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.



Do Locally Contractible, Path-Connected Groups have Accessible Bases?
For your restricted question the answer is affirmative. It follows from the following theorem of [Yagasaki](https://doi.org/10.1016/S0166-8641(99)00130-3):


**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.


