Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Which finite groups act smoothly on a given $4$-manifold?   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.

19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat