSIONS WITH OPERATORS;242
17.1;1. Introduction;242
17.2;2. Vector cohomology groups;243
17.3;3. Normalized co-chains;245
17.4;4. Application to;246
17.5;5. (X, Y)-kernels;248
17.6;6. Lyndon's normalization;249
17.7;REFERENCES;251
18;CHAPTER 11. ON THE 3-TYPE OF A COMPLEX;252
18.1;1. Introduction;252
18.2;2. Crossed Sequences;253
18.3;4. Mappings of Complexes with Operators;256
18.4;5. Proof of Theorem;258
18.5;6. A Sufficiency Theorem;258
19;CHAPTER 12. NOTE ON COHOMOLOGY SYSTEMS;260
19.1;1. Introduction;260
19.2;2. The block invariants;262
19.3;3. Two combinatorial operations;264
19.4;REFERENCES;267
20;CHAPTER 13. THE SECONDARY BOUNDARY OPERATOR;268
20.1;1. The Sequence;268
20.2;2. The Group G(A);270
20.3;3. G(A) and Cohomology;271
20.4;4. The Calculation of S4(K);272
20.5;5. An2-polyhedra;273
21;CHAPTER 14. ALGEBRAIC HOMOTOPY THEORY;274
21.1;REFERENCES;276
22;CHAPTER 15. A CERTAIN EXACT SEQUENCE;278
22.1;Introduction;278
22.2;CHAPTER I. THE SEQUENCE S(C, A);280
22.3;CHAPTER II. THE GROUP G(A);287
22.4;CHAPTER III. THE SEQUENCE S(K);298
22.5;CHAPTER IV. THE PONTRJAGIN SQUARES;310
22.6;CHAPTER V. THE SEQUENCE OF A GENERAL SPACE;325
22.7;APPENDIX A. ON SPACES DOMINATED BY COMPLEXES;334
22.8;APPENDIX B. ON SEPARATION COCHAINS;335
22.9;REFERENCES;336
23;CHAPTER 16. ON THE THEORY OF OBSTRUCTIONS;338
23.1;1. Introduction;338
23.2;2. General remarks on obstructions;338
23.3;3. The secondary boundary operator;341
23.4;4. Characteristic classes;344
23.5;5. Squaring operations;345
23.6;6. The relative extension theorem;349
23.7;7. Homotopy classification;350
23.8;8. Proof of (5.5) :dp0 = p1d;352
23.9;REFERENCES;353
24;CHAPTER 17. THE G-DUAL OF A SEMI-EXACT COUPLE;356
24.1;1. Introduction;356
24.2;2. Two theorems;360
24.3;3. Analysis of Hn(G), IIn(G);363
24.4;4. Semi-exact couples;368
24.5;5. Invariance of S(a);375
24.6;6. The couple;381
24.7;7. The low-dimensional groups;385
24.8;REFERENCES;387
25;CHAPTER 18. ON THE (n+2)-TYPE OF AN (n–1)-CONNECTED COMPLEX n>4);388
25.1;1. The main theorems;388
25.2;2. A lemma;397
25.3;3. Two theorems;397
25.4;4. Proof of (1.7);402
25.5;5. Proof of (1.8);403
25.6;6. Proof of (1.15);403
25.7;7. Proof of (1.19);405
25.8;8. Note on Ext(Q, G);405
25.9;REFERENCES;409
26;CHAPTER 19. ON THE EXACT COUPLE OF A CW-TRIAD;412
26.1;1. Introduction;412
26.2;2. Basic conventions;413
26.3;3. The homomorphisms ., µ;414
26.4;4. An exact sequence;418
26.5;5. An exact couple;419
26.6;6. An addition theorem;421
26.7;REFERENCES;423
27;CHAPTER 20. ON THE SECOND NON-VANISHING HOMOTOPY GROUPS OF PAIRS AND TRIADS;424
27.1;1. Introduction;424
27.2;2. On Ext (Q, G);424
27.3;3. The groups pr+1(Y), pr+1(K, L);428
27.4;4. Calculation of pm+n+1(A X ZB,A V B);430
27.5;5. Calculation of pm+n(X; A, B);433
27.6;6. The homomorphism .;436
27.7;REFERENCES;438
28;CHAPTER 21. THE FIRST NON-VANISHING GROUP OF AN (n + 1)-AD;440
28.1;1. Introduction;440
28.2;2. Preliminaries;440
28.3;3. Products in ( n + 1 )-ads;444
28.4;4. Complete (w+1)-ads;448
28.5;5. The main theorem;450
28.6;6. Certain triads;453
28.7;7. An algebraic lemma;456
28.8;8. Homotopy groups of (X, B);457
28.9;9. Two geometrical lemmas;460
29;REFERENCES;461
30;CONTENTS OF VOLUMES I TO IV;464