  Let $X$ be a smooth manifold belonging to one of these three collections:
acyclic manifolds (compact or not, possibly with boundary), compact connected
manifolds (possibly with boundary) with nonzero Euler characteristic, integral
homology spheres. We prove that $Diff(X)$ is Jordan. This means that there
exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an
abelian subgroup whose index in $G$ is at most $C$. Using a result of Randall
and Petrie we deduce that the automorphism groups of connected, non necessarily
compact, smooth real affine varieties with nonzero Euler characteristic are
Jordan.
