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First order rigidity of manifold diffeomorphism groups

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NPCW06 - Non-positive curvature and applications

Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that for two real numbers r,s ≥ 1, and for two smooth closed manifolds M and N, the Cr diffeomorphism group of M is elementarily equivalent to the Cs diffeomorphism group of N if and only if r=s and M is diffeomorphic to N. In the case of r=s=0, we can even weaken the hypothesis so that the manifolds are only assumed to be compact, dropping the smoothability or closedness hypotheses. This strengthens (1) Whittaker’s theorem (1963) on the C0 case, and (2) Takens—Filipkiewicz theorems (1982) on Cp case with an integer p. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS)

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