Join products and indices of \(k\)-fields over differentiable fiber bundles.
The intrinsic join product is used to study indices of \(k\)-fields with finite singularities defined on the total space of a differentiable fibre bundle. This is expressed as the product of the indices of \(k\)-fields with finite singularities given on the fiber and the base. The computations of the join product of the generators of \(\pi_{n-1}(V_{n,k})\) for \(1<k \leq 4\) are presented in the case of the real Stiefel manifold \(\pi_{n-1}(V_{n,k})\) of \(k\)-frames in \(\mathbb R^{n}\). The indices of k-fields with finite singularities on a smooth closed manifold \(M^{n}\) are expressed precisely in terms of generators of \(\pi_{n-1}(V_{n,k})\) for \(1<k \leq 4\) when the indices depend only on the oriented homotopy type of \(M^{n}\). Finally, direct proofs of some results of \textit{J. C. Becker} and \textit{R. E. Schultz} [Q. J. Math., Oxf. II. Ser. 33, 385--403 (1982; Zbl 0512.55015)] concerning the existence of differentiable fiber bundles are obtained.
scientific article; zbMATH DE number 2037259
scientific article