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In mathematics, the mapping torus in topol … In mathematics, the mapping torus in topology 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: The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold, Mf will be a manifold of dimension one higher, and it is said to "fiber over the circle". As a simple example, let be the circle, and be the inversion , 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 Mf 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 Mf is hyperbolic if and only if f is a pseudo-Anosov homeomorphism of S.f f is a pseudo-Anosov homeomorphism of S.
, En mathématiques et plus particulièrement en topologie, le tore d'application, dit aussi mapping torus ou encore tore de suspension, d'un homéomorphisme d'un espace topologique est l'espace produit quotienté par la relation d'équivalence .
, In der Mathematik sind Abbildungstori topologische Räume, mit denen topologische Abbildungen beschrieben werden.
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| rdfs:comment |
In mathematics, the mapping torus in topol … In mathematics, the mapping torus in topology 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: The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold, Mf will be a manifold of dimension one higher, and it is said to "fiber over the circle". As a simple example, let be the circle, and be the inversion , then the mapping torus is the Klein bottle.hen the mapping torus is the Klein bottle.
, In der Mathematik sind Abbildungstori topologische Räume, mit denen topologische Abbildungen beschrieben werden.
, En mathématiques et plus particulièrement en topologie, le tore d'application, dit aussi mapping torus ou encore tore de suspension, d'un homéomorphisme d'un espace topologique est l'espace produit quotienté par la relation d'équivalence .
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| rdfs:label |
Abbildungstorus
, Mapping torus
, Tore d'application
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