TITLE: Automorphism of Smooth manifolds
QUESTION [6 upvotes]: Suppose $M$ is a smooth manifold and $x,y \in M$ are two points. Is there always a
diffeomorphism $\phi: M \rightarrow M$ with $\phi(x)= y$ ?

REPLY [9 votes]: Here's another argument, exploiting the differentiable structure a bit more (in the sense that I don't see a straightforward way to adapt it to the topological/PL case). It might be the standard argument to show that things work locally, but it works globally as well.
Take a smooth, embedded path $\gamma: [0,1]\to M$ so that $\gamma(0)=p$ and $\gamma(1)=q$.  Since $\gamma$ is embedded, we can push forward $\partial/\partial t$ and get a vector field $W$ on the image of $\gamma$. Let $V$ be a compactly supported extension of $W$ to all of $M$.
Then it's easy to check that the flow of $V$ at time 1 is a diffeomorphism of $M$ that sends $p$ to $q$ (and is isotopic to the identity).
This argument readily generalises to $k$-transitivity, too: if $\dim M\ge 3$, we can choose generic paths $\gamma_i$ simultaneously, and use the same exact argument. If $M$ is a surface, we need to use the fact that a single embedded path doesn't disconnect $M$.