Minimal actions of homeomorphism groups

Yonatan Gutman¹

¹ Institute of Mathematics The Hebrew University Jerusalem, Israel

Fundamenta Mathematicae, Tome 198 (2008) no. 3, pp. 191-215

Résumé

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.

Détail

DOI : 10.4064/fm198-3-1

Keywords: closed manifold dimension higher hilbert cube following uspenskij consider action homeo equipped compact open topology mit phi subset space maximal chains equipped vietoris topology restricts action subset mit phi space maximal chains continua action minimal transitive shows action homeo homeo universal minimal space homeo transitive improving result uspenskij additionally above dim geq characterize minimal subspaces space closed subsets only minimal subspace mit phi dim geq homeo strongly proximal