First order rigidity of manifold homeomorphism groups

Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We establish that if the homeomorphism groups of two compact connected manifolds are elementarily equivalent, then the manifolds are homeomorphic. This generalizes Whittaker's theorem on isomorphic homeomorphism groups (Ann. of Math. 1963) without relying on it. Our result also confirms a conjecture of M. Rubin (1989). We also show the analogous results for measure-preserving homeomorphism groups. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS).