Elementary equivalence and diffeomorphism groups of smooth manifolds

2025/07/10 by Kim, Sang-hyun, Koberda, Thomas, González, J. de la Nuez
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.2507.07427

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∈[1,∞)∪\0,∞\, 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.

Citations

• Uniform first order interpretation of the second order theory of countable groups of homeomorphisms
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Related

• Algebraic Structure of Diffeomorphism Groups of One-Manifolds
• Actions of groups of diffeomorphisms on one-manifolds by C¹ diffeomorphisms
• Actions of groups of diffeomorphisms on one-manifolds
• Reconstructing maps out of groups
• A simplification problem in manifold theory