Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms **with compact support**, turning it into a (locally-)compact topological group?

(My question is motivated by the fact that the isometries of a (locally-)compact metric space with finitely many connected components form a (locally-)compact topological group, and I'd like to see how far this analogy can be pushed.)