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.