Generalized Hairy ball theorem

Question

I am looking for a reference about generalized hairy ball theorems. That is, the existence or non-existence of $C^{\infty}$ vector fields that vanish nowhere on a given manifold, $M$.

For which $M$ do we have results on this?

Answer 1

We can reduce our considerations to $M$ connected by considering the question on each connected component.

A smooth vector field on a smooth manifold $M$ is precisely a smooth section of $TM$. So your question can be rephrased: when does $TM$ admit a nowhere-zero section? This is a problem which can be solved using an area of algebraic topology called obstruction theory. What it shows is that there is a single obstruction to the existence of such a section of $TM$, namely $\mathfrak{o}(TM) \in H^n(M; \mathbb{Z})$ where $n = \dim M$. More precisely, such a section of $TM$ exists if and only if $\mathfrak{o}(TM) = 0$.

If $M$ is not closed, then $H^n(M; \mathbb{Z}) = 0$ so $\mathfrak{o}(TM) = 0$ and hence $TM$ admits a nowhere-zero section. If $M$ is closed and orientable, then $H^n(M; \mathbb{Z}) \cong \mathbb{Z}$ and $\mathfrak{o}(TM)$ is called the Euler class of $TM$, usually denoted $e(TM)$. The Euler class satisfies $\langle e(TM), [M]\rangle = \chi(M)$, so it follows that if $M$ is closed and orientable, then $TM$ admits a nowhere-zero section if and only if $\chi(M) = 0$. The same is true if you remove the orientability hypothesis, but the story is a bit more complicated.