We know that (see Hatcher's vector bundles and K-theory Prop. 3.22) the Euler class of an orientable vector bundle or rank $r$, $E\to M$ is the first obstruction to the existence of a never vanishing section of $E$ and thus belongs to $H^r(M,\mathbb{Z})$ .
It follows that if we consider $E= TM$ the tangent bundle $r=\dim (M) =: n$ and thus the Euler class $e(TM)$ is non just the first obstruction but all the obstruction.
Consequently if $e(TM)=0$ there exists a non vanishing section of $TM$
This seems a bit strong as would imply that the Euler characteristic is zero iif we have a never vanishing vector field.
Is my argument correct?