This is the first systematic account of fibre bundles, covering the development of the subject from its birth in 1935 until the most recent period. Most of the essential aspects are treated, the only major omission being the homology theory first developed by Gysin for sphere bundles and later generalized by Leray, Hirsch, and others to general fibre bundles. The book presupposes little knowledge of algebraic topology from the reader. It is divided into three parts. Part I deals with the general theory. Although there is yet no generally accepted definition of a fibre bundle, the author starts with one which seems to be best suited for applications. Numerous examples are given, ranging from the tensor bundles of a differentiable manifold to covering spaces. Notions such as principal bundle, associated bundles, cross-section, etc. are introduced. Perhaps the most important result is the covering homotopy theorem, which has many consequences. Part II is devoted to the homotopy theory of bundles. A clear and concise treatment of the homotopy groups is given, the first one in book form. As one of its most interesting applications, information is obtained on the topological properties of spheres. Topics include the homotopy groups of spheres and rotation groups, the fibering of spheres by spheres, classification of sphere bundles over spheres, vector fields over spheres, etc. Part III treats the cohomology theory of bundles. The basic notion is the obstruction cohomology class defined in the stepwise extension of a cross section. This is not a cohomology class in the ordinary sense and the author achieved the required generalization in a natural way by introducing the notion of a bundle of coefficients. The most important invariants so introduced are the Stiefel-Whitney characteristic classes of a sphere bundle. The book concludes with sections on quadratic forms on manifolds and complex analytic manifolds.
Reviewer: Chern, S. [form MathSciNet]