Elementary equivalence and diffeomorphism groups of smooth manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914266890633216 |
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| author | Kim, Sang-hyun Koberda, Thomas González, J. de la Nuez |
| author_facet | Kim, Sang-hyun Koberda, Thomas González, J. de la Nuez |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elementary equivalence and diffeomorphism groups of smooth manifolds Kim, Sang-hyun Koberda, Thomas González, J. de la Nuez Group Theory Geometric Topology Logic 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. |
| title | Elementary equivalence and diffeomorphism groups of smooth manifolds |
| topic | Group Theory Geometric Topology Logic |
| url | https://arxiv.org/abs/2507.07427 |