On Families of Continuous Vector Fields Over Spheres

Abstract

It has long been known that no continuous field of unit tangent vectors can exist on the 2n-dimensional sphere S2. On the other hand, if n = 2k 1, 4k 1, or 8k 1, there exist 1, 3, 7 everywhere independent continuous vector fields over S'. It was recently proved independently by B. Eckmann2 and the author3 that if n 1 (mod 4) every two vector fields over St are somewhere dependent. In the present paper an analogous result is obtained for the case n 3 (mod 8): every four vector fields over Sn are dependent at some point. The principal tools are the notion of fibre space developed by Huxewicz and Steenrod4 andthe results of Freudenthal5 on the homotopy groups of spheres. As a corollary to the above results and a result of C. Ehresmann,6 it is shown that no sphere of dimension > k can be a k-sphere bundle over any complex B if k = 2m, 4m + 1, or 8m + 3 with m > O.6