Is compactly supported de Rham cohomology always finite dimensional?

Question

I know that for a compact smooth manifold $M$, $\dim H^p_{dR}(M) < \infty$. I am trying to prove Poincaré duality, and in one of my steps so far, the compactly supported cohomology groups being finite dimensional would really simplify things. However, I don't know if $\dim H_c^n(M) < \infty$ on a noncompact manifold $M$. Is this true or false, and if so, can you provide references and/or a proof?

Answer 1

No, compactly-supported de Rham cohomology is not generally finite-dimensional, even if we assume (say) paracompactness: A manifold can have infinitely many path components (such as the integers, or the integers cross an arbitrary compact manifold); a connected surface can have infinitely many ends and/or handles; the complement in a compact smooth manifold of an infinite discrete set or an infinite disjoint union of compact submanifolds of codimension at least two is connected and has infinite-dimensional cohomology, etc., etc.