{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nStable homotopy theory is the study of homotopy invariants that are unchanged by suspension, such as the following groups.\nLet $X$ and $Y$ be [[pointed|T9DG8DDE75]] [[CW complexes|5W95QFHXPR]], with $X$ finite, and let $\\Sigma$ be the reduced [[suspension|2VD7GR64JJ]].\nLet $[A, B]_*$ be the set of pointed homotopy classes of pointed maps $A \\to B$.\nThe group of *stable maps* from $X$ to $Y$ is the colimit\n$$\\set{X, Y} = \\operatorname{colim}_{k} \\, [\\Sigma^k X, \\Sigma^k Y]_*,$$\ntaken along the maps $[\\Sigma^k X, \\Sigma^k Y]_* \\to [\\Sigma^{k+1} X, \\Sigma^{k+1} Y]_*$ that send $f$ to $\\Sigma f$.\nFor $n \\ge 0$, the $n$-th *stable homotopy group* of $Y$ is\n$$\\pi_n^s(Y) = \\operatorname{colim}_k \\, \\pi_{n+k}(\\Sigma^k Y) = \\set{S^n, Y}.$$\n\n## Remarks\n\nFor $k \\ge 2$ the set $[\\Sigma^k X, \\Sigma^k Y]_*$ is an abelian group, so the colimit is an abelian group.\nLet $Y$ be $(c-1)$-connected and let $X$ have dimension $d$.\nThe [[Freudenthal suspension theorem|7DM2KSS2DB]] shows that the maps in the colimit are bijective once $d + k \u003c 2(c + k) - 1$, that is, once $k \u003e d - 2c + 1$.\nSo the colimit is reached at a finite stage.\n\nExample: $\\pi_0^s(S^0) = \\Z$, since $\\pi_k(S^k) = \\Z$ for all $k \\ge 1$ and suspension preserves degree.\nAlso $\\pi_1^s(S^0) = \\Z/2$, generated by the stable class of the Hopf map $\\eta$.\n\nNon-example: the group $\\pi_3(S^2) \\cong \\Z$, generated by the Hopf map, is not a stable group.\nIts suspension lands in $\\pi_4(S^3) \\cong \\Z/2$, and the stable value $\\pi_1^s(S^0)$ is $\\Z/2$.\n\nToday stable homotopy theory is usually set in the stable homotopy category of spectra, where $\\set{X, Y}$ becomes a group of morphisms between suspension spectra.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "545EV0W4CE",
    "kind": "definition",
    "msc": [
      "55P42",
      "55Q10"
    ],
    "name": "Stable homotopy theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stable_homotopy_theory"
      }
    ],
    "tag": "5KERJ85BXJ",
    "uses": [
      "2VD7GR64JJ",
      "T9DG8DDE75",
      "5W95QFHXPR",
      "P976WYEFMB",
      "7DM2KSS2DB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/5KERJ85BXJ/",
  "links": [
    {
      "field": "defines",
      "target": "545EV0W4CE"
    },
    {
      "field": "uses",
      "target": "2VD7GR64JJ"
    },
    {
      "field": "uses",
      "target": "T9DG8DDE75"
    },
    {
      "field": "uses",
      "target": "5W95QFHXPR"
    },
    {
      "field": "uses",
      "target": "P976WYEFMB"
    },
    {
      "field": "uses",
      "target": "7DM2KSS2DB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/5KERJ85BXJ.md",
  "msc": [
    "55P42",
    "55Q10"
  ],
  "name": "Stable homotopy theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5KE/5KERJ85BXJ.md",
  "status": null,
  "tag": "5KERJ85BXJ",
  "uri": "m2://5KERJ85BXJ",
  "url": "https://math2.tamnd.com/wiki/stable-homotopy-theory-definition/",
  "writing": {
    "tag": "5KERJ85BXJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}