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.