Whitehead products and vector-fields on spheres

Abstract

In the theory of vector-spaces an ordered, ortho-normal set of r vectors is called an r-frame. Let Sn denote the unit sphere in euclidean (n+ 1)-space, where n ≥ 1. By an r-field on Sn is meant a continuous function which assigns to each point of Sn an r-frame in the tangent space at that point. If q < r we obtain a q-field from an r-field by suppressing the first r – q vectors of each r-frame. Certainly Sn admits a 0-field, and does not admit an (n+ 1)-field, since the tangent space is n-dimensional. An n-field on Sn is called a parallelism. Notice that an (n − 1)-field on Sn can always be extended to an n-field, since spheres are orientable. The problem is to determine the greatest value of r such that Sn admits an r-field.