{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $p \\colon \\tilde X \\to X$ be a [[covering map|3XZRRE165N]].\nA *deck transformation* of $p$ is a [[homeomorphism|0QT00PB5G2]] $f \\colon \\tilde X \\to \\tilde X$ such that $p \\circ f = p$.\nThe *deck transformation group* of $p$ is the set of all deck transformations of $p$, with composition of maps as the [[group|G5S1M9]] operation.\n\n## Remarks\n\nExample: for $p \\colon \\R \\to S^1$, $p(t) = (\\cos 2\\pi t, \\sin 2\\pi t)$, each translation $t \\mapsto t + n$ with $n \\in \\Z$ is a homeomorphism with $p(t + n) = p(t)$.\nThese are all the deck transformations, so the group is isomorphic to $\\Z$.\n\nNon-example: the translation $t \\mapsto t + \\tfrac12$ is a homeomorphism of $\\R$, but $p(t + \\tfrac12) = -p(t) \\ne p(t)$, so it is not a deck transformation.\n\nThe group is also called the covering transformation group and is written $\\operatorname{Deck}(p)$ or $G(\\tilde X)$.\nIf $\\tilde X$ is connected, a deck transformation that fixes one point is the identity.\nThe covering is called normal, or regular, when the group acts transitively on each fibre.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "BH32RQBX6A",
    "kind": "definition",
    "msc": [
      "57M10",
      "55R05"
    ],
    "name": "Deck transformation group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Covering_space"
      }
    ],
    "tag": "TWSZNFKBKA",
    "uses": [
      "3XZRRE165N",
      "0QT00PB5G2",
      "G5S1M9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/TWSZNFKBKA/",
  "links": [
    {
      "field": "defines",
      "target": "BH32RQBX6A"
    },
    {
      "field": "uses",
      "target": "3XZRRE165N"
    },
    {
      "field": "uses",
      "target": "0QT00PB5G2"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/TWSZNFKBKA.md",
  "msc": [
    "57M10",
    "55R05"
  ],
  "name": "Deck transformation group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TWS/TWSZNFKBKA.md",
  "status": null,
  "tag": "TWSZNFKBKA",
  "uri": "m2://TWSZNFKBKA",
  "url": "https://math2.tamnd.com/wiki/deck-transformation-group-definition/",
  "writing": {
    "tag": "TWSZNFKBKA",
    "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
      }
    ]
  }
}