Let $M$ be a closed $n$-manifold and $X\subseteq M$ some subspace (closed, if you want but in general not a manifold itself). Let $H^X_\ast\subseteq H_\ast(M)$ be the image of the map $H_\ast(X)\rightarrow H_\ast(M)$. Applying Poincaré-duality we obtain an isomorphism $$PD^{-1}_M:(H_\ast(M),H^X_\ast)\rightarrow (H^\ast(M),H{_X^\ast}),$$
where $H^\ast_X\subseteq H^\ast(M)$ is just the image of $H^X_\ast$ under $PD^{-1}_M$.
Now my question is: Is there some nice way to understand $H^\ast_X$? Maybe if $(M,X)$ is a pair of complexes? Explicitely following the maps seems not so lead to so much for me.