內容簡介

The notion of a fibre bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950, the definition of fibre bundle had been clearly formulated, the homotopy classification of fibre bundles achieved, and the theory of characteristic classes of fibre bundles developed by several mathematicians: Chern, Pontrjagin, Stiefel, and Whitney. Steenrod’’s book, which appeared in 1950, gavea coherent treatment of the subject up to that time.

About 1955, Miinor gave a construction ora universal fibre bundle for any topological group. This construction is also included in Part I along with an elementary proof that the bundle is universal.
 

目錄

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 Map0 (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, I]
　　10. The Cofunctor k
　　11. The Milnor Construction
　　12. Homotopy Classification of Numerable Principal G-Bundles
　　13. Homotopy Classification of Principal G-Bundles over
C W-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
……
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