$n$-transitivity of $\operatorname{Diff}(M)$ acting on a smooth manifold $M$

Question

Let $M$ be a smooth connected manifold and $\operatorname{Diff}(M)$ the set of diffeomorphisms from $M$ to $M$. I would like to show that this group acts $n$-transitively on $M$.

I started by showing transitivity. I looked at the orbit of one point and showed that is must be both open and closed (relying on the homogeneity of Euclidean space to whom $M$ is locally diffeomorphic and "globalising" via partitions of unity).The result thus follows from connectedness.

Is there some nice way to adapt this argument to obtain $n$-transitivity? Maybe an induction?

Thanks

Answer 1

Hint:for every $x\in M$ Use bump function to construct diffeomorphisms which are equal to the identity on the complement of a neighborhood of $x$ and act transitively on a neighborhood of $x$.

You have a neighborood $U$ of $x$ which is diffeomorphic to a ball $B(0,c)$ of $R^n$. You can find a bump function $f$ defined on $U$ such that the restriction of $f$ to $B(0,c/4)$ is $1$ and $f(y)=0$ if $\|y\|>c/2$. Let $u\in R^n$. You can define the vector field $X_u=fu$ which extend to $M$, and the flow $\phi^u_t$ of $fu$.

Answer 2

Let me outline a construction that works the same way for $n = 1$ and $n > 1$. Assume $M$ is connected.

For motivation, consider first the case $n = 1$ and let $p, q \in M$. Show first that $p, q$ can be connected by an embedded curve $\gamma \colon [0,1] \rightarrow M$ with $\gamma(0) = p, \gamma(1) = q$ (so that the image $\gamma([0,1])$ is a closed one-dimensional embedded manifold with boundary). On $[0,1]$ (with coordinate $t$) with we have vector field $\frac{d}{dt}$ whose time-one flow take $0$ to $1$. If we set $X = \dot{\gamma}(t)$ on $\gamma([0,1])$ and extend $X$ arbitrary to a compactly supported vector field $\tilde{X}$ on $M$ with $\tilde{X}(\gamma(t)) = X(\gamma(t)) = \dot{\gamma}(t)$ then by construction, the curve $\gamma$ is an integral curve of $\tilde{X}$ with $\gamma(0) = p, \gamma(1) = q$ so the time-one flow of $\tilde{X}$ is a global diffeomorphism of $M$ taking $p$ to $q$.

Now assume $n > 1$ and $\dim M > 1$ (otherwise, the result is not true). Show that given distinct $p_1, \dots, p_n$ and distinct $q_1, \dots, q_n$, we can find disjoint embedded curves $\gamma_i \colon [0,1] \rightarrow M$ with $\gamma_i(0) = p_i$ and $\gamma_i(1) = q_i$. Then we have a well-defined vector field on the closed, disconnected, embedded submanifold $\sqcup \gamma_i([0,1])$ (given on each $\gamma_i([0,1])$ like before by $\dot{\gamma_i}(t)$) and so by the extension lemma, we can extend it to a compactly supported global vector field on $M$ whose time-one flow will give us the required result.