I am trying to prove that the action of $Homeo(\mathbb{R}²)$ on the plane is $n-$transitive for all $n$.
Given $\{p_1,...,p_n\}$ points of the plane, the idea was trying to prove that the action of the subgroup of $Homeo(\mathbb{R}^2)$ that fixes this points is transitive on $\mathbb{R}^2 - \{p_1,...,p_n\} $ and that this is valid for any $n$. I try using polynomials to write maps that has no effect in the fixed points (i.e the polynomials has a root here) and that are basicly traslation far away from this points. The problem is that I failed to get a bijection.
For simplicity, I tried writing my maps as complex maps, from $\mathbb{C}$ to $\mathbb{C}$
Also I would like to extend this reasoning to $S^2$, but I am convinced that this is just an adaption
Any ideas?
Thanks