About: Mapping torus

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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.

Attributes	Values
rdfs:label	Abbildungstorus (de) Tore d'application (fr) Mapping torus (en)
rdfs:comment	In der Mathematik sind Abbildungstori topologische Räume, mit denen topologische Abbildungen beschrieben werden. (de) 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 . (fr) 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. (en)
dcterms:subject	Homeomorphisms Geometric topology General topology
Wikipage page ID	21034384 (xsd:integer)
Wikipage revision ID	1085351077 (xsd:integer)
Link from a Wikipage to another Wikipage	Cartesian product Mathematics Circle Homeomorphisms Closed manifold Deep result Surface (topology) Topology William Thurston 3-manifold Fiber bundle Closed interval Geometric topology Hyperbolic manifold General topology Homeomorphism Manifold Pseudo-Anosov map Topological space
sameAs	Mapping torus Mapping torus Mapping torus Mapping torus Mapping torus Mapping torus
dbp:wikiPageUsesTemplate	dbt:Reflist
has abstract	In der Mathematik sind Abbildungstori topologische Räume, mit denen topologische Abbildungen beschrieben werden. (de) 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. (en) 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 . (fr)
prov:wasDerivedFrom	wikipedia-en:Mapping_torus?oldid=1085351077&ns=0
page length (characters) of wiki page	1786 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Mapping_torus
is Link from a Wikipage to another Wikipage of	Open book decomposition Geometrization conjecture Surface bundle Suspension (dynamical systems) Fiber bundle Floer homology JSJ decomposition Thurston norm Space-filling curve Nielsen–Thurston classification Surface bundle over the circle Solvmanifold
is foaf:primaryTopic of	wikipedia-en:Mapping_torus

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