Vector fields on spheres

In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.

Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in ${\displaystyle n}$-dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that if ${\displaystyle \rho (n)}$ is the Radon-Hurwitz number of the n-sphere (definition below), then there were at least ${\displaystyle \rho (n)-1}$ such fields. Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found. Hence ${\displaystyle \rho (n)-1}$ is the exact number of pointwise linearly independent vector fields that exist on an (${\displaystyle n-1}$)-dimensional sphere.