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

Mundet i Riera, Ignasi; Sáez–Calvo, Carles

doi:10.1090/tran/8518

Which finite groups act smoothly on a given $4$-manifold?
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Trans. Amer. Math. Soc. 375 (2022), 1207-1260 Request permission

Abstract:

We prove that for any closed smooth $4$-manifold $X$ there exists a constant $C$ with the property that each finite subgroup $G<Diff(X)$ 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 $Diff(X)$ to be Jordan, meaning that there exists a constant $C$ such that any finite subgroup $G<Diff(X)$ 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.

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