First order rigidity of homeomorphism groups of manifolds

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.