{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a group, let $F$ be a [[free group|0KTW26C80J]], and let $R$ be a [[normal subgroup|0CB0DX5BVV]] of $F$ with $G \\cong F/R$.\nWrite $[A, B]$ for the subgroup generated by the [[commutators|0PXW0BN7DW]] $aba^{-1}b^{-1}$ with $a \\in A$ and $b \\in B$.\nThen the second integral [[homology group|CFAG3VRXEA]] of $G$ is\n$$H_2(G, \\Z) \\cong \\frac{R \\cap [F, F]}{[F, R]}.$$\n\n## Remarks\n\nIdea of the proof: the short exact sequence $1 \\to R \\to F \\to G \\to 1$ gives the exact sequence of low-degree terms\n$$H_2(F, \\Z) \\to H_2(G, \\Z) \\to R / [F, R] \\to F / [F, F] \\to G / [G, G] \\to 0.$$\nSince $F$ is free, $H_2(F, \\Z) = 0$, so $H_2(G, \\Z)$ is the kernel of $R / [F, R] \\to F / [F, F]$, which is $(R \\cap [F, F]) / [F, R]$.\n\nIn particular, the right side does not depend on the chosen [[presentation|0JGEQXQ4AC]] of $G$.\n\nExample: $G = \\Z^2 = \\langle a, b \\mid aba^{-1}b^{-1} \\rangle$, with $F$ free on $a, b$.\nHere $R = [F, F]$, because the normal closure of $aba^{-1}b^{-1}$ in $F$ is the commutator subgroup $[F, F]$.\nSo $H_2(\\Z^2, \\Z) \\cong [F, F] / [F, [F, F]]$, which is infinite cyclic, generated by the image of $aba^{-1}b^{-1}$.\nThis matches the homology of the torus.\n\nExample: $G = \\Z / m\\Z$, with $F = \\Z$ and $R = m\\Z$.\nThen $[F, F]$ is trivial, so $H_2(\\Z/m\\Z, \\Z) = 0$.\n\n## Explanations\n\n### L1 human\n\nHopf's formula computes a subtle invariant of a group from any description of the group by generators and relations.\nIt looks at the relations that are products of commutators, and it divides out the ones that every presentation has automatically.\nThe answer is the same for every presentation, which is surprising at first sight.\n\n### L2 human\n\nFor a finite group $G$, the group $H_2(G, \\Z)$ is the Schur multiplier, which controls the central extensions of $G$ and its projective representations.\nHopf's formula gives a purely algebraic way to compute it.\nThe exact sequence in the proof is the five-term sequence of the [[Lyndon–Hochschild–Serre spectral sequence|VC3QNG6TKR]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "CFAG3VRXEA"
    ],
    "kind": "statement",
    "msc": [
      "20J06",
      "20F05"
    ],
    "name": "Hopf's formula",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Schur_multiplier"
      }
    ],
    "tag": "S31M1ZYJRK",
    "uses": [
      "0KTW26C80J",
      "0CB0DX5BVV",
      "0KYX2N69J0",
      "0ASGSTXSBW",
      "0PXW0BN7DW",
      "CFAG3VRXEA",
      "0JGEQXQ4AC"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/S31M1ZYJRK/",
  "links": [
    {
      "field": "about",
      "target": "CFAG3VRXEA"
    },
    {
      "field": "uses",
      "target": "0KTW26C80J"
    },
    {
      "field": "uses",
      "target": "0CB0DX5BVV"
    },
    {
      "field": "uses",
      "target": "0KYX2N69J0"
    },
    {
      "field": "uses",
      "target": "0ASGSTXSBW"
    },
    {
      "field": "uses",
      "target": "0PXW0BN7DW"
    },
    {
      "field": "uses",
      "target": "CFAG3VRXEA"
    },
    {
      "field": "uses",
      "target": "0JGEQXQ4AC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/S31M1ZYJRK.md",
  "msc": [
    "20J06",
    "20F05"
  ],
  "name": "Hopf's formula",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S31/S31M1ZYJRK.md",
  "status": null,
  "tag": "S31M1ZYJRK",
  "uri": "m2://S31M1ZYJRK",
  "url": "https://math2.tamnd.com/wiki/hopfs-formula/",
  "writing": {
    "tag": "S31M1ZYJRK",
    "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
      }
    ]
  }
}