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Could someone point me to a standard reference for the fact that the top cohomology $H^n(M,A)$ of an $n$-dimensional manifold $M$ is non-trivial for local coefficients $A$ if and only if the manifold is compact?

EDIT: It seems that there are some issues when $M$ is non-orientable. I would like to include the non-orientable case. I figure the result uses (twisted) Poincaré duality and some kind of pairing between the $n$th cohomology and compactly supported cohomology in degree $0$.

I am not sure of its validity, but I am looking for (a reference for) an isomorphism $H_c^0\cong H^n$ which holds for local systems.

Thread on MathOverflow.

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    You need the hypothesis of orientability as well. If no one does before me, I'll get a reference once I get to work.2012-07-05

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As I pointed out in the comments, you need the hypothesis of orientability. For example, $\mathbb{R}P^2$ is nonorientable and compact, but has $H_2(\mathbb{R}P^2;\mathbb{Z}) = 0$.

For a reference for homology, see theorem 3.26, page 236 of Hatcher's Algebraic topology book. For cohomololgy, try Corollary 3.39 on page 250 of the same book.

His book is freely available from his own website. See here.

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    To answer your question, I think that the top cohomology is non-trivial, when the manifold is compact and the coefficients are a locally constant sheaf, which are twisted by the orientation character. In the case that the manifold is oriented, the orientation character is trivial and then the coefficients are not twisted, which corresponds to the situation described in Hatcher.2012-07-07