First order rigidity of homeomorphism groups of manifolds

Abstract

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

60 pages. To appear in Communications of the AMS