If M is a connected manifold then the set of orientation preserving homeomorphisms of M that are isotopic to the identity acts $n$-transitively on M for all positive $n\in\mathbb{N}$. I know several ways to prove this. I do not want to include a proof of this well known fact in my article, instead I am looking for a reference to a book or article where this is stated and proved (preferentially in a style that is not discouraging for the reader). Up to now I have not found such a reference.
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You need to assume that $M$ has dimension $\ge 2$, because $n$-transitivity is false if $M=S^1$ and $n \ge 3$. – Lee Mosher May 20 '22 at 01:46
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Good point @Lee_Mosher. I only need dimension 2 actually, so a ref that only covers d=2 is ok for me. – Arnaud May 20 '22 at 08:31