Elementary equivalence and diffeomorphism groups of smooth manifolds
Sang-hyun Kim, Thomas Koberda, J. de la Nuez González
TL;DR
The paper proves a first-order rigidity phenomenon for smooth manifold diffeomorphism groups: if Diff^r_0(M) and Diff^s_0(N) (or their volume-preserving counterparts) are elementarily equivalent under suitable hypotheses (M closed, dim≥2), then r=s and M,N are C^r-diffeomorphic. The authors develop an interpretability framework, introducing ACT and AGAPE structures, and show these are uniformly interpretable in the group language, enabling reconstruction of the manifold and its smooth structure from group-theoretic data. The two main components—Manifold Recognition and Regularity Detection—together imply that elementary equivalence rigidly determines both the regularity and the underlying smooth manifold, extending Takens–Filipkiewicz to non-integer regularities. The results advance the program of first-order rigidity for geometric structures and reveal how the first-order theory of diffeomorphism groups encodes deep geometric information, including potential exotic smooth structures and the precise regularity of ambient diffeomorphisms.
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.