The Topology of Fibre Bundles. (PMS-14), Volume 14 (Princeton Landmarks in Mathematics and Physics)

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. Contents Part I. The General Theory of Bundles 1. Introduction 2. Coordinate bundles and fibre bundles 3. Construction ofabundle from coordinate transformations 4. The product bundle 5. The Ehresmann-Feldbau definition of bundle 6. Differentiable manifolds and tensor bundles 7. Factor spaces of groups 8. The principal bundle and the principal map 9. Associated bundles and relative bundles 10. The induced bundle 11. Homotopies of maps of bundles 12. Construction of cross-sections 13. Bundles havingatotally disconnected group 14. Covering sp aces Part II. The Homotopy Theory of Bundles 15. Homotopygroups 16. The operations ofnon 7rn 17. The homotopy sequence ofabundle 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 orthogonalg roups 23. Acharacteristic map for the bundle Rn+1 over Sn 24. Acharacteristic map for the bundle Un over S2n-1 25. The homotopy groups of miscellaneous manifolds 26. Sphere bundles over spheres 27. The tangent bundle of Sn 28. On the non-existence of fiberings of spheres by spheres Part III. The Cohomology Theory of Bundles 29. The stepwise extension ofacross-section 30. Bundles of coefficients 31. Cohomology groups based on a bundle of coefficients 32. The obstruction cocycle 33. The difference cochain 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. Appendix Bibliography Index