The Topology of Fibre Bundles

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