I'm reminded of: "The hairy ball theorem of algebraic topology states that there is no non-vanishing continuous tangent vector field on the 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 p such that f(p) = 0.

This is famously stated as "you can't comb a hairy ball flat", or sometimes, "you can't comb the hair on a billiard ball". It was first proved in 1912 by Brouwer.

In fact from a more advanced point of view it can be shown that the sum at the zeros 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. This is a consequence of the Poincaré–Hopf theorem. In the case of the 2-torus, the Euler characteristic is 0; and it is possible to 'comb a hairy doughnut flat'. In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero."

But you won't forget that the sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the Gauss map from the boundary of Neps to the n-1-dimensional sphere will you? Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold M. Technique: cut away all zeroes of the vector field with small neighborhoods. Then use the fact that the boundary of an n-dimensional manifold to an n-1-dimensional sphere, that can be extended to the whole n-dimensional manifold, is zero.

 

But IIRC doesn't the hairy ball theorem go on to imply that (as well as being able to comb hairy doughnuts flat in three dimensions) you can also comb hairy balls flat in spaces of 2,4,6 ... etc dimensions? It's only in spaces of an odd number of dimensions that you can't. Which all goes to make playing tennis in four spatial dimensions much more satisfactory in my view.

Actually, the above is used as the bass in comb and paper bands, although not often seen, owing to the paucity of C & P bands these days.
Although this (below) may be used for HIP performances