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



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> 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.
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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?



On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces
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.


