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  "body": "\n## Statement\n\nLet $M$ be a compact [[manifold with boundary|7DS6B2BFK8]] of dimension $n$ with boundary $\\partial M$, and let $R$ be a commutative ring with $1$.\nAssume that the interior of $M$ is $R$-[[oriented|913WEQERBE]], and let $[M] \\in H_n(M, \\partial M; R)$ be the corresponding relative [[fundamental class|ZEA0PEM3B2]].\nLet $H^k$ be [[singular cohomology|6N7BC70CX0]], and use [[relative homology|4BVWH0BN28]] and relative cohomology for the pair $(M, \\partial M)$.\nThen for every $k$, the [[cap product|N7SXBQ7WK0]] with $[M]$ gives isomorphisms\n$$H^k(M; R) \\to H_{n-k}(M, \\partial M; R) \\qquad\\text{and}\\qquad H^k(M, \\partial M; R) \\to H_{n-k}(M; R).$$\n\n## Remarks\n\nThe relative fundamental class is the class whose restriction to $H_n(M, M \\setminus \\set{x})$ is the local orientation at each interior point $x$.\nThe proof uses a collar of the boundary to compare $M$ with its interior.\nIt then applies the duality theorem behind [[Poincaré duality|9VY90HJVZX]] to the interior, a noncompact manifold without boundary.\nWhen $\\partial M = \\emptyset$, both isomorphisms reduce to Poincaré duality.\n\nExample: for the disc $M = D^n$ with $\\partial M = S^{n-1}$, $H^0(D^n) \\cong \\Z$ and $H_n(D^n, S^{n-1}) \\cong \\Z$.\nAlso $H^n(D^n, S^{n-1}) \\cong \\Z$ matches $H_0(D^n) \\cong \\Z$, and all other groups vanish on both sides.\n\n## Explanations\n\n### L1 human\n\nPoincaré duality pairs holes of complementary dimensions on a closed manifold.\nFor a manifold with an edge, the same pairing works if one side of the pairing is measured relative to the edge.\nFor a disc, the single point-like piece matches the whole disc counted modulo its boundary circle.\n\n### L2 human\n\nLefschetz duality is the form of Poincaré duality used for compact manifolds with boundary, such as knot exteriors and cobordisms.\nTake a compact oriented $(2m+1)$-manifold $M$ and a field $F$.\nLefschetz duality and the long exact sequence of $(M, \\partial M)$ show that the kernel of $H_m(\\partial M; F) \\to H_m(M; F)$ has half the dimension of $H_m(\\partial M; F)$.\nIt is the basic tool in the study of manifolds with boundary and of bordism.\n",
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