I came across the Poincare-Alexander-Lefschetz duality here: If
$M$ is a closed (compact and no boundary) manifold, $B\subset A\subset M$ are closed subsets, then
$$H^k(M-B，M-A)\cong H_{n-k}(A,B)$$
I looked up the reference he mentioned there, this theorem is stated in the Theorem 8.3, p351. But I don't find this statement in Hatcher's book, maybe we can deduce this theorem from Poincare duality, Lefschetz duality in the book. But when we do the exercison, we will always encounter open sets and closed sets.
(1) Why do $A,B$ need to be closed sets ? If $B,A$ are closed sets, then $M-A,M-B$ are open sets on the LHS. So we put closed sets in homology and open sets in cohomology?
(2) Can I do following operation: If $B\subset A\subset M$ are open subsets, we have $M-A\subset M-B\subset M$ are closed subsets, then
$$H^k(A，B)\cong H_{n-k}(M-B，M-A).$$
(3) In fact, I am trying to see how to deduce following isomorphism from PAL duality:
$$H_1(M-\Sigma)\cong H^{n-1}(M,\Sigma)$$
where $\Sigma$ is closed two surface in the closed $n$-manifold $M$. I tried but it seems not work：$\Sigma\subset
M\subset M$, then we get $H^k(M-\Sigma)\cong H_{n-k}(M, \Sigma)$. Thanks for the help.
Best Answer
Let me address the isomorphism which you are after: $$H_{n-k}(M-\Sigma)\cong H^{k}(M,\Sigma),$$ where $n$ is the dimension of $M$.
Let $A$ be an open tubular neighborhood of $\Sigma$ in $M$ (a disk bundle over $\Sigma$). Then, by excision, and since $A$ deformation-retracts to $\Sigma$, we obtain: $$ H^k(M-A,\partial (M-A))\cong H^k(M,A)\cong H^k(M,\Sigma). $$ Using the APL duality theorem, we obtain: $$ H^k(M,\Sigma)\cong H^k(M-A,\partial (M-A)) \cong H_{n-k}(M-A)\cong H_{n-k}(M-\Sigma). $$ Now, taking $k=n-1$: $$ H^{n-1}(M,\Sigma)\cong H_1(M-\Sigma). $$