Degree proof of the hairy ball theorem
= Degree proof of the hairy ball theorem
A nowhere-zero tangent <vector field> on $S^n$ can be normalized to a unit field $w(x)$ perpendicular to $x$. Then $\cos(\pi t)x+\sin(\pi t)w(x)$ is a homotopy from the identity to the <antipodal map>. Their <mapping degrees> are $1$ and $(-1)^{n+1}$, respectively, contradicting <homotopy invariance of mapping degree> when $n$ is positive and even. For $S^0$ the tangent spaces are zero, so every tangent field is zero directly.