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The hairy ball theorem of algebraic topology states that, in layman's terms, "one cannot comb the hair on a ball in a smooth manner". One way to understand this theorem is to picture the hairs on a tennis ball: any attempt to make them "smooth" in a mathematical sense will leave a spot where two hairs point in "drastically" different directions. This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: 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. It was first proved in 1912 by Brouwer, see *.
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 donut flat'.
Cyclone consequences
Another, meteorological rather than veterinary, approach to this theorem involves thinking of wind as a vector defined at every point continuously over the surface of a planet with an atmosphere. As an idealisation, take wind to be a two-dimensional vector: suppose that relative to the planetary diameter of the Earth, its vertical (i.e., non-tangential) motion is negligible.
One scenario, in which there is absolutely no wind (air movement), corresponds to a field of zero-vectors. This scenario is uninteresting from the point of view of this theorem, and physically unrealistic (there will always be wind). In the case where there is at least some wind, the Hairy Ball Theorem dictates that there must be at least one point on a spherical planet at all times, with no wind at all. This corresponds to the above statement that there will be always be p such that f(p) = 0.
In a physical sense, this zero-wind point will be the center of a cyclone or anticyclone. (Like the swirled hairs on the tennis ball, the wind will spiral around and out from this zero-wind point.) In brief, then, the Hairy Ball Theorem dictates that, given at least some wind on Earth, there must at all times be a cyclone somewhere. For a toroidal planet that would no longer be true.
The significance of the hairy ball theorem is that it asserts that there will be always be p such that f(p) = 0, and that this is true, without having to know any of the underlying dynamics of the wind (except for some basic physical arguments to rule out discontinuities).
Lefschetz connection
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 have fixed points (since the Lefschetz number is nonzero). 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.
Corollary
A consequence of the hairy ball theorem is that any continuous function that maps a sphere into itself has either a fixed point or a point that maps onto its own antipodal point. This can be seen by transforming the function into a tangential vector field as follows.
Let s be the function mapping the sphere to itself, and let v be the tangential vector function to be constructed. For each point p, construct the stereographic projection of s(p) with p as the point of tangency. Then v(p) is the displacement vector of this projected point relative to p. According to the hairy ball theorem, there is a p such that v(p) = 0, so that s(p) = p.
This argument breaks down only if there exists a point p for which s(p) is the antipodal point of p, since such a point is the only one that cannot be stereographically projected onto the tangent plane of p.
Higher dimensions
The connection with the Euler characteristic χ suggests the correct generalisation: the 2n-sphere has no non-vanishing vector field for n ≥ 1. The difference in even and odd dimension is that the Betti numbers of the m-sphere are 0 except in dimensions 0 and m. Therefore their alternating sum χ is 2 for m even, and 0 for m odd.
See also
Differential topology | Fixed points | Mathematical theorems
"Hairy ball theorem"