Can I comb unoriented hair on a ball?

Question

I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball. (I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that the vector field is tangential to $S^2$ at every point if you prefer the Euclidean point of view.) Is this still impossible if the hair is unoriented? By unoriented hair I mean that instead of looking at a tangent vector at every point, I am looking at a tangent line. That is, is there a rank one subbundle of $TS^2$ (which is a line bundle on $S^2$)? Geometrical intuition suggests that there is not, but that is far from proof.

I am mostly interested in $S^2$, but I expect the answer is the same for all even-dimensional spheres just like in the oriented case.

Answer 1

An alternative way to see why this isn't true is to fix a Riemannian metric. If such a subbundle existed, then restricting to those vectors of norm $1$ would give a two-sheeted covering of $S^2$. If it were disconnected, there would be a nowhere zero vector field on $S^2$, a contradiction. If it were connected, $S^2$ would admit a connected two-sheeted covering, a contradiction with it being simply connected.

Answer 2

No. Line bundles on $S^2$ are classified by $H^1(S^2;\mathbb{Z}/2)=0$ which means that there is only the trivial line bundle $L$ on $S^2$. But then $L\oplus L$ is the trivial bundle hence not the tangent bundle of $S^2$.

An interesting question is if $TS^n$ splits as a direct sum $E\oplus F$ of low dimensional bundles. If $n$ is odd the Euler characteristic is zero, so there is a non-zero vector field. So let us assume $n$ is even.

If $E$ and $F$ are required to be orientable bundles this is not possible: The Euler class is multiplicative and $e(TS^n)$ is twice the generator of $H^n(S^n;\mathbb{Z})$ if $n$ is even and in particular it is non-zero. But if $E$ and $F$ are proper subbundles the euler classes $e(E)$ and $e(F)$ lie in some $H^i(S^n;\mathbb{Z})$ for $0<i<n$ hence $e(E)\cup e(F)=0$ which contradicts the fact that $e(E)\cup e(F)=e(TS^n)\not=0$.

A bundle is orientable iff its determinant bundle has a section, i.e. is a trivial line bundle. But we have seen that the sphere only admits the trivial line bundle. This implies that every bundle on $S^n$ is orientable and the previous argument shows that the tangent bundle cannot split at all.

This discussion btw does not imply that there are no non-trivial bundles on $S^n$ of rank $r<n$, and indeed there are some.