{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $S^1 = \\set{z \\in \\C : \\abs{z} = 1}$ be the unit [[circle|060W5CBZP6]], with base point $1$.\nFor $n \\in \\Z$, let $\\omega_n \\colon [0, 1] \\to S^1$ be the loop $\\omega_n(s) = e^{2\\pi i n s}$.\nThen the map $\\Z \\to \\pi_1(S^1, 1)$, $n \\mapsto [\\omega_n]$, is a [[group isomorphism|08MG7H8JW5]] from the additive group of [[integers|0C4JS9RF5M]] to the [[fundamental group|S1XK3HXHFX]] of $S^1$.\nIn particular $\\pi_1(S^1, 1) \\cong \\Z$, generated by the class of the loop that goes once around the circle.\n\n## Remarks\n\nThe proof uses the [[covering map|3XZRRE165N]] $p \\colon \\R \\to S^1$, $p(t) = e^{2\\pi i t}$.\nBy the [[path lifting property|HSWWF35BEX]], a loop $\\gamma$ at $1$ lifts to a unique path $\\tilde\\gamma$ in $\\R$ with $\\tilde\\gamma(0) = 0$, and $\\tilde\\gamma(1)$ is an integer, the degree of $\\gamma$.\nHomotopy lifting shows that path homotopic loops have the same degree, so the inverse map $[\\gamma] \\mapsto \\tilde\\gamma(1)$ is well defined.\nThe lift of $\\omega_n$ is $s \\mapsto ns$.\nThe lift of a concatenation is obtained by translating and joining lifts, which gives the homomorphism property.\nThe degree is the topological form of the [[winding number|0NQSV2EWDR]].\nConsequences include the Brouwer fixed point theorem in dimension $2$ and the fundamental theorem of algebra.\n\n## Explanations\n\n### L1 human\n\nEvery loop on a circle that starts and ends at a fixed point winds around the circle some whole number of times, counting clockwise turns as negative.\nTwo such loops can be deformed into each other exactly when they wind the same number of times.\nGoing around $m$ times and then $n$ times is going around $m + n$ times, so the loops behave like the integers under addition.\n\n### L2 human\n\nThis is the first nontrivial computation of a fundamental group, and the model for computations through covering spaces.\nIt shows that the circle is not contractible and that it is not a retract of the closed disk.\nWith the [[Seifert–van Kampen theorem|S17DY98VF2]] it gives the fundamental groups of wedges of circles, of surfaces and of many CW complexes.\nIt is also the case $n = 1$ of the computation $\\pi_n(S^n) \\cong \\Z$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "S1XK3HXHFX",
      "060W5CBZP6"
    ],
    "kind": "statement",
    "msc": [
      "55Q52",
      "57M05"
    ],
    "name": "Fundamental group of the circle",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fundamental_group"
      }
    ],
    "tag": "6SQXPE0JPN",
    "uses": [
      "S1XK3HXHFX",
      "060W5CBZP6",
      "0C4JS9RF5M",
      "08MG7H8JW5"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/6SQXPE0JPN/",
  "links": [
    {
      "field": "about",
      "target": "S1XK3HXHFX"
    },
    {
      "field": "about",
      "target": "060W5CBZP6"
    },
    {
      "field": "uses",
      "target": "S1XK3HXHFX"
    },
    {
      "field": "uses",
      "target": "060W5CBZP6"
    },
    {
      "field": "uses",
      "target": "0C4JS9RF5M"
    },
    {
      "field": "uses",
      "target": "08MG7H8JW5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/6SQXPE0JPN.md",
  "msc": [
    "55Q52",
    "57M05"
  ],
  "name": "Fundamental group of the circle",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6SQ/6SQXPE0JPN.md",
  "status": null,
  "tag": "6SQXPE0JPN",
  "uri": "m2://6SQXPE0JPN",
  "url": "https://math2.tamnd.com/wiki/fundamental-group-of-the-circle/",
  "writing": {
    "tag": "6SQXPE0JPN",
    "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
      }
    ]
  }
}