A Hausdorff topological group topology on a group $G$ is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on $G$. For every compact metrizable space $X$ containing an open $n$-cell, $n\ge2$, the homeomorphism group $H(X)$ has no minimum Hausdorff group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space $X$ containing a dense open one-manifold, $H(X)$ has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.