Hairy ball theorem

From Academic Kids

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 group of diffeomorphisms 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 Earth is approximately a ball, and at each point on the surface, wind has a direction. It follows from the theorem that there is always a place where the air is perfectly still.

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