Minimum topological group topologies

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≥2, the homeomorphism group H(X) has no minimum 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.