{
  "id": "enwiki-001377316804",
  "slug": "mapping-torus",
  "title": "Mapping torus",
  "qid": "",
  "lead": "In mathematics, specifically in topology, the mapping torus of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism: M_f =\\frac{(I \\times X)}{(1,x)\\sim (0,f(x))} The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold, M f will be a manifold of dimension one higher, and it is said to \"fiber over the circle\". As a simple example, let X be the circle, and f be the inversion e^{ix} \\mapsto e^{-ix} , then the mapping torus is the Klein bottle. Mapping tori of surface homeomorphisms play a key role in the theory of 3-manifolds and have been intensely studied. If S is a closed surface of genus g ≥ 2 and if f is a self-homeomorphism of S, the mapping torus M f is a closed 3-manifold that fibers over the circle with fiber S. A deep result of Thurston states that in this case the 3-manifold M f is hyperbolic if and only if f is a pseudo-Anosov homeomorphism of S.",
  "body": "In mathematics, specifically in topology, the mapping torus of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism: M_f =\\frac{(I \\times X)}{(1,x)\\sim (0,f(x))} The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold, M f will be a manifold of dimension one higher, and it is said to \"fiber over the circle\". As a simple example, let X be the circle, and f be the inversion e^{ix} \\mapsto e^{-ix} , then the mapping torus is the Klein bottle. Mapping tori of surface homeomorphisms play a key role in the theory of 3-manifolds and have been intensely studied. If S is a closed surface of genus g ≥ 2 and if f is a self-homeomorphism of S, the mapping torus M f is a closed 3-manifold that fibers over the circle with fiber S. A deep result of Thurston states that in this case the 3-manifold M f is hyperbolic if and only if f is a pseudo-Anosov homeomorphism of S. ==References==",
  "categories": [
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    "20477234"
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  "license": "CC BY-SA 4.0",
  "source_uri": "https://en.wikipedia.org/w/index.php?oldid=1377316804",
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  "created_at": null,
  "updated_at": 1790624759.0,
  "claims": [],
  "contradictions": [],
  "confidence": 0.6,
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    "type": "webpage",
    "title": "Mapping torus",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/mapping-torus",
    "issued": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "accessed": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001377316804",
    "isBasedOn": "https://en.wikipedia.org/w/index.php?oldid=1377316804",
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