Homotopy theory

HOMOTOPY THEORY......Page 1
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 6
Contents......Page 8
List of Special Symbols and Abbreviations......Page 13
2. The extension problem......Page 16
3. The method of algebraic topology......Page 18
4. The retraction problem......Page 20
5. Combined maps......Page 22
6. Topological identification......Page 23
7. The adjunction space......Page 24
8. Homotopy problem and classification problem......Page 26
9. The homotopy extension property......Page 28
10. Relative homotopy......Page 30
11. Homotopy equivalences......Page 32
12. The mapping cylinder......Page 33
13. A generalization of the extension problem......Page 35
14. The partial mapping cylinder......Page 36
15. The deformation problem......Page 37
16. The lifting problem......Page 39
A. Elementary properties of retracts......Page 40
D. Dugundji's extension of Tietze's theorem......Page 41
F. Spaces obtained by topological identification......Page 42
H. Extension properties......Page 43
J. Separable ANR's and compact ANR's......Page 44
M. The general homotopy extension theorem......Page 45
P. Relation between partial mapping cylinder and adjunction space......Page 46
S. Some strong deformation retracts......Page 47
V. A general expansion theorem......Page 48
2. The exponential map p: R → S¹......Page 50
3. Classification of the maps S¹ → S¹......Page 52
4. The fundamental group......Page 54
5. Simply connected spaces......Page 57
6. Relation between π1(X, x0) and H1(X)......Page 59
7. The Bruschlinsky group......Page 62
8. The Hopf theorems......Page 67
9. The Hurewicz theorem......Page 71
A. The fundamental group of a connected simplicial complex......Page 72
D. The degree of a suspended maps......Page 74
2. Covering homotopy property......Page 76
3. Definition of fiber space......Page 77
4. Bundle spaces......Page 80
5. Hopf fiberings of spheres......Page 81
6. Algebraically trivial maps X → S²......Page 83
7. Liftings and cross-sections......Page 84
8. Fiber maps and induced fiber spaces......Page 86
9. Mapping spaces......Page 88
10. The spaces of paths......Page 93
11. The space of loops......Page 94
12. The path lifting property......Page 97
13. The fibering theorem for mapping spaces......Page 98
14. The induced maps in mapping spaces......Page 100
15. Fiberings with discrete fibers......Page 101
16. Covering spaces......Page 104
17. Construction of covering spaces......Page 108
A. Sliced fiber spaces......Page 112
B. Local path lifting property......Page 113
D. Homogeneous spaces......Page 114
G. Stiefel manifolds......Page 115
I. Elementary properties of mapping spaces......Page 116
L. Maps on topological products......Page 117
O. The space of curves......Page 118
P. Generalized covering spaces......Page 119
R. The covering spaces of the torus......Page 120
T. Maps of a surface into a surface......Page 121
2. Absolute homotopy groups......Page 122
3. Relative homotopy groups......Page 125
4. The boundary operator......Page 127
5. Induced transformations......Page 128
6. The algebraic properties......Page 129
7. The exactness property......Page 130
8. The homotopy property......Page 132
9. The fibering property......Page 133
11. Homotopy systems......Page 134
12. The uniqueness theorem......Page 136
13. The group structures......Page 138
14. The role of the basic point......Page 140
15. Local system of groups......Page 144
16. n-Simple spaces......Page 146
B. The equivalence theorem......Page 150
C. Properties of the homotopy system......Page 151
D. The role of the basic point in the relative homotopy groups......Page 152
F. The Whitehead product......Page 153
G. Homotopy groups of H-spaces......Page 154
H. Semi-simplicial complexes......Page 155
J. Complete semi-simplicial complexes......Page 156
K. Homotopy groups of Kan complexes......Page 157
2. Homotopy groups of the product of two spaces......Page 158
3. The one-point union of two spaces......Page 160
4. The natural homomorphisms from homotopy groups to homology groups......Page 161
5. Direct sum theorems......Page 165
6. Homotopy groups of fiber spaces......Page 167
7. Homotopy groups of covering spaces......Page 169
8. The n-connective fiberings......Page 170
9. The homotopy sequence of a triple......Page 174
10. The homotopy groups of a triad......Page 175
11. Freudenthal's suspension......Page 177
A. The homotopy addition theorem......Page 179
C. The Hurewicz theorem......Page 181
D. The Whitehead theorem......Page 182
F. Spaces of homotopy type (π, n)......Page 183
G. The realizability theorems......Page 184
H. Topological realization of semi-simplicial complexes......Page 185
J. Induced cellular maps......Page 186
K. Admissible subcomplexes of S(X)......Page 187
L. The Eilenberg subcornplexes of S(X)......Page 189
2. The extension index......Page 190
3. The obstruction c^ n+1 (g)......Page 191
4. The difference cochain......Page 193
5. Eilenberg's extension theorem......Page 195
6. The obstruction sets for extension......Page 196
7. The homotopy problem......Page 197
8. The obstruction d^n (f, g; h t)......Page 198
9. The group R^n (K, L; f)......Page 199
10. The obstruction sets for homotopy......Page 200
11. The general homotopy theorem......Page 201
12. The classification problem......Page 202
13. The primary obstructions......Page 203
14. Primary extension theorems......Page 205
16. Primary classification theorems......Page 206
B. The fundamental group of a semi-simplicial complex......Page 208
C. Generalizing the obstruction theory by using Čech cohomology theory......Page 209
D. Generalizing the obstruction theory by using the singular complex......Page 210
E. The obstruction theory of deformation......Page 212
F. Maps into a space of homotopy type (π, n)......Page 213
H. The complex K(π) of a group π......Page 214
I. Contractible complexes on which π acts freely......Page 215
K. Computation of H m (π; G) and H^m (π; G)......Page 216
M. The influence of π n (X)......Page 218
2. The cohomotopy set π^m (X, A)......Page 220
3. The induced transformations......Page 221
4. The coboundary operator......Page 223
5. The group operation in π^m (X, A)......Page 224
6. The cohomotopy sequence of a triple......Page 229
7. An important lemma......Page 231
8. The statement (6)......Page 234
9. The statement (5)......Page 235
11. Relations with cohomology groups......Page 237
12. Relations with homotopy groups......Page 239
A. Generalizations to compact pairs......Page 241
D. Connection with the obstruction......Page 242
F. Relations between induced homomorphisms......Page 243
2. Differential groups......Page 244
3. Graded and bigraded groups......Page 246
4. Exact couples......Page 247
5. Bigraded exact couples......Page 249
6. Regular couples......Page 251
7. The graded groups R(C) and S(C)......Page 253
8. The fundamental exact sequence......Page 255
9. Mappings of exact couples......Page 257
10. Filtered differential groups......Page 259
11. Filtered graded differential groups......Page 260
12. Mappings of filtered graded d-groups......Page 263
A. Direct construction of the spectral sequence of a filtered differential group......Page 264
C. Multiplicative structures......Page 265
E. The homotopy exact couple......Page 266
F. The cohomotopy exact couple......Page 269
G. The Gamma functor of abelian groups......Page 270
H. Transgression and suspension......Page 271
I. Properties of Steenrod squares......Page 273
2. Cubical singular homology theory......Page 274
3. A filtration in the group of singular chains in a fiber space......Page 277
4. The associated exact couple......Page 278
5. The derived couple......Page 281
6. Homology with arbitrary coefficients......Page 284
7. The spectral homology sequence......Page 286
8. Proof of Lemma A......Page 287
9. Proof of Lemma B......Page 289
10. Proof of Lemmas C and D......Page 290
11. The Poincaré polynomials......Page 292
12. Gysin's exact sequences......Page 295
13. Wang's exact sequences......Page 297
14. Truncated exact sequences......Page 299
15. The spectral sequence of a regular covering space......Page 300
16. A theorem of P. A. Smith......Page 302
17. Influence of the fundamental group on homology and cohomology groups......Page 303
18. Finite groups operating freely on S^r......Page 305
B. The spectral cohomology sequence......Page 307
D. An isomorphism theorem......Page 308
F. Relations between the cohomology algebras of a space and its space of loops......Page 309
H. The cohomology algebra of Λ(S^n)......Page 310
I. The connective fiber spaces of S³......Page 311
2. The Definition of Classes......Page 312
4. The C-Notions on Abelian Groups......Page 313
6. Applications of Classes to Fiber Spaces......Page 315
7. Applications to n-connective fiber spaces......Page 319
8. The generalized Hurewicz theorem......Page 320
9. The relative Hurewicz theorem......Page 321
10. The Whitehead theorem......Page 322
C. On Perfectness and Completeness......Page 323
H. On Induced Homomorphisms......Page 324
2. The suspension theorem......Page 326
3. The canonical map......Page 328
4. Wang's isomorphism ρ*......Page 329
5. Relation between ρ* and I #......Page 330
6. The triad homotopy groups......Page 331
7. Finiteness of higher homotopy groups of odd-dimensional spheres......Page 332
8. The iterated suspension......Page 333
9. The p-primary components of π m (S³)......Page 334
10. Pseudo-projective spaces......Page 336
11. Stiefel manifolds......Page 338
13. The p-primary components of homotopy groups of even-dimensional spheres......Page 340
14. The Hopf invariant......Page 341
15. The groups π n+1 (S^n) and π n+2 (S^n)......Page 343
16. The groups π n+3 (S^n)......Page 344
17. The groups π n+4 (S^n)......Page 345
18. The groups π n+r (S^n), 5 ≤ r ≤ 15......Page 347
B. On relative (n+1)-cells......Page 348
C. Other definitions of the Hopf invariant......Page 349
D. The delicate suspension theorem......Page 350
Papers......Page 352
Index......Page 358