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  "body": "\n## Summary\n\nThe cup product is a multiplication $H^k(X; R) \\times H^l(X; R) \\to H^{k+l}(X; R)$ on [[singular cohomology|6N7BC70CX0]] with coefficients in a commutative ring $R$.\nIt is associative, has a unit, and is graded commutative.\nIt makes cohomology into a ring, the [[cohomology ring|RDT4785M4D]], which carries much more information than the cohomology groups alone.\n\n## Explanations\n\n### L1 human\n\nCohomology classes can be multiplied, and the degrees add.\nTake a $k$-cochain and an $l$-cochain, and a simplex of dimension $k + l$.\nTest the first on the front face of dimension $k$, test the second on the back face of dimension $l$, and multiply the results.\nOn the torus, the two classes that measure winding around each circle multiply to a class that measures the whole surface.\nOn the wedge of two circles and a sphere the same groups appear, but every product of two degree $1$ classes is zero.\nSo the cup product tells the two spaces apart.\n",
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