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First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical one-one correspondence between $C_k(M)$ and $C^n-k(M)$ where I'm considering the cellular chain and cochain groups. My question is can I deduce the Poincare duality theorem by analyzing carefully the behavior of boundary and coboundary maps? But I don't see where is the conidtion orientable needed.

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So I think the issue might be that to get the dual homology data you need to orient the belt disks (which become the cores of the dual presentation) in a way determined by the original homology data. The original data include orientations for the cores of your handles. To orient the belt disks you need an orientation for the whole manifold. Then you can say "orient the belt so that the core-orientation followed by the belt-orientation is the chosen orientation for the ambient manifold." I may have a better answer in a few days but I think that's the right idea. –  Tim kinsella Apr 9 '13 at 8:36
I think it's really cool. Looking forward for your answer. –  lee Apr 9 '13 at 15:13
You can see the details in Schwarz' book. –  Thomas Rot May 4 at 2:03

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