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…mehr
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.
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.
Inhaltsangabe
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
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
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