{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "9EA7KJ50AE"
    },
    {
      "field": "about",
      "target": "DE29ZC399Z"
    },
    {
      "field": "about",
      "target": "ZD7BNADK4H"
    },
    {
      "field": "uses",
      "target": "9EA7KJ50AE"
    },
    {
      "field": "uses",
      "target": "DE29ZC399Z"
    },
    {
      "field": "uses",
      "target": "ZD7BNADK4H"
    }
  ],
  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], viewed as a discrete group.\nA *classifying space* of $G$ is a path-connected [[CW complex|5W95QFHXPR]] $X$ such that:\n\n1. The [[fundamental group|S1XK3HXHFX]] $\\pi_1(X)$ is isomorphic to $G$.\n2. The [[homotopy group|P976WYEFMB]] $\\pi_n(X)$ is trivial for every $n \\ge 2$.\n\nSuch a space is written $BG$.\n\n## Remarks\n\nA classifying space of $G$ is the same as an [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(G, 1)$.\nAny two classifying spaces of $G$ are [[homotopy equivalent|XZG6MTSWCN]].\nEquivalently, $X$ is path-connected, $\\pi_1(X) \\cong G$, and its [[universal cover|Y1YPE6PM52]] is [[contractible|R5WP8Z69Q8]]; then $G$ acts freely on the universal cover and $X$ is the quotient.\n\nExample: the circle $S^1$ is a classifying space of $\\Z$.\nIts fundamental group is $\\Z$, and its universal cover $\\R$ is contractible, so $\\pi_n(S^1) = 0$ for $n \\ge 2$.\nIn the same way, the torus is a classifying space of $\\Z^2$, and the infinite real projective space $\\R P^\\infty$ is one of $\\Z/2\\Z$.\n\nNon-example: the real projective plane $\\R P^2$ has fundamental group $\\Z/2\\Z$, but its universal cover is the sphere $S^2$, so $\\pi_2(\\R P^2) \\cong \\pi_2(S^2) \\cong \\Z$.\nClause 2 fails, so $\\R P^2$ is not a classifying space of $\\Z/2\\Z$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "WA796ZCNRX",
    "kind": "definition",
    "msc": [
      "55R35",
      "20J06"
    ],
    "name": "Classifying space of a group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Classifying_space"
      }
    ],
    "tag": "7XRBR7AV3C",
    "uses": [
      "G5S1M9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/7XRBR7AV3C/",
  "links": [
    {
      "field": "defines",
      "target": "WA796ZCNRX"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7XRBR7AV3C.md",
  "msc": [
    "55R35",
    "20J06"
  ],
  "name": "Classifying space of a group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7XR/7XRBR7AV3C.md",
  "status": null,
  "tag": "7XRBR7AV3C",
  "uri": "m2://7XRBR7AV3C",
  "url": "https://math2.tamnd.com/wiki/classifying-space-of-a-group-definition/",
  "writing": {
    "tag": "7XRBR7AV3C",
    "kind": "definition",
    "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
      }
    ]
  }
}