I don't understand words after "the latter because". What is $\partial_*$? And why $H(S^n - P, Q) = 0$ implies that $\partial_*$ is isomorphism? It is Glen Bredon "Topology and geometry" VIII.8.15. Hope for your help.
Just remind statement of Poincaré-Alexander-Lefschetz duality
Let M closed manifold and $B \subset A \subset M$ closed sets. Then there is isomorphism $H^k(X-B,X-A) = H_{n-k}(A,B)$.
