Seifert fibered spaces
E911225
Seifert fibered spaces are three-dimensional manifolds that can be decomposed into a disjoint union of circles arranged in a highly structured, fibered way over a two-dimensional orbifold.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Seifert fibered spaces canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11214928 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
Target entity: Seifert fibered spaces Context triple: [Dehn surgery, canProduce, Seifert fibered spaces]
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A.
Foliations of Three-Manifolds Which Are Circle Bundles
"Foliations of Three-Manifolds Which Are Circle Bundles" is William Thurston’s influential 1972 doctoral dissertation in geometric topology, where he developed foundational ideas about the structure and classification of foliations on 3-manifolds.
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B.
Dehn surgery
Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
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C.
Thurston hyperbolization theorem
The Thurston hyperbolization theorem is a fundamental result in 3-manifold topology that characterizes when certain 3-manifolds admit complete hyperbolic structures, forming a cornerstone of Thurston’s geometrization program.
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D.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
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E.
“Braids, Links, and Mapping Class Groups”
“Braids, Links, and Mapping Class Groups” is a foundational monograph in low-dimensional topology that systematically develops the theory of braids, links, and mapping class groups and their interrelations.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: Seifert fibered spaces Target entity description: Seifert fibered spaces are three-dimensional manifolds that can be decomposed into a disjoint union of circles arranged in a highly structured, fibered way over a two-dimensional orbifold.
-
A.
Foliations of Three-Manifolds Which Are Circle Bundles
"Foliations of Three-Manifolds Which Are Circle Bundles" is William Thurston’s influential 1972 doctoral dissertation in geometric topology, where he developed foundational ideas about the structure and classification of foliations on 3-manifolds.
-
B.
Dehn surgery
Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
-
C.
Thurston hyperbolization theorem
The Thurston hyperbolization theorem is a fundamental result in 3-manifold topology that characterizes when certain 3-manifolds admit complete hyperbolic structures, forming a cornerstone of Thurston’s geometrization program.
-
D.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
-
E.
“Braids, Links, and Mapping Class Groups”
“Braids, Links, and Mapping Class Groups” is a foundational monograph in low-dimensional topology that systematically develops the theory of braids, links, and mapping class groups and their interrelations.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
3-manifold
ⓘ
fiber bundle (up to singular fibers) ⓘ mathematical concept ⓘ |
| appearsIn |
Thurston eight geometries
NERFINISHED
ⓘ
Thurston geometrization conjecture NERFINISHED ⓘ |
| baseDimension | 2 ⓘ |
| baseOrbifoldProperty |
base orbifold may have cone points
ⓘ
base orbifold may have reflector curves ⓘ |
| baseSpace | 2-dimensional orbifold ⓘ |
| classifiedBy |
Euler number of the fibration
ⓘ
Seifert invariants NERFINISHED ⓘ base orbifold ⓘ multiplicities of singular fibers ⓘ |
| constructionMethod | Dehn filling on a circle bundle over a surface ⓘ |
| covers | circle bundle over a surface (in some cases) ⓘ |
| dimension | 3 ⓘ |
| fiber |
S^1
ⓘ
circle ⓘ |
| field |
3-manifold theory
ⓘ
geometric topology ⓘ |
| fundamentalGroupProperty | contains infinite cyclic normal subgroup generated by a regular fiber ⓘ |
| generalizes | circle bundles over closed surfaces ⓘ |
| hasComponent |
regular fibers
ⓘ
singular fibers ⓘ |
| hasExample |
3-sphere with Hopf fibration
ⓘ
S^2 × S^1 ⓘ lens spaces ⓘ torus bundles over S^1 ⓘ |
| hasFundamentalGroup | group fitting into short exact sequence with Z as normal subgroup ⓘ |
| hasInvariant |
Seifert volume (in geometric cases)
ⓘ
orbifold Euler characteristic of the base ⓘ |
| hasStructure |
decomposition into disjoint union of circles
ⓘ
locally trivial circle fibration away from singular fibers ⓘ |
| introducedBy | Herbert Seifert NERFINISHED ⓘ |
| introducedInYear | 1933 ⓘ |
| isSubsetOf |
Haken 3-manifolds (in many cases)
ⓘ
prime 3-manifolds ⓘ |
| localModel |
S^1-bundle over a 2-dimensional disk for regular fibers
ⓘ
fibered solid torus with nontrivial gluing for singular fibers ⓘ |
| namedAfter | Herbert Seifert NERFINISHED ⓘ |
| orientationProperty | can be orientable or non-orientable ⓘ |
| specialCaseOf | graph manifold ⓘ |
| supportsGeometry |
ilde{SL_2(R)}-geometry (in some cases)
GENERATED
ⓘ
E^3-geometry (in some cases) GENERATED ⓘ Nil-geometry (in some cases) GENERATED ⓘ S^2 × R-geometry (in some cases) GENERATED ⓘ S^3-geometry (in some cases) GENERATED ⓘ |
| usedIn |
JSJ decomposition of 3-manifolds
ⓘ
classification of closed orientable 3-manifolds with infinite fundamental group ⓘ |
How these facts were elicited
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You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Subject: Seifert fibered spaces Description of subject: Seifert fibered spaces are three-dimensional manifolds that can be decomposed into a disjoint union of circles arranged in a highly structured, fibered way over a two-dimensional orbifold.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.