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

## TL;DR

This paper establishes bounds on finite group actions on smooth 4-manifolds, proving Jordan properties for symplectomorphism and automorphism groups under certain conditions, and identifying structural subgroups within finite groups of diffeomorphisms.

## Contribution

It introduces new bounds on finite subgroups of diffeomorphism groups of 4-manifolds and proves Jordan properties for symplectic and almost complex cases.

## Key findings

- Finite subgroups have abelian or nilpotent subgroups with bounded index.
- Conditions for Diff(X) to be Jordan include topological and geometric criteria.
- Symplectomorphism and automorphism groups of 4-manifolds are Jordan under these conditions.

## 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.