Vector Fields on Spheres

Abstract

The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. For each n, let Sn-I be the unit sphere in euclidean n-space Rn. A vector field on Sn-1 is a continuous function v assigning to each point x of Sn-1 a vector v(x) tangent to Sn-1 at x. Given r such fields v1, v2, ..., Vr, we say that they are linearly independent if the vectors v1(x), v2(x), *--, vr(x) are linearly independent for all x. The problem, then, is the following: for each n, what is the maximum number r of linearly independent vector fields on Sn-i? For previous work and background material on this problem, we refer the reader to [1, 10, 11, 12, 13, 14, 15, 16]. In particular, we recall that if we are given r linearly independent vector fields vi(x), then by orthogonalisation it is easy to construct r fields wi(x) such that w1(x), w2(x), *I * , wr(x) are orthonormal for each x. These r fields constitute a cross-section of the appropriate Stiefel fibering. The strongest known positive result about the problem derives from the Hurwitz-Radon-Eckmann theorem in linear algebra [8]. It may be stated as follows (cf. James [13]). Let us write n = (2a + 1)2b and b = c + 4d, where a, b, c and d are integers and 0 < c < 3; let us define p(n) = 2c + 8d. Then there exist p(n) 1 linearly independent vector fields on Sn-1. It is the object of the present paper to prove that the positive result stated above is best possible.