# Action of diffeomorphism group on non-vanishing vector fields

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.

• On an oriented 3-manifold this is the cohomotopy set $[M, S^2]$. For this, see this article. – Mike Miller Dec 5 '18 at 19:20
• I know that $\pi_0(\Gamma(TM\setminus 0))$ is the cohomotopy set $[M,S^2]$, but I don't see how this article answers the question about the action of $\text{Diff}(M)$. – ThorbenK Dec 5 '18 at 20:23
• That's going to be hard to answer without knowing something about the specific mapping class groups. That article only calculates the cohomotopy set, so it remains to determine the action. The action on $H_1 M$ is clear, but one needs to say something about the action on the fibers. I think, for instance, that if $M$ is a homology sphere the action on $[M, S^2] \cong \Bbb Z$ is by $\pm 1$ depending on whether or not the diffeomorphism is orientation preserving. – Mike Miller Dec 5 '18 at 20:46

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, 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.
• Is the action on $\pi_0(\Gamma(TM\setminus 0))$ for isometries of a manifold whose first de rham cohomology vanishes trivial, because in this case we can nicely identify vector fields with 1-forms? – ThorbenK Dec 6 '18 at 8:27
• That confused me earlier. Cobordism classes of framed links correspond to $[M, S^2]$, which is of course also the space of homotopy classes of non-vanishing vector fields. But there are different actions at play here right? The action on framed links is just given by precompoaition, while the push forward action on vector fields also twists the S^2 fibers of the tangent bundle so it does not just act via precompoaition. – ThorbenK Dec 7 '18 at 7:21
• Yes, the connection with maps to $S^2$ depends on a framing. One can ensure that the automorphisms preserve framing (trivialization) by pulling back a framing from downstairs. – Ian Agol Dec 7 '18 at 15:12