Join products and indices of k-fields over differentiable fiber bundles

We use the intrinsic join product to express the index of a k-field with finite singularities over the total space of a smooth fiber bundle F↪E→B as the product of the indices of k-fields with finite singularities given on the fiber F and the base B. We present tables for the join product of generators of πn−1(Vn,k) for 1<k⩽4 where Vn,k denotes the real Stiefel manifold of k-frames in Rn. The indices of k-fields with finite singularities on a smooth closed manifold Mn are expressed precisely in terms of generators of πn−1(Vn,k) for 1<k⩽4 when the indices depend only on the oriented homotopy type of Mn. We obtain direct proofs of some results of Becker and Schultz [Quart. J. Math. Oxford (2) 33 (1982) 385] concerning the existence of differentiable fiber bundles as applications.