{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[CW complex|5W95QFHXPR]] that is the union of two subcomplexes $A$ and $B$ whose intersection $C = A \\cap B$ is nonempty and connected.\nLet $m, n \\ge 0$, and assume that the pair $(A, C)$ is $m$-connected and the pair $(B, C)$ is $n$-connected.\nThen, for a base point in $C$, the map $\\pi_i(A, C) \\to \\pi_i(X, B)$ of [[relative homotopy groups|ZHHMDZWA7F]] induced by inclusion has the following properties:\n\n1. it is an isomorphism for $i \u003c m + n$;\n2. it is surjective for $i = m + n$.\n\n## Remarks\n\nHere a pair $(Y, Z)$ is $k$-connected if every path component of $Y$ meets $Z$ and $\\pi_i(Y, Z, z) = 0$ for $1 \\le i \\le k$ and all $z \\in Z$, as for [[n-connected spaces|1A943BWBQT]].\nThe theorem is also called the homotopy excision theorem.\nFor homology, excision gives $H_i(A, C) \\cong H_i(X, B)$ in all degrees; for homotopy groups excision holds only in the range above.\nThe proof reduces to the case where $A$ and $B$ are obtained from $C$ by attaching cells.\nIt then works cell by cell with a general position argument in the cubes $I^i$.\nApplied to the two cones $A$ and $B$ that form the suspension of an $(n-1)$-connected complex, it gives the [[Freudenthal suspension theorem|7DM2KSS2DB]].\n\n## Explanations\n\n### L1 human\n\nIn homology, gluing two pieces together behaves predictably: what happens near the seam can be cut out.\nFor homotopy groups this fails in general, and new homotopy classes can appear after gluing.\nThe theorem says that the failure only starts in high dimensions.\nThe more connected the pieces are, the higher it starts.\n\n### L2 human\n\nThe theorem is the reason homotopy theory becomes homology-like in the stable range.\nIt implies Freudenthal suspension, and it compares a cofibre sequence with a fibre sequence in a range of degrees.\nA version for cubical diagrams of spaces, due to Goodwillie, is a basic tool in the calculus of functors.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "ZHHMDZWA7F"
    ],
    "kind": "statement",
    "msc": [
      "55Q05",
      "55P40"
    ],
    "name": "Blakers–Massey theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4924632",
      "wikipedia": "Blakers–Massey theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Blakers%E2%80%93Massey_theorem"
      }
    ],
    "tag": "QF764XKBA3",
    "uses": [
      "5W95QFHXPR",
      "ZHHMDZWA7F",
      "1A943BWBQT"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/QF764XKBA3/",
  "links": [
    {
      "field": "about",
      "target": "ZHHMDZWA7F"
    },
    {
      "field": "uses",
      "target": "5W95QFHXPR"
    },
    {
      "field": "uses",
      "target": "ZHHMDZWA7F"
    },
    {
      "field": "uses",
      "target": "1A943BWBQT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/QF764XKBA3.md",
  "msc": [
    "55Q05",
    "55P40"
  ],
  "name": "Blakers–Massey theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QF7/QF764XKBA3.md",
  "status": null,
  "tag": "QF764XKBA3",
  "uri": "m2://QF764XKBA3",
  "url": "https://math2.tamnd.com/wiki/blakers-massey-theorem/",
  "writing": {
    "tag": "QF764XKBA3",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}