TITLE: Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?
QUESTION [9 upvotes]: A version of the "hairy ball" theorem, due probably to Chern, says that the Euler-characteristic of a closed (i.e. compact without boundary) manifold $M$ can be computed as follows.  Choose any vector field $\vec v \in \Gamma(\mathrm T M)$.  If $p$ is a zero of $\vec v$, then the matrix of first derivatives at $p$ makes sense as a linear map $\partial\vec v|_p : \mathrm T_p M \to \mathrm T_p M$.  By perturbing $\vec v$ slightly, assume that at every zero,  $\partial\vec v|_p$ is invertible.  Then $\chi(M) = \sum_{\vec v(p)=0} \operatorname{sign}\bigl( \det \bigl(\partial\vec v|_p\bigr)\bigr)$.
I am curious about the following potential converse: "If $M$ is closed and connected and $\chi(M) = 0$, then $M$ admits a nowhere-vanishing vector field."
Surely the above claim is false, or else I would have learned it by now, but I am not sufficiently creative to find a counterexample.  Moreover, I can easily see an outline of a proof in the affirmative, which I will post as an "answer" below, in the hopes that an error can be pointed out.  Thus my question:

What is an example of a compact, connected, boundary-free manifold with vanishing Euler characterstic that does not admit a nowhere-vanishing vector field?  (Or does no such example exist?)

REPLY [10 votes]: That a compact manifold M with vanishing Euler characteristic has a nonvanishing vector field was proved by Heinz Hopf,  Vektorfelder in Mannifaltigkeiten, Math. Annalen  95 (1925), 340-367.  A pretty convincing "intuitive proof" was outlined by Norman Steenrod in his book Fibre Bundles (Theorem 39.7), using a smooth triangulation and obstruction theory.  For a complete proof he refers to page 549 of the 1935 book Topologie by P. Alexandroff and H. Hopf