[Math] De Rham cohomology for non-compact manifolds


Let $M$ be a non-compact differential manifold. It is true that in general $H^q_c(M) \neq H^q(M)$, where $H^q_c$ is the de Rham's cohomolgy with compact support group and $H^q$ is the usual de Rham's cohomology group.

We have just begun the subject, so I don't have much confidence with it. I wanted to ask: is it true that for any non-compact $M$ there exists a $q$ for which $H^q_c (M) \neq H^q(M)$? Or are there any examples of non-compact manifolds for which the two cohomologies are the same for all $q$?

EDIT: ok, from the comments I gathered that if such an example exists it must be non-orientable (a reference to a proof would be nice, even though I think the resul is quite non-elementary). The question still remains open though (that's my main reason to editing: I think this question didn't get enough attention).

Best Answer

Let $M$ be a connected non-compact manifold. Then $H_c^0(M)\cong 0$, and $H^0(M)\cong\mathbb{R}$.

Let $f:M\rightarrow \mathbb{R}$ be a function with $df=0$. Then $f$ is constant (this uses the connectedness of $M$). If $f$ is assumed to be compactly supported, this constant must be zero. If $f$ is not assumed to be compactly supported, all constants occur.

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