Homotopy Theory (eBook)

Front Cover 1
Homotopy Theory, Volume 8 4
Copyright Page 5
Contents 8
PREFACE 6
LIST OF SPECIAL SYMBOLS AND ABBREVIATIONS 13
CHAPTER I. MAIN PROBLEM AND PRELIMINARY NOTIONS 16
1. Introduction 16
2. The extension problem 16
3. The method of algebraic topology 18
4. The retraction problem 20
5. Combined maps 22
6. Topological identification 23
7. The adjunction space 24
8. Homotopy problem and classification problem 26
9. The homotopy extension property 28
10. Relative homotopy 30
11. Homotopy equivalences 32
12. The mapping cylinder 33
13. A generalization of the extension problem 35
14. The partial mapping cylinder 36
15. The deformation problem 37
16. The lifting problem 39
17. The most general problem 40
Exercises 40
CHAPTER II. SOME SPECIAL CASES OF THE MAIN PROBLEMS 50
1. Introduction 50
2. The exponential map p: R . S1 50
3. Classification of the maps S1 . S1 52
4. The fundamental group 54
5. Simply connected spaces 57
6. Relation between p1(X, x0) and H1 ( X ) 59
7. The Bruschlinsky group 62
8. The Hopf theorems 67
9. The Hurewicz theorem 71
Exercises 72
CHAPTER III. FIBER SPACES 76
1. Introduction 76
2. Covering homotopy property 76
3. Definition of fiber space 77
4. Bundle spaces 80
5. Hopf fiberings of spheres 81
6. Algebraically trivial maps X . S2 83
7. Liftings and cross-sections 84
8. Fiber maps and induced fiber spaces 86
9. Mapping spaces 88
10. The spaces of paths 93
11. The space of loops 94
12. The path lifting property 97
13. The fibering theorem for mapping spaces 98
14. The induced maps in mapping spaces 100
15. Fiberings with discrete fibers 101
16. Covering spaces 104
17. Construction of covering spaces 108
Exercises 112
CHAPTER IV. HOMOTOPY GROUPS 122
1. Introduction 122
2. Absolute homotopy groups 122
3. Relative homotopy groups 125
4. The boundary operator 127
5. Induced transformations 128
6. The algebraic properties 129
7. The exactness property 130
8. The homotopy property 132
9. The fibering property 133
10. The triviality property 134
11. Homotopy systems 134
12. The uniqueness theorem 136
13. The group structures 138
14. The role of the basic point 140
15. Local system of groups 144
16. n-Simple spaces 146
Exercises 150
CHAPTER V. THE CALCULATION OF HOMOTOPY GROUPS 158
1. Introduction 158
2. Homotopy groups of the product of two spaces 158
3. The one-point union of two spaces 160
4. The natural homomorphisms from homotopy groups to homology groups 161
5. Direct sum theorems 165
6. Homotopy groups of fiber spaces 167
7. Homotopy groups of covering spaces 169
8. The n-connective fiberings 170
9. The homotopy sequence of a triple 174
10. The homotopy groups of a triad 175
11. Freudenthal's suspension 177
Exercises 179
CHAPTER VI. OBSTRUCTION THEORY 190
1. Introduction 190
2. The extension index 190
3. The obstruction cn+1 (g) 191
4. The difference cochain 193
5. Eilenberg's extension theorem 195
6. The obstruction sets for extension 196
7. The homotopy problem 197
8. The obstruction dn(f, g ht)
9. The group Rn(K,L f)
10. The obstruction sets for homotopy 200
11. The general homotopy theorem 201
12. The classification problem 202
13. The primary obstructions 203
14. Primary extension theorems 205
15. Primary homotopy theorems 206
16. Primary classification theorems 206
17. The characteristic element of Y 208
Exercises 208
CHAPTER VII. COHOMOTOPY GROUPS 220
1. Introduction 220
2. The cohomotopy set pm( X, A ) 220
3. The induced transformations 221
4. The coboundary operator 223
5. The group operation in pm( X, A ) 224
6. The cohomotopy sequence of a triple 229
7. An important lemma 231
8. The statement (6) 234
9. The statement (5) 235
10. Higher cohomotopy groups 237
11. Relations with cohomology groups 237
12. Relations with homotopy groups 239
Exercises 241
CHAPTER VIII. EXACT COUPLES AND SPECTRAL SEQUENCES 244
1. Introduction 244
2. Differential groups 244
3. Graded and bigraded groups 246
4. Exact couples 247
5. Bigraded exact couples 249
6. Regular couples 251
7. The graded groups R( G ) and S( G ) 253
8. The fundamental exact sequence 255
9. Mappings of exact couples 257
10. Filtered differential groups 259
11. Filtered graded differential groups 260
12. Mappings of filtered graded d-groups 263
Exercises 264
CHAPTER IX. THE SPECTRALSEQUENCE OF A FIBER SPACE 274
1. Introduction 274
2. Cubical singular homology theory 274
3. A filtration in the group of singular chains in a fiber space 277
4. The associated exact couple 278
5. The derived couple 281
6. Homology with arbitrary coefficients 284
7. The spectral homology sequence 286
8. Proof of Lemma A 287
9. Proof of Lemma B 289
10. Proof of Lemmas C and D 290
11. The Poincaré polynomials 292
12. Gysin’s exact sequences 295
13. Wang’s exact sequences . 297
14. Truncated exact sequences 299
15. The spectral sequence of a regular covering space 300
16. A theorem of P. A. Smith 302
17. Influence of the fundamental group on homology and cohomology groups 303
18. Finite groups operating freely on Sr 305
Exercises 307
CHAPTER X. CLASSES OF ABELIAN GROUPS 312
1. Introduction 312
2. The definition of classes 312
3. The primary components of abelian groups 313
4. The G-notions on abelian groups 313
5. Perfectness and completeness 315
6. Applications of classes to fiber spaces 315
7. Applications to n-connective fiber spaces 319
8. The generalized Hurewicz theorem 320
9. The relative Hurewicz theorem 321
10. The Whitehead theorem 322
Exercises 323
CHAPTER XI. HOMOTOPY GROUPS OF SPHERES 326
1. Introduction 326
2. The suspension theorem 326
3. The canonical map 328
4. Wang's isomorphism p* 329
5. Relation between p* and i# 330
6. The triad homotopy groups 331
7. Finiteness of higher homotopy groups of odd-dimensional spheres 332
8. The iterated suspension 333
9. The p-primary components of pm(S3) 334
10. Pseudo-projective spaces 336
11. Stiefel manifolds 338
12. Finiteness of higher homotopy groups of even-dimensional spheres 340
13. The p-primary components of homotopy groups of even- dimensional spheres 340
14. The Hopf invariant . 341
15. The groups pn+1(Sn) and pn+2(Sn) 343
16. The groups pn+3(Sn) 344
17. The groups pn+4(Sn) 345
18.The groups pn+r(Sn), 5 = Y = 15 347
Exercises 348
BIBLIOGRAPHY 352
INDEX 358