{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$ and $k \\ge 1$, and let $\\pi_k(S^n)$ be the [[homotopy group|P976WYEFMB]] of the $n$-sphere.\n\n1. If $k \\ne n$, and it is not the case that $n$ is even and $k = 2n - 1$, then $\\pi_k(S^n)$ is a finite group.\n2. If $n$ is even, then $\\pi_{2n-1}(S^n) \\cong \\Z \\oplus F$ for a finite abelian group $F$.\n\n## Remarks\n\nSince $\\pi_n(S^n) \\cong \\Z$, the theorem says exactly which [[homotopy groups of spheres|YD6VAXXSBM]] are infinite: $\\pi_n(S^n)$, and $\\pi_{2n-1}(S^n)$ for even $n$.\nThe proof uses the Serre spectral sequence and Serre's theory of classes of abelian groups.\nFor odd $n$, it shows that a map $S^n \\to K(\\Z, n)$ representing a generator induces isomorphisms on $\\pi_k \\otimes \\Q$.\nFor even $n$, the Whitehead product $[\\iota_n, \\iota_n] \\in \\pi_{2n-1}(S^n)$ has Hopf invariant $\\pm 2$, so it has infinite order.\nExample: $\\pi_3(S^2) \\cong \\Z$ is the case $n = 2$, $k = 3$, with $F = 0$, and $\\pi_7(S^4) \\cong \\Z \\oplus \\Z/12$.\nIn particular, every stable homotopy group $\\pi_k^s$ with $k \\ge 1$ is finite.\n\n## Explanations\n\n### L1 human\n\nMost homotopy groups of spheres are finite: a map from one sphere to another, repeated enough times, can be undone.\nThere are only two kinds of exceptions, where an infinite family of different maps exists.\nOne is the wrapping of a sphere around itself, counted by the degree.\nThe other occurs for spheres of even dimension $n$, in dimension $2n - 1$, like the Hopf map $S^3 \\to S^2$.\n\n### L2 human\n\nThe theorem says that the rational homotopy groups of spheres are very simple, even though the integral groups are not.\nIt is a starting point of rational homotopy theory.\nIt reduces the study of homotopy groups of spheres to their $p$-primary parts, one prime at a time.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "YD6VAXXSBM"
    ],
    "kind": "statement",
    "msc": [
      "55Q40",
      "55Q45"
    ],
    "name": "Serre's finiteness theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres"
      }
    ],
    "tag": "Y40ZRJJNH0",
    "uses": [
      "YD6VAXXSBM",
      "P976WYEFMB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/Y40ZRJJNH0/",
  "links": [
    {
      "field": "about",
      "target": "YD6VAXXSBM"
    },
    {
      "field": "uses",
      "target": "YD6VAXXSBM"
    },
    {
      "field": "uses",
      "target": "P976WYEFMB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y40ZRJJNH0.md",
  "msc": [
    "55Q40",
    "55Q45"
  ],
  "name": "Serre's finiteness theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y40/Y40ZRJJNH0.md",
  "status": null,
  "tag": "Y40ZRJJNH0",
  "uri": "m2://Y40ZRJJNH0",
  "url": "https://math2.tamnd.com/wiki/serres-finiteness-theorem/",
  "writing": {
    "tag": "Y40ZRJJNH0",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}