A form of Lefschetz duality

Question

Let W be a manifold with boundary such that \partial W is a union of two compact manifold A,B attached along their boundary. Does poincare duality hold for (W,A) and (W,B)?

Answer 1

Yes. Let's assume $W$ is oriented. It has a fundamental class $[W]\in H_n(W)$, and by Lefschetz duality the cap product with $[W]$ produces isomorphisms $H^p(W,A\cup B)\to H_{n-p}(W)$ and $H^p(W)\to H_{n-p}(W,A\cup B)$. If you think about it, it also produces a map $H^p(W,A)\to H_{n-p}(W,B)$. The latter is an isomorphism by a five-lemma argument. The sequence $$ \dots \to H^p(W,A\cup B)\to H^p(W,A)\to H^p(A\cup B,A)\to \dots $$ gets mapped to the sequence $$ \dots \to H_{n-p}(W)\to H_{n-p}(W,B)\to H_{n-p-1}(B)\to . $$ (The excision isomorphism $H^p(A\cup B,A)\cong H_p(B,A\cap B)$ is involved.)