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  "body": "\n## Summary\n\nThe Toda bracket is a secondary composition operation on homotopy classes of maps.\nIt takes three maps $f$, $g$, $h$ such that $g \\circ f$ and $h \\circ g$ are null-homotopic.\nIt produces a set of maps out of the suspension of the source of $f$.\nToda introduced it to compute the [[homotopy groups of spheres|YD6VAXXSBM]].\nIt remains a basic tool for finding relations in the [[stable homotopy groups of spheres|HGZAYPQBXW]].\n\n## Explanations\n\n### L1 human\n\nIf $g \\circ f$ is homotopic to a constant map, there is a homotopy that witnesses this.\nIf also $h \\circ g$ is null, there is a second witnessing homotopy.\nComparing the two witnesses for the triple composite $h \\circ g \\circ f$ gives a loop of maps, and this loop is a map out of a suspension.\nThe Toda bracket is the set of all maps that arise from such choices.\nFor example, in the stable homotopy groups of spheres the bracket $\\langle 2, \\eta, 2 \\rangle$ is the single element $\\eta^2$, where $\\eta$ is the stable Hopf map.\n",
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