Homotopy theory

Cover......Page 1
Title page......Page 3
Copyright page......Page 4
PREFACE......Page 5
Contents......Page 7
LIST OF SPECIAL SYMBOLS AND ABBREVIATIONS......Page 12
2. The extension problem......Page 15
3. The method of algebraic topology......Page 17
4. The retraction problem......Page 19
5. Combined maps......Page 21
6. Topological identification......Page 22
7. The adjunction space......Page 23
8. Homotopy problem and classification problem......Page 25
9. The homotopy extension property......Page 27
10. Relative homotopy......Page 29
11. Homotopy equivalences......Page 31
12. The mapping cylinder......Page 32
13. A generalization of the extension problem......Page 34
14. The partial mapping cylinder......Page 35
15. The deformation problem......Page 36
16. The lifting problem......Page 38
Exercises......Page 39
2. The exponential map $p: R \\to S^1$......Page 49
3. Classification of the maps $S^1 \\to S^1$......Page 51
4. The fundamental group......Page 53
5. Simply connected spaces......Page 56
6. Relation between $\\pi_1(X,x_0)$ and $H_1(X)$......Page 58
7. The Bruschlinsky group......Page 61
8. The Hopf theorems......Page 66
9. The Hurewicz theorem......Page 70
Exercises......Page 71
2. Covering homotopy property......Page 75
3. Definition of fiber space......Page 76
4. Bundle spaces......Page 79
5. Hopf fiberings of spheres......Page 80
6. Algebraically trivial maps $X \\to S^2$......Page 82
7. Liftings and cross-sections......Page 83
8. Fiber maps and induced fiber spaces......Page 85
9. Mapping spaces......Page 87
10. The spaces of paths......Page 92
11. The space of loops......Page 93
12. The path lifting property......Page 96
13. The fibering theorem for mapping spaces......Page 97
14. The induced maps in mapping spaces......Page 99
15. Fiberings with discrete fibers......Page 100
16. Covering spaces......Page 103
17. Construction of covering spaces......Page 107
Exercises......Page 111
2. Absolute homotopy groups......Page 121
3. Relative homotopy groups......Page 124
4. The boundary operator......Page 126
5. Induced transformations......Page 127
6. The algebraic properties......Page 128
7. The exactness property......Page 129
8. The homotopy property......Page 131
9. The fibering property......Page 132
11. Homotopy systems......Page 133
12. The uniqueness theorem......Page 135
13. The group structures......Page 137
14. The role of the-basic point......Page 139
15. Local system of groups......Page 143
16. $n$-Simple spaces......Page 145
Exercises......Page 149
2. Homotopy groups of the product of two spaces......Page 157
3. The one-point union of two spaces......Page 159
4. The natural homomorphisms from homotopy groups to homology groups......Page 160
5. Direct sum theorems......Page 164
6. Homotopy groups of fiber spaces......Page 166
7. Homotopy groups of covering spaces......Page 168
8. The $n$-connective fiberings......Page 169
9. The homotopy sequence of a triple......Page 173
10. The homotopy groups of a triad......Page 174
11. Freudenthal\'s suspension......Page 176
Exercises......Page 178
2. The extension index......Page 189
3. The obstruction $c^{n+1}(g)$......Page 190
4. The difference cochain......Page 192
5. Eilenberg\'s extension theorem......Page 194
6. The obstruction sets for extension......Page 195
7. The homotopy problem......Page 196
8. The obstruction $d^n{f,g;h_t)$......Page 197
9. The group $R^n(K,L;f)$......Page 198
10. The obstruction sets for homotopy......Page 199
11. The general homotopy theorem......Page 200
12. The classification problem......Page 201
13. The primary obstructions......Page 202
14. Primary extension theorems......Page 204
16. Primary classification theorems......Page 205
Exercises......Page 207
2. The cohomotopy set $\\pi^m(X,A)$......Page 219
3. The induced transformations......Page 220
4. The coboundary operator......Page 222
5. The group operation in $\\pi^m(X,A)$......Page 223
6. The cohomotopy sequence of a triple......Page 228
7. An important lemma......Page 230
8. The statement (6)......Page 233
9. The statement (5)......Page 234
11. Relations with cohomology groups......Page 236
12. Relations with homotopy groups......Page 238
Exercises......Page 240
2. Differential groups......Page 243
3. Graded and bigraded groups......Page 245
4. Exact couples......Page 246
5. Bigraded exact couples......Page 248
6. Regular couples......Page 250
7. The graded groups $R(\\mathcal{C})$ and $S(\\mathcal{C})$......Page 252
8. The fundamental exact sequence......Page 254
9. Mappings of exact couples......Page 256
10. Filtered differential groups......Page 258
11. Filtered graded differential groups......Page 259
12. Mappings of filtered graded $d$-groups......Page 262
Exercises......Page 263
2. Cubical singular homology theory......Page 273
3. A filtration in the group of singular chains in a fiber space......Page 276
4. The associated exact couple......Page 277
5. The derived couple......Page 280
6. Homology with arbitrary coefficients......Page 283
7. The spectral homology sequence......Page 285
8. Proof of Lemma A......Page 286
9. Proof of Lemma B......Page 288
10. Proof of Lemmas C and D......Page 289
11. The Poincaré polynomials......Page 291
12. Gysin\'s exact sequences......Page 294
13. Wang\'s exact sequences......Page 296
Î4. Truncated exact sequences......Page 298
15. The spectral sequence of a regular covering space......Page 299
16. A theorem of P. A. Smith......Page 301
17. Influence of the fundamental group on homology and cohomo- logy groups......Page 302
18. Finite groups operating freely on $S^r$......Page 304
Exercises......Page 306
2. The definition of classes......Page 311
4. The $\\mathcal{C}$-notions on abelian groups......Page 312
6. Applications of classes to fiber spaces......Page 314
7. Applications to $n$-connective fiber spaces......Page 318
8. The generalized Hurewicz theorem......Page 319
9. The relative Hurewicz theorem......Page 320
10. The Whitehead theorem......Page 321
Exercises......Page 322
2. The suspension theorem......Page 325
3. The canonical map......Page 327
4. Wang\'s isomorphism $\\rho_\\ast$......Page 328
5. Relation between $\\rho_\\ast$ and $i_\\#$......Page 329
6. The triad homotopy groups......Page 330
7. Finiteness of higher homotopy groups of odd-dimensional spheres......Page 331
8. The iterated suspension......Page 332
9. The $\\mathcal{p}$-primary components of $\\pi_m(S^3)$......Page 333
10. Pseudo-projective spaces......Page 335
11. Stiefel manifolds......Page 337
13. The $\\mathcal{p}$-primary components of homotopy groups of even- dimensional spheres......Page 339
14. The Hopf invariant......Page 340
15. The groups $\\pi_{n+1}(S^n)$ and $\\pi_{n+2}(S^n)$......Page 342
16. The groups $\\pi_{n+3}(S^n)$......Page 343
17. The groups $\\pi_{n+4}(S^n)$......Page 344
18. The groups $\\pi_{n+r}(S^n)$, $5 \\leq r \\leq 15$......Page 346
Exercises......Page 347
BIBLIOGRAPHY......Page 351
INDEX......Page 357