hairy ball theorem


If X is a vector field on S2n, 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 S2n is not homotopicMathworldPlanetmath 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 χ(M)=vZ(X)ι(v) where Z(X) is the set of zeroes of X, and ι(v) is the index of x at v, and χ(M) is the Euler characteristicMathworldPlanetmath of M.

It is not difficult to show that S2n+1 has non-vanishing vector fields for all n. A much harder result of Adams shows that the tangent bundle of Sm is trivial if and only if n=0,1,3,7, corresponding to the unit spheres in the 4 real division algebrasMathworldPlanetmath.


First, the low tech proof. Assume that S2n has a unit vector field X. Then the antipodal map is homotopic to the identity ( 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 S2n is non-trivial, since any trivial bundleMathworldPlanetmath has a non-zero sectionPlanetmathPlanetmathPlanetmath. ∎


Now for the sledgehammer proof. Suppose X is a nonvanishing vector field on S2n. Then by the Poincaré-Hopf index theorem, the Euler characteristic of S2n is χ(X)=vX-1(0)ι(v)=0. But the Euler characteristic of S2k 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