{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $M$ be a closed oriented [[$3$-manifold|R20TB2KEQQ]].\nAn *open book decomposition* of $M$ is a pair $(B, \\pi)$ of the following kind.\nThe *binding* $B \\subset M$ is an oriented link, and $\\pi \\colon M \\setminus B \\to S^1$ is a fibration.\nEach fiber $\\pi^{-1}(\\theta)$ is the interior of a compact surface $\\Sigma_\\theta$, a *page*, with oriented boundary $\\partial \\Sigma_\\theta = B$.\nA cooriented positive contact structure $\\xi$ on $M$ is *supported* by $(B, \\pi)$ if $\\xi = \\ker \\alpha$ for some contact form $\\alpha$ with two properties.\nThe form $\\alpha$ is positive on the oriented binding $B$, and $d\\alpha$ is a positive area form on each page.\nThen:\n\n1. Every open book decomposition of $M$ supports a [[contact structure|HBQ8BZ7XB5]], and any two contact structures supported by the same open book are isotopic.\n2. Every positive contact structure on $M$ is supported by some open book decomposition.\n3. Two open book decompositions support isotopic contact structures if and only if they become isotopic after finitely many positive stabilizations of each.\n\nSo supporting gives a bijection between two sets.\nThe first is the set of positive contact structures on $M$ up to isotopy.\nThe second is the set of open books of $M$ up to isotopy and positive stabilization.\n\n## Remarks\n\nPart 1, the existence, goes back to Thurston and Winkelnkemper; Giroux proved the uniqueness and parts 2 and 3.\nThe proof of part 2 builds a contact cell decomposition of $M$ and uses a ribbon of its Legendrian $1$-skeleton as a page.\nA positive stabilization of an open book adds a $1$-handle to the page and composes the monodromy with a positive Dehn twist along a curve crossing the new handle once.\n\nExample: the open book of $S^3$ whose binding is an unknot and whose pages are discs supports the standard tight contact structure of $S^3$.\nThe open book with annulus pages and monodromy a negative Dehn twist about the core supports an overtwisted contact structure on $S^3$.\n\n## Explanations\n\n### L1 human\n\nAn open book describes a closed three-dimensional space as a family of surfaces, the pages, that all share the same boundary, the binding.\nThey look like the pages of a book turning around its spine.\nGiroux's theorem says that contact structures on such a space correspond exactly to open books, up to a simple move called positive stabilization.\nThis turns questions about twisting plane fields into questions about surfaces and their diffeomorphisms.\n\n### L2 human\n\nThe correspondence lets one study contact $3$-manifolds through the monodromy of an open book, an element of the mapping class group of the page.\nFor example, a contact structure is Stein fillable exactly when it is supported by some open book whose monodromy is a product of positive Dehn twists.\nThis is due to work of Giroux, of Loi and Piergallini, and of Akbulut and Ozbagci.\nIt also gives a route from contact structures to Heegaard Floer invariants, through the contact invariant of Ozsváth and Szabó.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "HBQ8BZ7XB5"
    ],
    "kind": "statement",
    "msc": [
      "57K33",
      "53D10"
    ],
    "name": "Giroux correspondence",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Open_book_decomposition"
      }
    ],
    "tag": "1HTW0F4ZZ7",
    "uses": [
      "HBQ8BZ7XB5",
      "R20TB2KEQQ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/1HTW0F4ZZ7/",
  "links": [
    {
      "field": "about",
      "target": "HBQ8BZ7XB5"
    },
    {
      "field": "uses",
      "target": "HBQ8BZ7XB5"
    },
    {
      "field": "uses",
      "target": "R20TB2KEQQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1HTW0F4ZZ7.md",
  "msc": [
    "57K33",
    "53D10"
  ],
  "name": "Giroux correspondence",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1HT/1HTW0F4ZZ7.md",
  "status": null,
  "tag": "1HTW0F4ZZ7",
  "uri": "m2://1HTW0F4ZZ7",
  "url": "https://math2.tamnd.com/wiki/giroux-correspondence/",
  "writing": {
    "tag": "1HTW0F4ZZ7",
    "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
      }
    ]
  }
}