geometrization conjecture
E255013
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| geometrization conjecture canonical | 3 |
| Geometrization program | 1 |
| Thurston geometries | 1 |
| Thurston’s hyperbolization theorem | 1 |
| geometrization conjecture (in dimension 3) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2325577 — 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: geometrization conjecture Context triple: [Ricci flow, usedInProofOf, geometrization conjecture]
-
A.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
B.
Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
-
C.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
-
D.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
-
E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- 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: geometrization conjecture Target entity description: 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.
-
A.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
B.
Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
-
C.
Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
-
D.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
-
E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
statement in 3-dimensional topology ⓘ |
| alsoKnownAs | Thurston’s geometrization conjecture ⓘ |
| canonicalGeometriesCount | 8 ⓘ |
| centralTo | classification of closed 3-manifolds ⓘ |
| concerns |
closed 3-manifolds
ⓘ
geometric structures on 3-manifolds ⓘ prime decomposition of 3-manifolds ⓘ |
| context | Thurston’s eight model geometries ⓘ |
| field |
3-manifold theory
ⓘ
geometric topology ⓘ topology ⓘ |
| formulatedIn | 1970s ⓘ |
| hasCanonicalGeometry |
ilde{SL_2(R)} geometry
ⓘ
Euclidean geometry ⓘ H^2 × R geometry ⓘ Nil geometry ⓘ S^2 × R geometry ⓘ Sol geometry ⓘ hyperbolic geometry ⓘ spherical geometry ⓘ |
| hasConsequence |
complete classification of closed orientable 3-manifolds up to geometric decomposition
ⓘ
most closed 3-manifolds admit hyperbolic structures ⓘ |
| implies | Poincaré conjecture ⓘ |
| includes | Poincaré conjecture as a special case ⓘ |
| influencedBy | Thurston’s work on hyperbolic 3-manifolds ⓘ |
| influences |
geometric group theory
ⓘ
low-dimensional topology ⓘ modern 3-manifold topology ⓘ |
| language | English ⓘ |
| proofCompletedIn |
2002
ⓘ
2003 ⓘ |
| proofMethod |
Ricci flow
ⓘ
surface form:
Ricci flow with surgery
analysis of singularities in Ricci flow ⓘ |
| proposedBy | William Thurston ⓘ |
| provedBy | Grigori Perelman ⓘ |
| recognizedBy |
Poincaré conjecture
ⓘ
surface form:
Clay Millennium Prize Problem on the Poincaré conjecture
Fields Medal invitation to Grigori Perelman ⓘ |
| relatedTo |
JSJ decomposition
ⓘ
Ricci flow ⓘ Ricci flow with surgery ⓘ prime decomposition theorem for 3-manifolds ⓘ |
| statesThat |
each piece of a decomposed closed 3-manifold admits one of eight model geometries
ⓘ
every closed 3-manifold can be decomposed into pieces that admit canonical geometric structures ⓘ |
| status | proven ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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.
Input
Subject: geometrization conjecture Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Thurston geometries
this entity surface form:
Thurston’s hyperbolization theorem
this entity surface form:
geometrization conjecture (in dimension 3)
this entity surface form:
Geometrization program