| Preface to the Third Edition Preface to the Second Edition Preface to the First Edition CHAPTER 1 Preliminaries on Homotopy Theory 1. Category Theory and Homotopy Theory 2. Complexes 3. The Spaces Map (X, Y) and Mapo (X, Y) 4. Homotopy Groups of Spaces 5. Fibre Maps PART Ⅰ THE GENERAL THEORY OF FIBRE BUNDLES CHAPTER 2 Generalities on Bundles 1. Definition of Bundles and Cross Sections 2. Examples of Bundles and Cross Sections 3. Morphisms of Bundles 4. Products and Fibre Products 5. Restrictions of Bundles and Induced Bundles 6. Local Properties of Bundles 7. Prolongation of Cross Sections Exercises CHAPTER 3 Vector Bundles 1. Definition and Examples of Vector Bundles 2. Morphisms of Vector Bundles 3. Induced Vector Bundles 4. Homotopy Properties of Vector Bundles 5. Construction of Gauss Maps 6. Homotopies of Gauss Maps 7. Functorial Description of the Homotopy Classification of Vector Bundles 8. Kernel, Image, and Cokernel of Morphisms with Constant Rank 9. Riemannian and Hermitian Metrics on Vector Bundles Exercises CHAPTER 4 General Fibre Bundles 1. Bundles Defined by Transformation Groups 2. Definition and Examples of Principal Bundles 3. Categories of Principal Bundles 4. Induced Bundles of Principal Bundles 5. Definition of Fibre Bundles 6. Functorial Properties of Fibre Bundles 7. Trivial and Locally Trivial Fibre Bundles 8. Description of Cross Sections of a Fibre Bundle 9. Numerable Principal Bundles over B x [0, 1] 10. The Cofunctor k 11. The Milnor Construction 12. Homotopy Classification of Numerable Principal G-Bundles 13. Homotopy Classification of Principal G-Bundles over CW-Complexes Exercises CHAPTER 5 Local Coordinate Description of Fibre Bundles 1. Automorphisms of Trivial Fibre Bundles 2. Charts and Transition Functions 3. Construction of Bundles with Given Transition Functions 4. Transition Functions and Induced Bundles 5. Local Representation of Vector Bundle Morphisms 6. Operations on Vector Bundles 7. Transition Functions for Bundles with Metrics Exercises CHAPTER 6 Change of Structure Group in Fibre Bundles 1. Fibre Bundles with Homogeneous Spaces as Fibres …… PART Ⅱ ELEMENTS OF K-THEORY PART Ⅲ CHARACTERISTIC CLASSES Appendix 1 Dold's Theory of Local Properties of Bundles Appendix 2 On the Double Suspension Bibliography Index |
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