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  "body": "\n## Definition\n\nLet $X$ be a space and let $R$ be a [[commutative ring|0310REJJY8]].\nLet $C^*(X; R)$ be the singular cochain complex, with coboundary $\\delta$ and [[cup product|JX47MS1R1H]] $\\smile$, so that $\\delta(x \\smile y) = \\delta x \\smile y + (-1)^{k} x \\smile \\delta y$ for $x$ of degree $k$.\nLet $u \\in H^p(X; R)$, $v \\in H^q(X; R)$ and $w \\in H^r(X; R)$ be classes in [[singular cohomology|6N7BC70CX0]] with $u \\smile v = 0$ and $v \\smile w = 0$.\n\n1. Choose cocycles $a, b, c$ representing $u, v, w$.\n2. Choose cochains $s$ of degree $p + q - 1$ and $t$ of degree $q + r - 1$ with $\\delta s = a \\smile b$ and $\\delta t = b \\smile c$.\n3. Then $m = s \\smile c + (-1)^{p+1} a \\smile t$ is a cocycle of degree $p + q + r - 1$.\n\nThe *Massey product* $\\langle u, v, w \\rangle$ is the set of all classes $[m] \\in H^{p+q+r-1}(X; R)$ obtained from all such choices.\n\n## Remarks\n\nThe cocycle condition is a direct check: $\\delta(s \\smile c) = a \\smile b \\smile c$ and $\\delta(a \\smile t) = (-1)^p a \\smile b \\smile c$, so $\\delta m = 0$.\nChanging $s$ or $t$ by a cocycle changes $[m]$ by an element of $H^{p+q-1}(X; R) \\smile w$ or of $u \\smile H^{q+r-1}(X; R)$.\nSo $\\langle u, v, w \\rangle$ is a coset of the subgroup $u \\smile H^{q+r-1}(X; R) + H^{p+q-1}(X; R) \\smile w$, called the indeterminacy.\nThe Massey product is said to vanish if it contains $0$.\nSign conventions differ between books; they change the set only by a sign.\n\nExample: if $u = 0$, take $a = 0$ and $s = 0$; then $m = 0$, so $\\langle 0, v, w \\rangle$ contains $0$ and vanishes.\n\nNon-example: on the torus $T^2$ with degree-one generators $x, y$, the triple $\\langle x, y, x \\rangle$ is not defined, because $x \\smile y \\neq 0$, so clause 2 has no solution $s$.\n\nHigher Massey products $\\langle u_1, \\ldots, u_n \\rangle$ are defined in the same way when the lower ones vanish compatibly.\nWith rational or real coefficients, all Massey products vanish on formal spaces, such as spheres and compact Kähler manifolds.\n",
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