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If M is a connected manifold of dimension $\geq 2$ 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.

Actually I only need it for dimension 2 so if your ref only covers dimension 2, I'll be happy with that.

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    $\begingroup$ Kind of pedantic but you do need an asterisk for 1-dimensional manifolds, because there the ordering of the points will be preserved $\endgroup$
    – Kevin Casto
    Commented May 18, 2022 at 15:13
  • $\begingroup$ Good point @KevinCasto, I have edited the question $\endgroup$
    – Arnaud Chéritat
    Commented May 18, 2022 at 16:01

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In [Ancel and Bellamy, On homogeneous locally conical spaces, Fund. Math. 241 (2018), no. 1, 1–15] it is shown that every homogeneous locally conical connected separable metric space that is not a $1$-manifold is strongly $n$-homogeneous for each $n\ge 2$.

Locally conical means that each point has a neighborhood homeomorphic to an open cone over a compact space. Manifolds are of course locally conical. It is a good elementary exercise that connected manifolds are homogeneous. (Hint: first prove it for points in the same chart, and then note that any two points can be joined by a chain of charts).

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  • $\begingroup$ Quite interesting, thanks. I will look closely. $\endgroup$
    – Arnaud Chéritat
    Commented May 19, 2022 at 6:47
  • $\begingroup$ Very nice result and reference. Almost what I am looking for. $\endgroup$
    – Arnaud Chéritat
    Commented May 19, 2022 at 9:10
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I suppose we induct on $n$ (the number of points) and appeal to some version of "Guggenheim's theorem". I started by looking in "Introduction to piecewise linear topology" by Rourke and Sanderson [1982] (see page 56 and after). In their references, I found "Extending piecewise-linear isotopies" by Hudson [1966] who in turn points at "Unpublished doctoral dissertation" by Irwin [1962].

Anyway, Hudson gives the result in codimension three (too bad!) but gives a complicated sufficient condition ("allowable locally unknotted") in codimension two (your situation). But I am sure that "a finite set of points in a surface" satisfy the condition...

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  • $\begingroup$ Thanks. That approach seems overkill. Btw what is Guggenheim's theorem? $\endgroup$
    – Arnaud Chéritat
    Commented May 18, 2022 at 20:48
  • $\begingroup$ "Any two PL n-balls in a connected PL n-manifold are PL ambient isotopic." The actual theorem statement is more general. Here is the correct spelling of the name: Victor Kurt Alfred Morris Gugenheim. Here are the relevant papers: Some theorems on piecewise linear embedding [1952 - the announcement] and "Piecewise Linear Isotopy and Embedding of Elements and Spheres (I, II)" [1953 - the proofs]. $\endgroup$
    – Sam Nead
    Commented May 19, 2022 at 9:16
  • $\begingroup$ There is an sequence of memories regarding VKAMG (very fondly) here: maths.ed.ac.uk/~v1ranick/surgery/uicc/vkamg.txt $\endgroup$
    – Sam Nead
    Commented May 19, 2022 at 9:22
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I think the following discussion (which includes references) may be interesting:

https://lamington.wordpress.com/2014/11/07/explosions-now-in-glorious-2d/

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  • $\begingroup$ It was interesting reading this piece of history. Thanks. Too far from what I need, though. $\endgroup$
    – Arnaud Chéritat
    Commented May 20, 2022 at 7:14
  • $\begingroup$ I think the lemma by Nietecki-Shub stated there gives exactly what you need (after showing that, say using convenient a gradient flow, you can take the two pairs of n-points to a small ball). $\endgroup$
    – rpotrie
    Commented May 21, 2022 at 15:39
  • $\begingroup$ It implies the result I want. But there is the distance complication. I will look in the article to see if they do the simple case but I doubt. $\endgroup$
    – Arnaud Chéritat
    Commented May 21, 2022 at 16:23
  • $\begingroup$ I read the two relevant statements (Lemmas 9 and 13) in the Nietecko-Shub paper and the proof of the second one. The proof has a few problems. Their constant 2pi is in fact the better constant 4 (actually this is good). They use state Lemma 9 with only one path but use it with several paths. The claim that the time-1 maps sends the slice 0 to the slice 2pi is likely wrong (this can be fixed but changes the estimate at the end ; maybe Lemma 9 can be adapted in fact). $\endgroup$
    – Arnaud Chéritat
    Commented May 21, 2022 at 18:04
  • $\begingroup$ Ok, sorry. I didn't know that there was some careless arguments in their proof. I had just read the blog post, and thought the reference could be useful. $\endgroup$
    – rpotrie
    Commented May 21, 2022 at 22:37

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