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Abstract

Let $X$ be a closed manifold of dimension $2$ or higher or the Hilbert cube. Following Uspenskij one can consider the action of ${\rm Homeo}(X)$ equipped with the compact-open topology on ${\mit\Phi}\subset 2^{2^{X}}$, the space of maximal chains in $2^{X}$, equipped with the Vietoris topology. We show that if one restricts the action to $M\subset {\mit\Phi}$, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of ${\rm Homeo}(X)$ on $U_{{{\rm Homeo}(X)}}$, the universal minimal space of ${\rm Homeo}(X)$, is not transitive (improving a result of Uspenskij). Additionally for $X$ as above with ${\rm dim}(X)\geq 3$ we characterize all the minimal subspaces of $V(M)$, the space of closed subsets of $M$, and show that $M$ is the only minimal subspace of ${\mit\Phi}$. For ${\rm dim}(X)\geq 3$, we also show that $(M,{\rm Homeo}(X))$ is strongly proximal.