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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $X$ be a pointed [[CW complex|5W95QFHXPR]] that is [[$(n-1)$-connected|1A943BWBQT]].\nLet $\\Sigma X$ be the reduced [[suspension|Q7RFNJH76Q]] of $X$.\nThe suspension homomorphism $E \\colon \\pi_i(X) \\to \\pi_{i+1}(\\Sigma X)$, $[f] \\mapsto [\\Sigma f]$, on [[homotopy groups|P976WYEFMB]] has the following properties:\n\n1. it is an isomorphism for $i \u003c 2n - 1$;\n2. it is surjective for $i = 2n - 1$.\n\n## Remarks\n\nThe usual proof applies the [[Blakers–Massey theorem|QF764XKBA3]] to the two cones that form $\\Sigma X$, whose intersection is a copy of $X$.\nFor spheres, $S^n$ is $(n-1)$-connected and $\\Sigma S^n = S^{n+1}$, so $\\pi_i(S^n) \\to \\pi_{i+1}(S^{n+1})$ is an isomorphism for $i \u003c 2n - 1$.\nHence $\\pi_{n+k}(S^n)$ does not depend on $n$ once $n \\ge k + 2$, which defines the [[stable homotopy groups of spheres|HGZAYPQBXW]].\nThe bound is sharp: for $n = 2$ and $i = 3$, the map $\\pi_3(S^2) \\cong \\Z \\to \\pi_4(S^3) \\cong \\Z/2$ is onto but not injective.\nThe theorem also gives $\\pi_n(S^n) \\cong \\Z$ for all $n \\ge 1$ by induction from $\\pi_1(S^1) \\cong \\Z$, using $\\pi_2(S^2) \\cong \\Z$ from the Hopf fibration at the first step.\n\n## Explanations\n\n### L1 human\n\nSuspending a space places it as the equator of a larger space made of two cones.\nEvery map of spheres into the space suspends to a map one dimension higher.\nThe theorem says that, in a range of dimensions, this loses no information and creates nothing new.\n\n### L2 human\n\nThe theorem is the starting point of stable homotopy theory.\nIt shows that the suspension sequence $\\pi_{n+k}(S^n) \\to \\pi_{n+k+1}(S^{n+1}) \\to \\cdots$ eventually becomes constant, so stable groups and spectra make sense.\nIn the stable range, homotopy groups behave like a homology theory, since excision holds there.\n",
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