Let $M$ be a connected topological manifold of dimension $n \geq 1$. It is well known that if $p,q \in M$ then there exists a homeomorphism $\phi : M \to M$ such that $\phi(p) = q$. This can be summed up by saying that every connected topological manifold is topologically homogeneous.
I have wondered a few times whether:
given points $m \in M$ and $p,q \in M \backslash \{m\}$ there exists a homeomorphism $f : M \to M$ which fixes $m$ and sends $p$ to $q$.
Now this is true when $M$ is a compact connected topological manifold of dimension $n \geq 2$. Here is why. Since $n \geq 2$, $M \backslash \{m\}$ is a connected manifold whose one point compactification is $M$. By topological homogeneity, there exists a homeomorphism $f : M \backslash \{m\} \to M \backslash \{m\}$ which maps $p$ to $q$. $f$ is a proper map, therefore it lifts to an automorphism $M \to M$ of the 1 point compactification which maps infinity to infinity, i.e it maps $m$ to $m$.
It is also true for $\mathbb{R}$. A dilation centered at $m$ does the trick.
is the above boxed statement true for other non compact manifolds?