Let $M$ be a $2$-dimensional orientable compact manifold. (For example, a Riemann surface.) We have an intersection pairing $$H_1(X,\def\Z{\mathbb{Z}}\Z) \times H_1(X,\Z)\to \Z$$ which is unimodular, which I think means that the induced homo $H_1(X,\Z) \to \operatorname{Hom}(H_1(X,\Z), \Z) \cong H^1(X,\Z)$ is an iso. The latter iso is the one from Poincaré duality.
Now take finitely many points $C \subset M$. Poincaré-Lefschetz duality says that $H_1(X,C;\Z) \cong H^1(X \setminus C,\Z)$. I'm looking for analog of the above intersection pairing, which should induce the iso from the previous sentence. It should be something like $$ H_1(X,C;\Z) \times H_1(X \setminus C, \Z) \to \Z \quad , $$ but I don't know if this works out. I don't even know if it's true that $H^1(X \setminus C, \Z) \cong \operatorname{Hom}(H_1(X \setminus C, \Z),\Z)$, since we are not in the compact case anymore.
It would be great if somebody with more knowledge in algebraic topology could help me out!