The Topology of Fibre Bundles
by Steenrod, NormanFormat: Paperback
Pub. Date: 1999-04-05
Publisher(s): Princeton Univ Pr
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Summary
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.
Author Biography
Norman Steenrod was a professor of mathematics at Princeton University.
Table of Contents
| The general theory of bundles | |
| Introduction | p. 3 |
| Coordinate bundles and fibre bundles | p. 6 |
| Construction of a bundle from coordinate transformations | p. 14 |
| The product bundle | p. 16 |
| The Ehresmann-Feldbau definition of bundle | p. 18 |
| Differentiable manifolds and tensor bundles | p. 20 |
| Factor spaces of groups | p. 28 |
| The principal bundle and the principal map | p. 35 |
| Associated bundles and relative bundles | p. 43 |
| The induced bundle | p. 47 |
| Homotopies of maps of bundles | p. 49 |
| Construction of cross-sections | p. 54 |
| Bundles having a totally disconnected group | p. 59 |
| Covering spaces | p. 67 |
| The homotopy theory of bundles | |
| Homotopy groups | p. 72 |
| The operations of Pi1 on Pi n | p. 83 |
| The homotopy sequence of a bundle | p. 90 |
| The classification of bundles over the n-sphere | p. 96 |
| Universal bundles and the classification theorem | p. 100 |
| The fibering of spheres by spheres | p. 105 |
| The homotopy groups of spheres | p. 110 |
| Homotopy groups of the orthogonal groups | p. 114 |
| A characteristic map for the bundle Rn+1 over S n | p. 118 |
| A characteristic map for the bundle Un over S 2n - 1 | p. 124 |
| The homotopy groups of miscellaneous manifolds | p. 131 |
| Sphere bundles over spheres | p. 134 |
| The tangent bundle of S n | p. 140 |
| On the non-existence of fiberings of spheres by spheres | p. 144 |
| The cohomology theory of bundles | |
| The stepwise extension of a cross-section | p. 148 |
| Bundles of coefficients | p. 151 |
| Cohomology groups based on a bundle of coefficients | p. 155 |
| The obstruction cocycle | p. 166 |
| The difference cochain | p. 169 |
| Extension and deformation theorems | p. 174 |
| The primary obstruction and the characteristic cohomology class | p. 177 |
| The primary difference of two cross-sections | p. 181 |
| Extensions of functions, and the homotopy classification of maps | p. 184 |
| The Whitney characteristic classes of a sphere bundle | p. 190 |
| The Stiefel characteristic classes of differentiable manifolds | p. 199 |
| Quadratic forms on manifolds | p. 204 |
| Complex analytic manifolds and exterior forms of degree 2 | p. 209 |
| Appendix | p. 218 |
| Bibliography | p. 223 |
| Index | p. 228 |
| Table of Contents provided by Publisher. All Rights Reserved. |
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