Continua having distal minimal actions by amenable groups

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.