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  "body": "\n## Definition\n\nLet $W$, $X$, $Y$, $Z$ be [[pointed spaces|T9DG8DDE75]], and let $f \\colon W \\to X$, $g \\colon X \\to Y$, $h \\colon Y \\to Z$ be pointed maps such that $g \\circ f$ and $h \\circ g$ are [[null-homotopic|SMEV5BSQ5S]].\nLet $CW = (W \\times [0, 1]) / (W \\times \\set{1} \\cup \\set{w_0} \\times [0, 1])$ be the reduced cone on $W$, with $W$ included as $W \\times \\set{0}$.\n\n1. Choose a pointed null-homotopy of $g \\circ f$, that is, a map $A \\colon CW \\to Y$ that restricts to $g \\circ f$ on $W$.\n2. Choose a pointed null-homotopy of $h \\circ g$, that is, a map $B \\colon CX \\to Z$ that restricts to $h \\circ g$ on $X$.\n3. The maps $h \\circ A \\colon CW \\to Z$ and $B \\circ Cf \\colon CW \\to Z$ both restrict to $h \\circ g \\circ f$ on $W$.\n   Gluing them along $W$ gives a map from two cones on $W$ glued along their bases, which is the reduced [[suspension|2VD7GR64JJ]] $\\Sigma W$, to $Z$.\n\nThe *Toda bracket* $\\langle h, g, f \\rangle$ is the set of [[homotopy classes|F2F143KQ2M]] of all maps $\\Sigma W \\to Z$ obtained in this way, over all choices of $A$ and $B$.\n\n## Remarks\n\nChanging the choices of $A$ and $B$ changes the result by an element of $h_* [\\Sigma W, Y] + (\\Sigma f)^* [\\Sigma X, Z]$.\nThis subgroup of $[\\Sigma W, Z]$ is the *indeterminacy*, and $\\langle h, g, f \\rangle$ is a coset of it.\nThe bracket depends only on the homotopy classes of $f$, $g$, $h$.\n\nExample: in the stable homotopy groups of spheres, let $2$ denote twice the identity and $\\eta$ the stable Hopf map in degree $1$.\nThen $2 \\eta = 0$, so $\\langle 2, \\eta, 2 \\rangle$ is defined, and it equals $\\set{\\eta^2}$ in degree $2$.\n\nNon-example: for the identity $\\iota$ of $S^n$, the composite $\\iota \\circ \\iota$ is not null-homotopic, so $\\langle \\iota, \\iota, \\iota \\rangle$ is not defined.\n\nSome authors use the opposite order of the gluing, which changes the bracket by a sign.\n",
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