Synopses & Reviews
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.
Synopsis
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.
Description
Includes bibliographical references (p. 223-227) and index.
Table of Contents
Part I. THE GENERAL THEORY OF BUNDLES
1. Introduction 3
2. Coordinate bundles and fibre bundles 6
3. Construction of a bundle from coordinate transformations 14
4. The product bundle 16
5. The Ehresmann-Feldbau definition of bundle 18
6. Differentiable manifolds and tensor bundles 20
7. Factor spaces of groups 28
8. The principal bundle and the principal map 35
9. Associated bundles and relative bundles 43
10. The induced bundle 47
11. Homotopies of maps of bundles 49
12. Construction of cross-sections 54
13. Bundles having a totally disconnected group 59
14. Covering spaces 67
Part II. THE HOMOTOPY THEORY OF BUNDLES
15. Homotopy groups 72
16. The operations of Pi1 on Pi n 83
17. The homotopy sequence of a bundle 90
18. The classification of bundles over the n-sphere 96
19. Universal bundles and the classification theorem 100
20. The fibering of spheres by spheres 105
21. The homotopy groups of spheres 110
22. Homotopy groups of the orthogonal groups 114
23. A characteristic map for the bundle Rn+1 over S n 118
24. A characteristic map for the bundle Un over S 2n - 1 124
25. The homotopy groups of miscellaneous manifolds 131
26. Sphere bundles over spheres 134
27. The tangent bundle of S n 140
28. On the non-existence of fiberings of spheres by spheres 144
Part III. THE COHOMOLOGY THEORY OF BUNDLES
29. The stepwise extension of a cross-section 148
30. Bundles of coefficients 151
31. Cohomology groups based on a bundle of coefficients 155
32. The obstruction cocycle 166
33. The difference cochain 169
34. Extension and deformation theorems 174
35. The primary obstruction and the characteristic cohomology class 177
36. The primary difference of two cross-sections 181
37. Extensions of functions, and the homotopy classification of maps 184
38. The Whitney characteristic classes of a sphere bundle 190
39. The Stiefel characteristic classes of differentiable manifolds 199
40. Quadratic forms on manifolds 204
41. Complex analytic manifolds and exterior forms of degree 2 209
Appendix 218
Bibliography 223
Index 228