hairy ball theorem

Theorem.

If $X$ is a vector field on $S^{\mathrm{2}\mathit{}n}$, then $X$ has a zero. Alternatively, there are no continuous unit vector field on the sphere. Moreover, the tangent bundle of the sphere is nontrivial as a bundle, that is, it is not simply a product.

There are two proofs for this. The first proof is based on the fact that the antipodal map on $S^{2n}$ is not homotopic to the identity map. The second proof gives the as a corollary of the Poincaré-Hopf index theorem.

Near a zero of a vector field, we can consider a small sphere around the zero, and restrict the vector field to that. By normalizing, we get a map from the sphere to itself. We define the index of the vector field at a zero to be the degree of that map.

Theorem (Poincaré-Hopf index theorem).

If $X$ is a vector field on a compact manifold $M$ with isolated zeroes, then $\chi \mathit{}\mathrm{(}M\mathrm{)}\mathrm{=}{\mathrm{\sum }}_{v\mathrm{\in }Z\mathit{}\mathrm{(}X\mathrm{)}}\iota \mathit{}\mathrm{(}v\mathrm{)}$ where $Z\mathit{}\mathrm{(}X\mathrm{)}$ is the set of zeroes of $X$, and $\iota \mathit{}\mathrm{(}v\mathrm{)}$ is the index of $x$ at $v$, and $\chi \mathit{}\mathrm{(}M\mathrm{)}$ is the Euler characteristic of $M$.

It is not difficult to show that $S^{2n+1}$ has non-vanishing vector fields for all $n$. A much harder result of Adams shows that the tangent bundle of $S^m$ is trivial if and only if $n=0,1,3,7$, corresponding to the unit spheres in the 4 real division algebras.

Proof.

First, the low tech proof. Assume that $S^{2n}$ has a unit vector field $X$. Then the antipodal map is homotopic to the identity (http://planetmath.org/AntipodalMapOnSnIsHomotopicToTheIdentityIfAndOnlyIfNIsOdd). But this cannot be, since the degree of the antipodal map is $-1$ and the degree of the identity map is $+1$. We therefore reject the assumption that $X$ is a unit vector field.

This also implies that the tangent bundle of $S^{2n}$ is non-trivial, since any trivial bundle has a non-zero section. ∎

Proof.

Now for the sledgehammer proof. Suppose $X$ is a nonvanishing vector field on $S^{2n}$. Then by the Poincaré-Hopf index theorem, the Euler characteristic of $S^{2n}$ is $\chi (X)={\sum }_{v\in X^{-1}(0)}\iota (v)=0$. But the Euler characteristic of $S^{2k}$ is $2$. Hence $X$ must have a zero. ∎

Title	hairy ball theorem
Canonical name	HairyBallTheorem
Date of creation	2013-03-22 13:11:33
Last modified on	2013-03-22 13:11:33
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	12
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 57R22
Synonym	porcupine theorem
Synonym	Poincaré-Hopf theorem
Defines	Poincaré-Hopf index theorem