The question is to prove Poincare Duality of the form $(H^p_c (M))^* = H^{n-p} (M)$ using direct limits and inverse limits, where $H^p (M)$ denotes the $p$-th de Rham cohomology group of $M$. The outline of the proof has been given, which is to be followed. In particular, no homology theory of any sort can be used. (In fact, I don't know any homology theory; this is why I cannot find any reference that is helpful at all, since most texts (e.g. Hatcher, Greenberg & Harper) build on homology theories. The only text that does not mention homology at all is Madsen & Tornehave, but it bypasses my problem with some trick that is not compatible with the outline.)
I have figured out most of the question, except the following part:
Let $M$ be a general manifold, with $M = \bigcup_i U_i$, where $U_i$ is open and $U_i \subset \overline{U_i} \subset U_{i+1}$. If Poincare Duality holds for each $U_i$, show that it holds for $M$ using direct limit and inverse limit.
For this my approach is to show that (1) $H^p_c (M) = \lim_{\rightarrow} H^p_c (U_i)$, and that (2) $H^p (M) = \lim_{\leftarrow} H^p (U_i)$, then since the dual of a direct limit is the inverse limit of the duals, the result would follow. (1) is easy to show, but I found (2) to be surprisingly hard.
The problem is, I can see that there is a natural map from $H^p (M)$ to $\lim_{\leftarrow} H^p (U_i)$ induced by inclusion of $U_i$ into $M$, namely the map that restricts $p$-forms to each $U_i$. I believe it is also surjective. However, I can't prove injectivity: It is apparent that if $[\omega] \in H^p (M)$ restricts to $[0]$ for each $U_i$, then it doesn't "see any holes anywhere" on any $U_i$, and hence $[\omega]$ must be $[0]$ itself, but I can't formalize this argument.
I looked up some references, and I have to admit I understood very little. I remember one even saying that this is not true in general, and there is some "derived functor" $\lim^1$ (?) involved.
Could someone please give me a pointer on this? It doesn't have to be the whole solution; I would appreciate any hint. Thank you!