Thurston hyperbolization theorem
E898489
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Thurston hyperbolic Dehn surgery theorem | 1 |
| Thurston hyperbolization theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991689 — 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.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Thurston hyperbolization theorem Context triple: [Kleinian group, relatedTo, Thurston hyperbolization theorem]
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A.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
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B.
Dehn lemma
The Dehn lemma is a fundamental result in 3-manifold topology that gives conditions under which a loop on the boundary of a 3-manifold bounds an embedded disk in the manifold.
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C.
Thurston’s classification of surface diffeomorphisms
Thurston’s classification of surface diffeomorphisms is a foundational theorem in low-dimensional topology that categorizes self-maps of surfaces into periodic, reducible, or pseudo-Anosov types, profoundly influencing the study of 3-manifolds and dynamical systems.
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D.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
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E.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Thurston hyperbolization theorem Target entity description: 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.
-
A.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
-
B.
Dehn lemma
The Dehn lemma is a fundamental result in 3-manifold topology that gives conditions under which a loop on the boundary of a 3-manifold bounds an embedded disk in the manifold.
-
C.
Thurston’s classification of surface diffeomorphisms
Thurston’s classification of surface diffeomorphisms is a foundational theorem in low-dimensional topology that categorizes self-maps of surfaces into periodic, reducible, or pseudo-Anosov types, profoundly influencing the study of 3-manifolds and dynamical systems.
-
D.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
E.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in 3-manifold topology ⓘ |
| appliesTo |
Haken 3-manifolds under suitable conditions
ⓘ
atoroidal Haken 3-manifolds with infinite fundamental group ⓘ many knot complements in S^3 ⓘ |
| assumes |
irreducible 3-manifolds
ⓘ
sufficiently large 3-manifolds in the Haken setting ⓘ |
| author | William P. Thurston NERFINISHED ⓘ |
| characterizes | when certain 3-manifolds admit complete hyperbolic metrics ⓘ |
| concerns |
3-manifolds
ⓘ
complete hyperbolic structures ⓘ geometric structures on 3-manifolds ⓘ hyperbolic 3-manifolds ⓘ |
| concludes |
3-manifold admits a complete finite-volume hyperbolic metric under its hypotheses
ⓘ
3-manifold is hyperbolic in the sense of Thurston’s geometrization ⓘ |
| field |
3-manifold theory
ⓘ
geometric topology ⓘ hyperbolic geometry ⓘ |
| formalizes | conditions under which a 3-manifold supports a hyperbolic geometry ⓘ |
| hasConsequence |
classification of many 3-manifolds as hyperbolic
ⓘ
existence of large families of hyperbolic 3-manifolds ⓘ |
| hasVersion |
hyperbolization theorem for Haken 3-manifolds
NERFINISHED
ⓘ
hyperbolization theorem for atoroidal Haken manifolds ⓘ |
| helpsProve | hyperbolicity of complements of many knots and links ⓘ |
| historicalPeriod | late 20th century ⓘ |
| implies |
existence of hyperbolic structures on many Haken 3-manifolds
ⓘ
many 3-manifolds have unique hyperbolic structures up to isometry ⓘ |
| importance |
cornerstone of modern 3-manifold topology
ⓘ
key component of the proof of geometrization for Haken manifolds ⓘ |
| influenced | Perelman’s work on geometrization ⓘ |
| inspired | subsequent generalizations of hyperbolization to other settings ⓘ |
| isCornerstoneOf | Thurston’s theory of 3-dimensional geometries NERFINISHED ⓘ |
| namedAfter | William P. Thurston NERFINISHED ⓘ |
| partOf | Thurston’s geometrization program NERFINISHED ⓘ |
| provedUsing | methods of low-dimensional topology and hyperbolic geometry ⓘ |
| relatedTo |
Haken’s work on 3-manifolds
ⓘ
JSJ decomposition of 3-manifolds ⓘ Mostow–Prasad rigidity theorem NERFINISHED ⓘ Thurston geometrization conjecture NERFINISHED ⓘ hyperbolic Dehn surgery theorem NERFINISHED ⓘ prime decomposition of 3-manifolds ⓘ |
| status | proved ⓘ |
| uses |
Haken hierarchy
NERFINISHED
ⓘ
Kleinian group theory NERFINISHED ⓘ Mostow rigidity NERFINISHED ⓘ character variety methods ⓘ hyperbolic Dehn surgery techniques ⓘ incompressible surfaces ⓘ pleated surfaces ⓘ |
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Input
Subject: Thurston hyperbolization theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Thurston hyperbolic Dehn surgery theorem