Connections, Curvature, and Cohomology V1 (eBook)

De Rham cohomology of manifolds and vector bundles

eBook Download: PDF

1972 | 1. Auflage
442 Seiten
Elsevier Science (Verlag)
978-0-08-087360-2 (ISBN)

Infos

Connections, curvature, and cohomology V1

Inhalt

Front Cover 1
Connections, Curvature, and Cohomology 4
Copyright Page 5
Contents 8
Preface 12
Introduction 14
Contents of Volumes II and III 20
Chapter 0. Algebraic and Analytic Preliminaries 22
1. Linear algebra 22
2. Homological algebra 28
3. Analysis and topology 33
Chapter I. Basic Concepts 36
1. Topological manifolds 36
2. Smooth manifolds 43
3. Smooth fibre bundles 59
Problems 62
Chapter II. Vector Bundles 65
1. Basic concepts 65
2. Algebraic operations with vector bundles 71
3. Cross-sections 80
4. Vector bundles with extra structure 85
5. Structure theorems 97
Problems 105
Chapter III. Tangent Bundle and Differential Forms 108
1. Tangent bundle 108
2. Local properties of smooth maps 120
3. Vector fields 127
4. Differential forms 136
5. Orientation 145
Problems 152
Chapter IV. Calculus of Differential Forms 162
1. The Opertors i,.,d 162
2. Smooth families of differential forms 174
3. Integration of n-forms 180
4. Stokes’ theorem 188
Problems 191
Chapter V. De Rham Cohomology 197
1. The axioms 197
2. Examples 204
3. Cohomology with compact supports 210
4. Poincaré duality 215
5. Applications of Poincaré duality 222
6. Kiinneth theorems 229
7. The De Rham theorem 238
Problems 249
Chapter VI. Mapping Degree 261
1. Global degree 261
2. The canonical map aM 273
3. Local degree 280
4. The Hopf theorem 287
Problems 294
Chapter VII. Integration over the Fibre 301
1. Tangent bundle of a fibre bundle 301
2. Orientation in fibre bundles 306
3. Vector bundles and sphere bundles 312
4. Fibre-compact carrier 316
5. Integration over the fibre 319
Problems 331
Chapter VIII. Cohomology of Sphere Bundles 337
1. Euler class 337
2. The difference class 346
3. Index of a cross-section at an isolated singularity 350
4. Index sum and Euler class 355
5. Existence of cross-sections in a sphere bundle 358
Problems 365
Chapter IX. Cohomology of Vector Bundles 373
1. The Thom isomorphism 373
2. The Thom class of a vector bundle 380
3. Index of a cross-section at an isolated zero 388
Problems 399
Chapter X. The Lefschetz Class of a Manifold 412
I . The Lefschetz isomorphism 412
2. Coincidence number 421
3. The Lefschetz coincidence theorem 426
Problems 435
Appendix A. The Exponential Map 439
References 444
Bibliography 446
Bibliography—Books 452
Notation Index 454
Index 456
Pure and Applied Mathematics 465

Produkt-Details

• Erscheint lt. Verlag: 31.7.1972
• Mitarbeit: Herausgeber (Serie): Werner Greub, Stephen Halperin, Ray Vanstone
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-087360-X / 008087360X
• ISBN-13: 978-0-08-087360-2 / 9780080873602