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)?



A form of Lefschetz duality
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.)


