I was browsing through the litterature, hoping to find sufficient and necessary conditions for a smooth manifold to have finite-dimensional de Rham cohomology, but I can't find any satisfactory answer. I wonder if anyone has ever encountered a paper, or a book, answering (possibly in part) the question. I am especially interested in real-coefficient cohomology, but I would appreciate answers related to cohomology with coefficients in any abelian group.

Obviously, I don't expect "compact manifold" as an answer; although this is a sufficient condition, it is far from answering the question.

per se, but rather about the algebraic topology of smooth manifolds. – Pete L. Clark Feb 17 '11 at 5:44