Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0) $ via $f. \phi(x) =Df\_{f^{-1}(x)}(\phi(f^{-1}(x))$. I'm interested in the induced action on $\pi\_0( \Gamma(TM\setminus 0))$, which evidently factors through $\pi\_0(\text{Diff}(M))$. 


The most interesting question for me at the moment is whether this induced action is faithful or transitive. Especially whether it is faithful or transitive if $M$ is an orientable $3$-manifold. But I'm also interested in more facts about this action and what kind of methods are helpful in understanding this action. 



Action of diffeomorphism group on non-vanishing vector fields
In general, the action of $\pi\_0(Diff(M))$ on $\pi\_0(\Gamma(TM\backslash 0))$ is not faithful. One can find hyperbolic homology 3-spheres with non-trivial isometry group. By a theorem of [Gabai](https://mathscinet.ams.org/mathscinet-getitem?mr=1895350), the mapping class group is the isometry group. But the action on $\pi\_0(\Gamma(TM\backslash 0))$ will be trivial. 


To find such examples, take a hyperbolic periodic knot, and do $1/n$ Dehn filling to get a homology 3-sphere with a cyclic action. 


