Continua having distal minimal actions by amenable groups

TL;DR

This paper proves that continua with distal minimal actions by finitely generated amenable groups have nontrivial first Čech cohomology, implying such spaces cannot be simply connected if homotopy equivalent to a CW complex.

Contribution

It establishes a link between group actions and topological properties of continua, showing nontrivial cohomology under certain dynamical conditions.

Findings

Nontrivial first Čech cohomology for continua with distal minimal group actions

Such continua cannot be simply connected if homotopy equivalent to a CW complex

Provides a topological obstruction related to group actions

Abstract

Let $X$ be a non-degenerate connected compact metric space. If $X$ admits a distal minimal action by a finitely generated amenable group, then the first \vCech cohomology group $ {\check H}^1(X)$ with integer coefficients is nontrivial. In particular, if $X$ is homotopically equivalent to a CW complex, then $X$ cannot be simply connected.