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.

5more comments