We study the topological characters of a continuum $X$ and the algebraic structures of a group $G$ that forbid $G$ from acting on $X$ distally and minimally. Explicitly, we obtain the following results: (1) Let $G$ be a lattice in ${/rm SL}(n, /mathbb R)$ with $n/geq 3$ and $/mathcal S$ be a closed surface. Then $G$ has no distal minimal action on $/mathcal S$. (2) If $X$ admits a distal minimal action by a finitely generated amenable group, then the first /v Cech 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.

 

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