Finite group actions on homology spheres and manifolds with nonzero Euler characteristic

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.