Theorems in topology | Differential topology

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 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 there were at least ρ(N)-1 such fields (see definition below). Adams applied homotopy theory and topological K-theory to prove that no more independent vector fields could be found. Hence ρ(N)-1 is the exact number of pointwise linearly independent vector fields that exist on an (N-1)-dimensional sphere. (Wikipedia).

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Related pages

Clifford algebra | Tangent bundle | Gram–Schmidt process | Coding theory | Linear independence | Hurwitz problem | Power of two | Homotopy theory | Hairy ball theorem | Mathematics | Orthonormal basis | Euclidean space | Differential topology | Orthogonal matrix | Quadratic form | Exotic sphere | Division algebra | Matrix (mathematics) | Frank Adams | Johann Radon | Topological K-theory