Lefschetz duality

Let \(M\) be a compact \(R\dash\)orientable \(n\dash\)manifold whose boundary is decomposed as \(\del M = A \union B\) with \(A, B\) compact \((n-1)\dash\)manifolds meeting in \(\del A = \del B = A \intersect B\). Then cap product with a fundamental class \([M]\in H_n(M, \del M; R)\) gives isomorphisms \[ D_M: H^k(M, A; R) \mapsvia{\sim} H_{n-k}(M, B; R) \] for all \(k\). The cases \(A = \del M\), \(B = \emptyset\) and \(A = \emptyset\), \(B = \del M\) give the isomorphisms \[ H^k(M, \del M; R) \cong H_{n-k}(M; R), \qquad H^k(M; R)\cong H_{n-k}(M, \del M; R) \] [@Hat02].