Recovering topological spaces from their auto-homeomorphism groups.

Given two topological spaces $X$ and $Y$ we ask whether any algebraic isomorphism $phi$ between the homeomorphism groups $H(X)$ and $H(Y )$ of $X$ and $Y$ is a conjugation by some homeomorphism between $X$ and $Y$. That is, we ask whether there is $\tau : X \cong Y$ such that $\phi(g) = \tau \circ g \circ \tau^{1}$ for all $g \in H(X)$. I shall survey the work and open questions in this area. The following theorem will be explained.

Theorem: Let $X$ and $Y$ be open subsets of locally convex metrizable
topological vector spaces $E$ and $F$ respectively and $phi$ be an isomorphism between $H(X)$ and $H(Y)$. Then there is a homeomorphism $\tau$ between $X$ and $Y$ such that $\phi(g) = \tau \circ g \circ \tau^{1}$ for every $g \in H(X)$.