I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 \cong H^1$ and $H_2 \cong H^0$ where $H_i$ ($H^i$) is the $i$th (co)homology group. I understand that this question is unlikely to give rise to an "if and only if" answer, but am hoping that in the special case of two dimensions there may be a collection of observations an expert could make to get something stronger than trivial "if" statements.
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$\begingroup$ That seems like a weak condition. Wouldn't you rather these isomorphisms be induced by a map from the chain complex to the dual chain complex? $\endgroup$– Qiaochu YuanCommented Jan 20, 2016 at 17:46
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$\begingroup$ Yes, but I don't want to specify a map. $\endgroup$– FreddyCommented Jan 20, 2016 at 18:13
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$\begingroup$ Well it feels rather weird to say that something like $0 \to \mathbb{Z} \xrightarrow{=} \mathbb{Z} \to 0 \to 0$ satisfies Poincaré duality... $\endgroup$– Najib IdrissiCommented Jan 20, 2016 at 19:55
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$\begingroup$ Well, the tautological iff looks sufficient to me, and it is induced by a chain map since, over a PID, any chain complex is quasi-isomorphic to its homology. $\endgroup$– Fernando MuroCommented Jan 20, 2016 at 22:28
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