{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "MTWNAJMH1D"
    },
    {
      "field": "uses",
      "target": "MTWNAJMH1D"
    },
    {
      "field": "defines",
      "target": "TWSZNFKBKA"
    }
  ],
  "body": "\n## Summary\n\nThe deck transformation group of a [[covering map|3XZRRE165N]] $p \\colon \\tilde X \\to X$ is the group of homeomorphisms of $\\tilde X$ that commute with $p$.\nThese maps permute the points of each fibre, and they measure the symmetry of the covering.\nFor the [[universal cover|Y1YPE6PM52]] of a reasonable space, the deck transformation group is isomorphic to the [[fundamental group|S1XK3HXHFX]] of the space.\n\n## Explanations\n\n### L1 human\n\nA deck transformation shuffles the sheets of a covering without moving anything in the base.\nFor the helix $\\R$ over the circle, shifting the whole helix up by one full turn is such a symmetry.\nThe shifts by an integer number of turns are all of them, so the deck transformation group is $\\Z$.\nThis matches the fact that a loop can wind around the circle any integer number of times.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "57M10",
      "55R05"
    ],
    "name": "Deck transformation group",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Covering_space"
      }
    ],
    "tag": "BH32RQBX6A"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/BH32RQBX6A/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/BH32RQBX6A.md",
  "msc": [
    "57M10",
    "55R05"
  ],
  "name": "Deck transformation group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BH3/BH32RQBX6A.md",
  "status": null,
  "tag": "BH32RQBX6A",
  "uri": "m2://BH32RQBX6A",
  "url": "https://math2.tamnd.com/wiki/deck-transformation-group/",
  "writing": {
    "tag": "BH32RQBX6A",
    "kind": "concept",
    "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
      }
    ]
  }
}