Thurston norm
E518459
The Thurston norm is a topological invariant on the second homology of a 3-manifold that measures the minimal complexity (via Euler characteristic) of embedded surfaces representing a given homology class.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Thurston norm canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425320 — 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 norm Context triple: [William Thurston, knownFor, Thurston norm]
-
A.
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.
-
B.
Dehn complex
The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
-
C.
Dehn invariant
The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
-
D.
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.
-
E.
Dehn function
The Dehn function is a mathematical tool in geometric group theory that measures the complexity of filling loops with discs in a space or group, quantifying the difficulty of solving the word problem.
- 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 norm Target entity description: The Thurston norm is a topological invariant on the second homology of a 3-manifold that measures the minimal complexity (via Euler characteristic) of embedded surfaces representing a given homology class.
-
A.
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.
-
B.
Dehn complex
The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
-
C.
Dehn invariant
The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
-
D.
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.
-
E.
Dehn function
The Dehn function is a mathematical tool in geometric group theory that measures the complexity of filling loops with discs in a space or group, quantifying the difficulty of solving the word problem.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
seminorm
ⓘ
topological invariant ⓘ |
| associatedWith |
Thurston’s geometrization program
ⓘ
linked to:
geometrization conjecture
Thurston’s work on norm on homology ⓘ |
| boundaryCondition | surfaces considered are properly embedded ⓘ |
| codomain | nonnegative real numbers ⓘ |
| definedFor | compact 3-manifolds with boundary ⓘ |
| definedOn |
H_2(M; \mathbb{R})
ⓘ
H_2(M; \mathbb{Z}) ⓘ second homology group of a 3-manifold ⓘ |
| definedUsing | minimal value of -\chi(S) over embedded surfaces S ⓘ |
| dependsOn | topology of the 3-manifold ⓘ |
| domain | homology classes in H_2(M; \mathbb{Z}) ⓘ |
| generalizes | notion of genus for knots to higher-dimensional homology classes ⓘ |
| ignores |
disk components of surfaces
ⓘ
spherical components of surfaces ⓘ |
| introducedBy |
William P. Thurston
ⓘ
linked to:
William Thurston
|
| introducedInContext | study of hyperbolic 3-manifolds ⓘ |
| introducedInField | 3-manifold topology ⓘ |
| invariantOf | oriented compact 3-manifolds ⓘ |
| invariantUnder | homeomorphisms of the 3-manifold ⓘ |
| mathematicsSubjectClassification |
57M27
ⓘ
57N10 ⓘ |
| measures |
complexity of embedded surfaces
ⓘ
minimal complexity of a homology class ⓘ |
| minimizes | sum of max(0, -\chi(S_i)) over components S_i ⓘ |
| property |
homogeneous on rays
ⓘ
is a seminorm, not necessarily a norm ⓘ satisfies triangle inequality ⓘ subadditive ⓘ |
| relatedInvariant |
Gromov norm
ⓘ
simplicial volume ⓘ |
| relatedTo |
fibered faces of the unit ball
ⓘ
fibrations of 3-manifolds over the circle ⓘ |
| unitBall | a finite-sided rational polyhedron in H_2(M; \mathbb{R}) ⓘ |
| unitBallProperty | symmetric about the origin ⓘ |
| usedIn |
classification of 3-manifolds
ⓘ
study of mapping tori ⓘ |
| usedToStudy |
fibered 3-manifolds
ⓘ
hyperbolic structures on 3-manifolds ⓘ taut foliations ⓘ |
| uses | Euler characteristic ⓘ |
| vanishesOn | classes represented by unions of embedded spheres and tori ⓘ |
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: Thurston norm Description of subject: The Thurston norm is a topological invariant on the second homology of a 3-manifold that measures the minimal complexity (via Euler characteristic) of embedded surfaces representing a given homology class.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.