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  "body": "\n## Definition\n\nLet $X$ be a topological space and let $R$ be a [[commutative ring|0310REJJY8]] with $1$.\nFor a singular simplex $\\sigma \\colon \\Delta^n \\to X$ with vertices $v_0, \\ldots, v_n$ and a cochain $\\varphi \\in C^k(X; R)$ with $k \\le n$, the *cap product* is the chain\n$$\\sigma \\frown \\varphi = \\varphi\\big(\\sigma|_{[v_0, \\ldots, v_k]}\\big) \\, \\sigma|_{[v_k, \\ldots, v_n]} \\in C_{n-k}(X; R).$$\nIt extends linearly to a bilinear map $C_n(X; R) \\times C^k(X; R) \\to C_{n-k}(X; R)$.\nIt satisfies $\\partial(\\sigma \\frown \\varphi) = (-1)^k \\big(\\partial\\sigma \\frown \\varphi - \\sigma \\frown \\delta\\varphi\\big)$.\nHence it induces a bilinear map $\\frown \\colon H_n(X; R) \\times H^k(X; R) \\to H_{n-k}(X; R)$ between [[singular homology|M0Q5CFJ7ZH]] and [[singular cohomology|6N7BC70CX0]], also called the cap product.\n\n## Remarks\n\nCap and [[cup product|JX47MS1R1H]] are related by $\\psi(\\sigma \\frown \\varphi) = (\\varphi \\smile \\psi)(\\sigma)$ for $\\varphi \\in C^k$ and $\\psi \\in C^{n-k}$.\nOn homology, $(c \\frown \\varphi) \\frown \\psi = c \\frown (\\varphi \\smile \\psi)$, so $H_\\bullet(X; R)$ is a module over $H^\\bullet(X; R)$.\nIt is natural in the form $f_*(c \\frown f^*\\varphi) = f_*(c) \\frown \\varphi$ for a continuous map $f$.\n\nExample: for $k = n$, $\\sigma \\frown \\varphi = \\varphi(\\sigma)\\, \\sigma(v_n)$ is a $0$-chain.\nIts image under the augmentation $C_0(X; R) \\to R$ is $\\varphi(\\sigma)$, so the cap product in top degree followed by augmentation is the Kronecker pairing of cohomology with homology.\n\nNon-example: for $k \u003e n$ the formula does not apply, since an $n$-simplex has no face of dimension $k$; the product $H_n \\times H^k \\to H_{n-k}$ is only defined for $k \\le n$.\n",
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