Gysin sequence Poincare dual to a long exact sequence of homology of a pair.

Question

I am trying to understand the Gysin morphism and its associated long exact sequence. I'll look at a simple example of $\mathbb{P}^n$ along with a smooth hypersurface $i : S \hookrightarrow \mathbb{P}^n$. Then we can define a map $G : H^j(S) \rightarrow H^{j+2}(\mathbb{P}^n)(-1)$ by taking the poincare dual of $i^* : H^i(\mathbb{P}^n) \rightarrow H^i(S)$. There is a long exact sequence: $$ \cdots \rightarrow H^n(\mathbb{P}^n) \rightarrow H^n(\mathbb{P}^n-S) \xrightarrow{res} H^{n-1}(S) \xrightarrow{G}H^{n+1}(\mathbb{P})\rightarrow \cdots $$ where $res$ is a residue map that can be defined topologically or on differential forms if we look at de Rham cohomology, but anyway I read that the above sequence should be Poincare dual to a long exact sequence of homology of the pair ($\mathbb{P}$,$\mathbb{P}^n-S$), but I can seem to show that this other sequence is Poincare dual to the first one. Any ideas?