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

Nancy de Souza Cardim, Maria Hermı́nia de P.L. Mello, Duane Randall, Mário Olivero Marques da Silva
2004 Topology and its Applications  
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 π n−1 (V n,k ) for 1 < k 4 where V n,k denotes the real Stiefel manifold of k-frames in R n . The indices of k-fields with finite singularities on a smooth closed manifold M n are
more » ... precisely in terms of generators of π n−1 (V n,k ) for 1 < k 4 when the indices depend only on the oriented homotopy type of M n . 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.
doi:10.1016/j.topol.2003.08.008 fatcat:oe6xlyuai5cptj2pp4lucbaoba