In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are topological spaces in a broad class of spaces $K$ and there is an isomorphism between $\mathrm{Homeo}(X)$ and $\mathrm{Homeo}(Y)$, then $X$ and $Y$ are homeomorphic" are proved. Moreover the following result is claimed
Assume $V=L$. If $X$ and $Y$ are second countable connected Euclidean manifolds and $\mathrm{Homeo}(X)$ is elementary equivalent to $\mathrm{Homeo}(Y)$, then $X$ and $Y$ are homeomorphic.
to appear in Second countable connected manifolds with elementarily equivalent homeomorphism groups are homeomorphic in the constructible universe. Unfortunately I cannot find any information on a paper with this title online. Has a proof of this theorem been published by Rubin? What is known about this result in $\mathsf{ZFC}$ without extra set theoretic assumptions?
Best Answer
There's a new paper that proves something even stronger for compact manifolds: https://arxiv.org/abs/2302.01481. I don't know about noncompact manifolds, though, and it seems neither do the authors.