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.
Read more about this topic: Hairy Ball Theorem
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