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Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\mathbb{R}^2)$, in the compact-open topology. In this context, it's the same as the topology of compact convergence. $G$ is metric since it's a subgroup of the orientation-preserving homeomorphisms of the (compact) sphere, namely the ones fixing infinity.

What I'd like to do is extend these local contractions to global homeomorphisms of $G$. I feel like this could be impossible in some weird cases, so for my purposes it's sufficient to have the following:

Does each point have a local basis of contractible neighborhoods whose boundaries are path-accessible?

In other words, for $x \in G$, is there an arbitrarily small, contractible nbhd $U$ of $x$ such that for every $y \in \partial(U)$, there is an embedding $f$ of $[0,1]$ into $G \setminus U$ with $f(0) = y$?

It seems like a big ask, but I actually think this is true. If not, is it at least true for $Aut(\mathbb{R}^2)$?

EDIT: Note that by the comments below, there is some historical issue with the definition of "locally contractible." The "geometric topologist's sense" is a weaker definition: For each nbhd $U$ of $x$ there is a nbhd $V \subset U$ that deforms to a point in $U$, not necessarily itself. Especially, $Aut(\mathbb{R}^2)$ may only satisfy the weaker version. Feel free to use either version for this question, though. In this case, we'll want the $V$ to have accessible boundary.

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  • $\begingroup$ Is it true that each point of $Aut(\mathbb{R}^2)$ has a local base of contractible neighbourhoods (this is not the standard definition of local contractibility)? $\endgroup$ – Tyrone Mar 29 at 13:04
  • $\begingroup$ Yes it's true; it's the standard definition as far as I know! What is your definition? $\endgroup$ – John Samples Mar 29 at 16:12
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    $\begingroup$ A common definition of local contractibility is that for all $x\in U\subseteq X$ ($U$ open) there is $x\in V\subseteq U$ ($V$ open) such that the inclusion $V\subseteq U$ is inessential (so $V$ is deformable in $U$ to a point). As far as I can tell from, say, Kirby-Edwards, this is as much as you can say in general about homeomorphism groups. I looked at Dyer-Hamstrom too for their work on $Aut(\mathbb{R}^2)$ and they seemed to be using the same definition. I'd be interested to know where the stronger condition is established (or if I'm just missing something ;)). $\endgroup$ – Tyrone Mar 29 at 16:35
  • $\begingroup$ It's the same, but it's not obvious, as it's special to this situation. See 1.11 Remark (3) here: maths.dur.ac.uk/users/mark.a.powell/… I think the definition you cite was used because it was better-suited to the other topologies (uniform and Whitney/"majorant") he wanted to compare to. $\endgroup$ – John Samples Mar 29 at 16:52
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    $\begingroup$ You know what, I think you might be right that this is a big issue. For some reason I was so sure that this was also true for the plane, but I'm beginning to doubt it. I posted an answer in this thread related to the topic, maybe you can look at it and see if this is the idea? Or if this isn't a proper way of understanding the obstruction to "full local contractibility": math.stackexchange.com/questions/4075662/… $\endgroup$ – John Samples Mar 30 at 9:21
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For your restricted question the answer is affirmative. It follows from the following theorem of Yagasaki:

Theorem. For any subpolyhedron $X$ in a connected 2-manifold $M$, the connected component of the group $H_X(M)$ of homeomorphisms of $M$ that are identity on $X$ is an $\ell_2$-manifold.

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    $\begingroup$ @JohnSamples Maybe this paper on the local contratibility of homeomorphism groups will be of some help? R. Geoghegan, On spaces of homeomorphisms, embeddings and functions – I, Topology 11 (1972), 159–177. $\endgroup$ – Taras Banakh Apr 1 at 7:44
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    $\begingroup$ @JohnSamples An $\ell_2$-manifold is a paracompact topological space that has a base of the topology consisting of sets homeomorphic to the separable Hilbert space $\ell_2$. So, locally an $\ell_2$-manifold behave like the Hilbert space. This yields all desirable properties of neighborhoods (like accessibility). $\endgroup$ – Taras Banakh Apr 1 at 8:37
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    $\begingroup$ @JohnSamples For the group $Aut(\mathbb R^2)$ one can take $X=\emptyset$ and conclude that $Aut(\mathbb R^2)=H_\emptyset(\mathbb R^2)$. But on the other hand, $Aut(\mathbb R^2)$ can be identified with the group $H_{X}(S^2)$ where $X$ is a singleton in the 2-sphere $S^2$. So, this general result of Yagasaki can be applied in many different ways. $\endgroup$ – Taras Banakh Apr 1 at 8:39
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    $\begingroup$ @JohnSamples Actually, it is a classical open problem if $H(M)$ is an $\ell_2$-manifold for any $n$-manifold $M$. The answer is known only in dimension $\le 2$. For higher dimension we only know that the homeomorphism group is locally contractible and stable under multiplication by $\ell_2$. The ANR-property is a wide open question. $\endgroup$ – Taras Banakh Apr 1 at 8:44
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    $\begingroup$ I came into this from plane dynamics/isotopies, I've avoided infinite-dimensional anything my whole life. Occasionally I've regretted it, this is one of those times because this is all very interesting and beautiful! $\endgroup$ – John Samples Apr 1 at 8:45

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