Universal minimal flows of homeomorphism groups of high-dimensional manifolds are not metrizable

TL;DR

This paper proves that the universal minimal flows of homeomorphism groups of high-dimensional manifolds and the Hilbert cube are not metrizable, revealing complex dynamical properties of these groups.

Contribution

It establishes non-metrizability of universal minimal flows for homeomorphism groups of manifolds of dimension two or higher and the Hilbert cube, answering Uspenskij's question.

Findings

Universal minimal flows of $ ext{Homeo}(X)$ are not metrizable for high-dimensional manifolds.

Minimal flows with maximal connected chains have meager orbits in dimensions three and above.

Provides new insights into the dynamics of homeomorphism groups of complex manifolds.

Abstract

Answering a question of Uspenskij, we prove that if $X$ is a closed manifold of dimension $2$ or higher or the Hilbert cube, then the universal minimal flow of $\mathrm{Homeo}(X)$ is not metrizable. In dimension $3$ or higher, we also show that the minimal $\mathrm{Homeo}(X)$-flow consisting of all maximal, connected chains in $X$ has meager orbits.