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We show that for any connected smooth manifold $M$ of dimension differentfrom $3$ the restriction of the compact-open topology to the diffeomorphismgroup of $M$ is minimal, i.e. the group does not admit a strictly coarserHausdorff group topology. This implies the minimality of the compact-opentopology on the homeomorphism group of $M$ in all dimensions different from $3$and $4$. In those cases for which in addition to all of this automaticcontinuity is known to hold, such as when $M$ is closed, one can conclude thatthe compact-open topology is the unique separable Hausdorff group topology onthe homeomorphism group.

The compact-open topology on the diffeomorphism or homeomorphism group of a smooth manifold without boundary is minimal in almost all dimensions