  We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth
manifold, possibly with boundary, satisfying $\chi(X)\neq 0$, then there exists
an integer $C\geq 1$ such that any finite group $G$ acting smoothly and
effectively on $X$ has an abelian subgroup $A$ satisfying $[G:A]\leq C$,
$\chi(X^A)=\chi(X)$, and $A$ can be generated by at most $2$ elements.
Furthermore, if $\chi(X)<0$ then $A$ is cyclic. This proves, for any such $X$,
a conjecture of Ghys. We also prove an analogous result for manifolds of
arbitrary dimension and non-vanishing Euler characteristic, but restricted to
pseudofree actions.
