Given a topological group $G$ , one may be interested in understanding its possible minimal actions. This can be partially achieved by calculating the universal minimal flow (UMF) of $G$. The UMF is the (unique) $G$-minimal system which factors onto all $G$-minimal systems. Answering a question of Uspenskij (2000), we prove that if $X$ is a closed manifold of dimension $2$ or higher or the Hilbert cube, then the universal minimal flow of $Homeo(X)$, the group of homeomorphisms of $X$ equipped with the compact-open topology, is not metrizable.
This is joint work with Todor Tsankov and Andy Zucker.