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2021-12-02

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cc-by-nc-nd (c) American Mathematical Society (AMS), 2021
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/195522

Which finite groups act smoothly on a given 4-manifold?

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https://doi.org/10.1090/tran/8518

Abstract

We prove that for any closed smooth 4-manifold $X$ there exists a constant $C$ with the property that each finite subgroup $G<\operatorname{Diff}(X)$ has a subgroup $N$ which is abelian or nilpotent of class 2 , and which satisfies $[G: N] \leq C$. We give sufficient conditions on $X$ for $\operatorname{Diff}(X)$ to be Jordan, meaning that there exists a constant $C$ such that any finite subgroup $G<\operatorname{Diff}(X)$ has an abelian subgroup $A$ satisfying $[G: A] \leq C$. Some of these conditions are homotopical, such as having nonzero Euler characteristic or nonzero signature, others are geometric, such as the absence of embedded tori of arbitrarily large self-intersection arising as fixed point components of periodic diffeomorphisms. Relying on these results, we prove that: (1) the symplectomorphism group of any closed symplectic 4-manifold is Jordan, and (2) the automorphism group of any almost complex closed 4-manifold is Jordan.

Subject

Transformacions (Matemàtica), Varietats (Matemàtica), Topologia de baixa dimensió, Varietats simplèctiques

Subject (English)

Transformations (Mathematics), Manifolds (Mathematics), Low-dimensional topology, Symplectic manifolds

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Articles publicats en revistes (Matemàtiques i Informàtica)

Citation

MUNDET I RIERA, Ignasi and SÁEZ CALVO, Carles. Which finite groups act smoothly on a given 4-manifold?. Transactions of the American Mathematical Society. 2021. Vol. 375, num. 2, pags. 1207-1260. ISSN 0002-9947. [consulted: 10 of August of 2026]. Available at: https://hdl.handle.net/2445/195522

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