Editing Hairy ball theorem (section)

==Higher dimensions==
The connection with the [[Euler characteristic]] χ suggests the correct generalisation: the [[N-sphere|2''n''-sphere]] has no non-vanishing vector field for {{nowrap|''n'' ≥ 1}}. The difference between even and odd dimensions is that, because the only nonzero [[Betti number]]s of the ''m''-sphere are b<sub>0</sub> and b<sub>m</sub>, their [[alternating sum]] χ is 2 for ''m'' even, and 0 for ''m'' odd.

Indeed it is easy to see that an odd-dimensional sphere admits a non-vanishing tangent vector field through a simple process of considering coordinates of the ambient even-dimensional [[Euclidean space]] <math>\mathbb{R}^{2n}</math> in pairs. Namely, one may define a tangent vector field to <math>S^{2n-1}</math> by specifying a vector field <math>v: \mathbb{R}^{2n} \to \mathbb{R}^{2n}</math> given by
:<math> v(x_1,\dots,x_{2n}) = (x_2, -x_1,\dots,x_{2n},-x_{2n-1}).</math>
In order for this vector field to restrict to a tangent vector field to the unit sphere <math>S^{2n-1}\subset \mathbb{R}^{2n}</math> it is enough to verify that the [[dot product]] with a [[unit vector]] of the form <math>x=(x_1,\dots,x_{2n})</math> satisfying <math>\|x\|=1</math> vanishes. Due to the pairing of coordinates, one sees
:<math> v(x_1,\dots,x_{2n}) \bullet (x_1,\dots,x_{2n}) = (x_2 x_1 - x_1 x_2) + \cdots + (x_{2n} x_{2n-1} - x_{2n-1} x_{2n}) = 0.</math>
For a 2''n''-sphere, the ambient Euclidean space is <math>\mathbb{R}^{2n+1}</math> which is odd-dimensional, and so this simple process of pairing coordinates is not possible. Whilst this does not preclude the possibility that there may still exist a tangent vector field to the even-dimensional sphere which does not vanish, the hairy ball theorem demonstrates that in fact there is no way of constructing such a vector field.