Prove Poincare duality theorem with Morse theory.

Question

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 co-boundary maps? But I don't see where is the condition orientable needed.