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  "body": "\n## Statement\n\nLet $M$ be a closed [[topological manifold|QQQ9B1DK4P]] of dimension $n$, that is, compact and without boundary.\nLet $R$ be a commutative ring with $1$, and assume that $M$ is $R$-[[oriented|913WEQERBE]], with [[fundamental class|ZEA0PEM3B2]] $[M] \\in H_n(M; R)$.\nThen for every $k$ the map\n$$D \\colon H^k(M; R) \\to H_{n-k}(M; R), \\qquad D(a) = [M] \\frown a,$$\ngiven by the [[cap product|N7SXBQ7WK0]], is an isomorphism from [[singular cohomology|6N7BC70CX0]] to [[singular homology|M0Q5CFJ7ZH]].\n\n## Remarks\n\nThe proof first shows a version with compactly supported cohomology for every $R$-oriented manifold, not only closed ones.\nIt holds for open subsets of $\\R^n$ that are convex, and it passes to unions by a Mayer–Vietoris argument and to increasing unions by a limit argument.\nFor a closed manifold, compactly supported cohomology is ordinary cohomology.\n\nEvery closed manifold is $\\Z/2$-oriented, so the theorem always holds with $\\Z/2$ coefficients.\nWith a field $F$ of coefficients, it implies that the pairing $H^k(M; F) \\times H^{n-k}(M; F) \\to F$, $(a, b) \\mapsto \\langle a \\smile b, [M] \\rangle$, is nondegenerate.\nIn particular, the [[Betti numbers|ST71ZV3FES]] satisfy $b_k(M) = b_{n-k}(M)$ for closed orientable $M$, and the [[Euler characteristic|RNDDWDAFJ5]] of a closed manifold of odd dimension is $0$.\n\nThe hypotheses are needed.\nWithout orientability: for the [[real projective plane|1DDYB6VVSR]], $H^0(\\R P^2; \\Z) = \\Z$ but $H_2(\\R P^2; \\Z) = 0$.\nWithout compactness: $H^0(\\R^n; \\Z) = \\Z$ but $H_n(\\R^n; \\Z) = 0$ for $n \\ge 1$.\n\n## Explanations\n\n### L1 human\n\nOn a closed oriented manifold of dimension $n$, the holes of dimension $k$ and the holes of dimension $n - k$ come in matching pairs.\nFor a closed orientable surface, the number of independent loops is even, because each loop has a partner loop that it crosses once.\nThe duality comes from cutting the manifold into cells and also into the dual cells, where each $k$-cell matches an $(n - k)$-cell crossing it.\n\n### L2 human\n\nThe theorem identifies cohomology and homology of closed oriented manifolds with a shift of degree $n$.\nIt turns the cup product into intersection of cycles, which gives the intersection form of a $4$-manifold and its unimodularity.\nIt yields the symmetry of Betti numbers and the vanishing of the Euler characteristic in odd dimensions.\nLefschetz duality extends it to compact manifolds with boundary, and Alexander duality is a consequence for complements of subsets of spheres.\n",
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