  For every compact, connected manifold $M$, we prove the existence of a
sentence $\phi_M$ in the language of groups such that the homeomorphism group
of another compact manifold $N$ satisfies $\phi_M$ if and only if $N$ is
homeomorphic to $M$. We prove the analogous statement for groups of
homeomorphisms preserving an Oxtoby--Ulam probability measure.
