In this edition, we have added two new chapters, Chapter 7 on the gauge group of a principal bundle and Chapter 19 on the definition of Chern classes by differential forms. These subjects have taken on special importance when we consider new applications of the fibre bundle theory especially to mathematical physics. For these two chapters, the author profited from discussions with Professor M. S. Narasimhan.

  The idea of using the term bundle for what is just a map, but is eventually a fibre bundle projection, is due to Grothendieck.

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 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

 ……