A Short Proof of the Hairy Ball Theorem (2016) [pdf]

Ooh go on, what do they say about Germans?

I lived in West Germany for some years back in the day and I don't recall the locals being too shy. Frankly the Germans and the Dutch seemed to have had a rather more ribald sense of humour than the "oo err Missus" efforts we Brits fielded back then.

To be fair we could robustly swear on telly after 2100, provided it didn't involve too many rude bits and you could not misspell one variant of King Canute's name or Matron would be jolly upset.

Anyway, I'm pretty sure someone called this the "dog's arse" (it has to go somewhere!)

Also known as the Combed Hedgehog Theorem (which i like a bit better)

Another 'reason' it works for odd-dimensional spheres is that the (2n - 1)-sphere can be identified with a certain subset of C^n (n-dimensional complex coordinate space) where your 'swap elements and multiply one by -1' idea is just multiplication by i, which, when you think of your vectors as being back in R^2n again, always produces something orthogonal to the original vector.

Even better, the (4n - 1)-sphere (so think of S^3, S^7, S^11, ...) can be thought of as a certain subset of H^n (same thing as before but with quaternions instead of complex numbers), where multiplication by i, j and k are available! And now in this case you have not only one nowhere-vanishing vector field on the sphere, but three everywhere pairwise orthogonal vector fields. This in particular shows that S^3 is 'parallelisable' — a property it shares with S^1 and means that there exists a continuous global choice of basis for each tangent space.

Well, all these closed surfaces you mention are (topologically) spheres. The theorem doesn’t apply to some other closed surfaces, like the torus, which does admit a continuous non-vanishing vector field.

I think there's a typo in the definition of C? It should say q in R^3, not S^2, right?

I'm finding it a little harder to visualize rotation numbers, though. My best attempt at a description is to imagine continuously tracing the curve '\gamma(t)', going through every point that it passes through, while looking top-down on it. At every point on the curve, the vector field 'v' produces a vector 'v(\gamma(t))' that begins at '\gamma(t)', lies flat on the sphere (i.e. is tangent to the sphere), and is of nonzero length. (The last assumption is the assumption we are making for contradiction).

The idea is that, as we trace the curve '\gamma(t)', we are constantly measuring the angle - with a positive-negative sign - between (a) the tangent vector 'v(\gamma(t))', and (b) the current velocity vector of '\gamma(t)'. As we trace the curve, if this angle rotates counterclockwise 0...90...180...270...0, we add "1" to our rotation number, and we subtract one for a clockwise rotation 0...-90...-180...-270...0.