First order rigidity of homeomorphism groups of manifolds

TL;DR

This paper establishes a logical characterization of the homeomorphism groups of compact manifolds, showing that each such group uniquely encodes the topology of its manifold, including measure-preserving cases.

Contribution

It introduces a first-order logical sentence that uniquely identifies the homeomorphism group of a given compact manifold, linking group properties to manifold topology.

Findings

Existence of a logical sentence characterizing homeomorphism groups

Unique identification of manifold topology via group properties

Extension to measure-preserving homeomorphism groups

Abstract

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.