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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 $v_0, \\ldots, v_n$, write $\\sigma|_{[v_i, \\ldots, v_j]}$ for its restriction to the face spanned by $v_i, \\ldots, v_j$.\nFor cochains $\\varphi \\in C^k(X; R)$ and $\\psi \\in C^l(X; R)$, the *cup product* $\\varphi \\smile \\psi \\in C^{k+l}(X; R)$ is\n$$(\\varphi \\smile \\psi)(\\sigma) = \\varphi\\big(\\sigma|_{[v_0, \\ldots, v_k]}\\big) \\cdot \\psi\\big(\\sigma|_{[v_k, \\ldots, v_{k+l}]}\\big)$$\nfor every singular simplex $\\sigma \\colon \\Delta^{k+l} \\to X$.\nIt satisfies $\\delta(\\varphi \\smile \\psi) = \\delta\\varphi \\smile \\psi + (-1)^k \\varphi \\smile \\delta\\psi$.\nHence it induces a bilinear map $\\smile \\colon H^k(X; R) \\times H^l(X; R) \\to H^{k+l}(X; R)$ on [[singular cohomology|6N7BC70CX0]], also called the cup product.\n\n## Remarks\n\nThe cup product on cohomology is associative, the class of the constant $0$-cochain $1$ is a unit, and $a \\smile b = (-1)^{kl}\\, b \\smile a$ for $a \\in H^k$ and $b \\in H^l$.\nGraded commutativity holds on cohomology but not on cochains.\nIt is natural: $f^*(a \\smile b) = f^*(a) \\smile f^*(b)$ for a continuous map $f$.\nThe same formula gives relative products such as $H^k(X; R) \\times H^l(X, A; R) \\to H^{k+l}(X, A; R)$.\n\nExample: for the torus $T^2 = S^1 \\times S^1$, $H^1(T^2; \\Z) \\cong \\Z^2$ with generators $\\alpha$ and $\\beta$ dual to the two circles.\nThen $\\alpha \\smile \\beta$ generates $H^2(T^2; \\Z) \\cong \\Z$, and $\\alpha \\smile \\alpha = 0$.\n\nNon-example: the pointwise product $(\\varphi\\psi)(\\sigma) = \\varphi(\\sigma)\\psi(\\sigma)$ only makes sense when $\\varphi$, $\\psi$ and $\\sigma$ all have the same dimension.\nSo it does not define a product $C^k \\times C^l \\to C^{k+l}$.\nThe cup product avoids this by splitting each simplex into a front face and a back face.\n",
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