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 $H^k(X-B,X-A) = H_{n-k}(A,B)$.

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    $\begingroup$ It's the degree-shifting boundary map in the relative long exact sequence of the triple $(S^n-P, S^n - K, Q)$. $\endgroup$
    – user98602
    Mar 26, 2017 at 22:54
  • $\begingroup$ Thank you! Just forget about that LES, only remembered for $A \subset X$ LES $\to H_k(A) \to H_k(X) \to H_k(X,A) \to$, but now its perfectly clear! $\endgroup$ Mar 26, 2017 at 23:08


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