Elementary equivalence and diffeomorphism groups of smooth manifolds

TL;DR

This paper proves that elementary equivalence of diffeomorphism groups of smooth manifolds implies the manifolds are diffeomorphic with the same regularity, extending previous results to elementary equivalence and volume-preserving groups.

Contribution

It establishes that elementary equivalence of diffeomorphism groups determines the manifolds and their smoothness class, generalizing known isomorphism results to elementary equivalence and volume-preserving cases.

Findings

Elementary equivalence implies diffeomorphism of manifolds.

The regularity class of the diffeomorphism groups must match.

Results extend to volume-preserving diffeomorphism groups in higher dimensions.

Abstract

Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and $M$ and $N$ are $C^r$--diffeomorphic. This strengthens a previously known result by Takens and Filipkiewicz, which asserts that for integer regularities, a group isomorphism between diffeomorphism groups of closed manifolds necessarily arises from a diffeomorphism of the underlying manifolds. We prove an analogous result for groups of diffeomorphisms preserving smooth volume forms, in dimension at least two.