Vector Bundles - Vol 1 (eBook)

Front Cover 1
Vector Bundles: Foundations and Stiefel–Whitney Classes 4
Copyright Page 5
Contents 8
Preface 12
Introduction 14
Chapter I. Base Spaces 18
0. Introduction 18
1. The Category of Base Spaces 18
2. Some Simplicial Spaces 21
3. More Simplicial Spaces 24
4. Weak Simplicial Spaces 28
5. CW Spaces 32
6. Smooth Manifolds 36
7. Grassmann Manifolds 47
8. Some More Coverings 52
9. The Mayer–Vietoris Technique 54
10. Remarks and Exercises 60
Chapter II. Fiber Bundles 70
0. Introduction 70
1. Fibre Bundles and Fiber Bundles 73
2. Coordinate Bundles 79
3. Bundles over Contractible Spaces 86
4. Pullbacks along Homotopic Maps 88
5. Reduction of Structure Groups 93
6. Polar Decompositions 99
7. The Leray–Hirsch Theorem 106
8. Remarks and Exercises 110
Chapter III. Vector Bundles 118
0. Introduction 118
1. Real Vector Bundles 118
2. Whitney Sums and Products 121
3. Riemannian Metrics 125
4. Sections of Vector Bundles 129
5. Smooth Vector Bundles 132
6. Vector Fields and Tangent Bundles 139
7. Canonical Vector Bundles 153
8. The Homotopy Classification Theorem 160
9. More Smooth Vector Bundles 171
10. Orientable Vector Bundles 173
11. Complex Vector Bundles 182
12. Realifications and Complexifications 185
13. Remarks and Exercises 191
Chapter IV. Z/2 Euler Classes 204
0. Introduction 204
1. The Z/2 Thom Class U. e Hn(E, E*) 205
2. Properties of Z/2 Thom Classes 207
3. Z/2 Euler Classes 209
4. Gysin Sequences and H*(RP8 Z/2)
5. The Splitting Principle 214
6. Axioms for Z/2 Euler Classes 218
7. Z/2 Euler Classes of Product Line Bundles 219
8. The Z/2 Thom Class V. e Hn(P. +1, P.) 222
9. Remarks and Exercises 224
Chapter V. Stiefel–Whitney Classes 228
0. Introduction 228
1. Multiplicative Z/2 Classes 229
2. Whitney Product Formulas 232
3. Orientability 239
4. The Rings H*(Gm(R8) Z/2)
5 . Axioms for Stiefel–Whitney Classes 246
6. Dual Classes 250
7. Remarks and Exercises 251
Chapter VI. Unoriented Manifolds 262
0. Introduction 262
1. Z/2 Fundamental Classes 264
2. Z/2 Poincaré–Lefschetz Duality 269
3. Multiplicative Z/2 Classes of RPn 271
4. Nonimmersions and Nonembeddings 273
5. Stiefel–Whitney Numbers 278
6. Stiefel–Whitney Genera 287
7. Z/2 Thom Forms 289
8. The Thom–Wu Theorem 298
9. Remarks and Exercises 300
References 316
Glossary of Notation 364
Index 372
Pure Applied Mathematics 385