The classification of sphere bundles

Abstract

Whitney has introduced [13] the general notion of one space A being a fibre bundle over a second space B. It is a topological condition on a function f mapping A onto B which insures that f shall have a high degree of smoothness. It is the topological counterpart of the analytic requirement, in case A and B are differentiable manifolds, that f shall have a Jacobian of maximum rank at every point. Fibre bundles should prove of importance since they arise in many connections. The factor spaces of a Lie group form a lattice of fibre bundles (see Theorem 1). The spaces of tensors over a manifold are fibre bundles [9]. Among the simpler bundles are those for which the fibres (inverse images of points of B) are k-spheres. One of the main problems with which Whitney has concerned himself is the classification of the k-sphere bundles over a given space B. In the present paper, this problem is reduced to a familiar problem of topology. A factor space Ml of the rotation group of a (k + 1 + 1)-sphere is selected, and it is proved that the equivalence classes of k-sphere bundles over a complex B are in 1-1 correspondence with the homotopy classes of maps of B in Ml. In addition, the homotopy groups of MI are computed for dimensions ?6. This leads to a complete solution of the classification problem when B is a sphere of dimension ? 6. Sections 2 and 3 contain definitions and discussion of fibre bundles and related concepts. Sections 4, 5 and 6 contain the statements of the principal results without proofs. Sections 7 and 8 are concerned with showing that covering homotopies exist in a fibre bundle. With this mechanism and its consequences at hand, the proofs of the main results are given in the remaining sections.