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  "body": "\n## Summary\n\nA Massey product is a secondary operation on cohomology.\nIt takes three classes $u, v, w$ with $uv = 0$ and $vw = 0$ and produces a set of classes of degree one less than that of $u v w$.\nIt is defined from the [[cup product|JX47MS1R1H]] on cochains, not just on cohomology.\nMassey products detect structure that the [[cohomology ring|RDT4785M4D]] alone does not see, such as the linking of the Borromean rings.\n\n## Explanations\n\n### L1 human\n\nIf $uv = 0$ in cohomology, then on cochains the product $a b$ is the coboundary of some cochain $s$.\nThe same holds for $vw$ with a cochain $t$.\nThe two ways of filling in, $s$ against $c$ and $a$ against $t$, combine into a new cocycle.\nIts class is the Massey product $\\langle u, v, w \\rangle$, defined up to some choices.\nIn the Borromean rings, any two of the three rings are unlinked, so in the complement all cup products of the degree-one classes vanish.\nBut a triple Massey product is nonzero, and it shows that the three rings are linked together.\n",
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