S^2 isn't a special case though: Brouwer's showed the theorem can be easily extended to high dimensions, hence today we usually consider the more general statement that there is a nonzero tangent vector field on the n-sphere S^n iff n is odd.
Not only does it generalize to higher n, it also shows a bit more: not only that the lack of such vector field for an even n, but the also the existence of such for odds.
Heard a mathematician friend call this the “hairy sphere theorem” once. At first I thought he was being a prude, but now I appreciate that the theorem is about spheres, as opposed to balls.
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 am confused how we can define a rotation number of the map from S^1 to R^3 defined at the end of the second paragraph. R^3 is nullhomotopic, after all...
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[ 3.5 ms ] story [ 23.1 ms ] threadNot only does it generalize to higher n, it also shows a bit more: not only that the lack of such vector field for an even n, but the also the existence of such for odds.
I didn't quite understand the curves that they are constructing on S^2. Some figures would be nice.