  Let $X$ be a compact smooth manifold, possibly with boundary. Denote by
$X_1,\dots,X_r$ the connected components of $X$. Assume that the integral
cohomology of $X$ is torsion free and supported in even degrees. We prove that
there exists a constant $C$ such that any finite group $G$ acting smoothly and
effectively on $X$ has an abelian subgroup $A$ of index at most $C$, which can
be generated by at most $\sum_i[\dim X_i/2]$ elements, and which satisfies
$\chi(X_i^A)=\chi(X_i)$ for every $i$. This proves, for all such manifolds $X$,
a conjecture of \'Etienne Ghys. An essential ingredient of the proof is a
result on finite groups by Alexandre Turull and the author which uses the
classification of finite simple groups.
