Vector Fields on Spheres

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.