Hairy ball theorem
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".
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.
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.
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.
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.
For more see vector fields on spheres.