Theorems in algebraic topology | Fixed points (mathematics) | Theorems in differential topology
Hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0). The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher dimensions in 1912 by Luitzen Egbertus Jan Brouwer. The theorem has been expressed colloquially as "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut". (Wikipedia).
