Home > Hairy ball theorem

The hairy ball theorem of algebraic topology states that, in layman's terms, "one cannot comb the hair on a ball smooth".

This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. Less briefly, if f is a continuous function that assigns a vector in R³ to every point p on a sphere, and for all p the vector f(p) is a tangent direction to the sphere at p, then there is at least one p such that f(p) = 0.

In fact from a more advanced point of view it can be shown that the sum at the zeroes of such a vector field of a certain 'index' must be 2, the Euler characteristic of the 2-sphere; and that therefore there must be at least some zero. In the case of the 2- torus, the Euler characteristic is 0; and it is possible to 'comb a hairy torus flat'.

There is a closely-related argument from algebraic topology, using the Lefschetz fixed point theorem. Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on homology of the identity mapping) is 2. By integrating a vector field we get (at least a small part of) a one-parameter groupIn mathematics. a one-parameter group usually means a group homomorphism :φ: R → G from the real line R as additive group, to some other topological group G which is also continuous. That means that it is not in fact a group, strictly speaking; i of diffeomorphismIn mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. Here is definition Given two differentiable manifolds M and N a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth. Two manifolds M and N as on the sphere; and all of the mappings in it are homotopic to the identity. Therefore they all have Lefschetz number 2, also. Hence they are not without fixed points (which means Lefschetz number 0). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general Poincaré-Hopf index theorem .

One surprising consequence of the hairy ball theorem: The EarthEarth also known as the Earth or Terra is the planet on which we live, the third planet outward from the Sun. It is the largest of the solar system's terrestrial planets, and the only planetary body that modern science confirms as harbouring life. The pla is approximately a ball, and at each point on the surface, windFor the 1928 film, see The Wind. Wind in the most general sense, is the movement of air. It occurs at all scales, from local breezes generated by heating of land surfaces and lasting tens of minutes to global winds resulting from solar heating of the plan has a direction. It follows from the theorem that there is always a cyclone somewhere on the Earth's surface.