We will talk about recent work in which we prove that the restriction of the compact-open topology to the diffeomorphism group of a manifold without boundary of dimension different from 3 is a minimal element of the lattice of Hausdorff group topologies on the group. If the dimension is also different from 4 it follows that the same holds for the compact-open topology on the homeomophism group, which combined with K. Mann's automatic continuity results implies the latter admits a unique separable Hausdorff group topology