First order rigidity of homeomorphism groups of manifolds

Kim, Sang-hyun; Koberda, Thomas; de la Nuez González, J.

doi:10.1090/cams/47

First order rigidity of homeomorphism groups of manifolds
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by Sang-hyun Kim, Thomas Koberda and J. de la Nuez González;

Comm. Amer. Math. Soc. 5 (2025), 144-194

DOI: https://doi.org/10.1090/cams/47

Published electronically: May 12, 2025

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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 an analogous statement for groups of homeomorphisms preserving Oxtoby–Ulam probability measures.

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First order rigidity of homeomorphism groups of manifolds
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Abstract: