Vector fields on spheres

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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,^[1] by direct construction using Clifford algebras, that there were at least ${\displaystyle \rho (n)-1}$ such fields (see definition below). Adams applied homotopy theory and topological K-theory^[2] 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.