Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically.

It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike.

Table of Contents

Part I. The General Theory of Bundles

1. Introduction

2. Coordinate bund les and fibre bundles

3. Construction of a bundle from coordinate transformations

4. The product bund le

5. The Ehresmann-Feldbau definition of bund le

6. Differentiable manifolds and tensor bund les

7. Factor spaces of groups

8. The principal bund le and the principal map

9. Associated bund les and relative bundles

10. The induced bund le

11. Homotopies of maps of bundles

12. Construction of cross-sections

13. Bundles having a totally disconnected group

14. Covering spaces

Part II. The Homotopy Theory of Bundles

15. Homotopygroups

16. The operations of ...

17. The Homotopy sequence of a bundle

18. The classification of bundles over the n-sphere

19. Universal bundles and the classification theorem

20. The fibering of spheres by spheres

21. The homotopy groups of spheres

22. Homotopy groups of the orthogonal groups

23. A characteristic map for the bundle Rn+ 1 over Sn

24. A characteristic map for the bundle Un over S2n-1

25. The homotopy goups of miscellaneous manifolds

26. Sphere bundles over spheres

27. The tangent bundle of Sn

28. On the no n-existence of fiberings of spheres by spheres

Part Ill. The Cohomology Theory of Bundles

29. The stepwise extension of a cross-section

30. Bundles of coefficients

31. Cohomology groups based on a bund le of coefficients

32. The obstruction cocycle

33. The difference co chain

34. Extension and deformation theorems

35. The primary obstruction and the characteristic cohomology class

36. The primary difference of two cross-sections

37. Extensions of functions, and the homotopy classification of maps

38. The Whitney characteristic classes of a sphere bundle

39. The Stiefel characteristic classes of differentiable manifolds

40. Quadratic forms on manifolds

41. Complex analytic manifolds and exterior forms of degree 2.

About the Author

Norman Steenrod was a professor of Mathematics at Princeton University and a member of the National Academy Of Sciences. He also edited Reviews in Topology.