Dehn–Lickorish theorem
E912779
The Dehn–Lickorish theorem is a fundamental result in low-dimensional topology stating that the mapping class group of a closed, orientable surface is generated by finitely many Dehn twists.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dehn–Lickorish theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11215038 — 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: Dehn–Lickorish theorem Context triple: [Dehn twist, appearsIn, Dehn–Lickorish theorem]
-
A.
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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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 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.
Lickorish
Lickorish is a mathematician known for his influential contributions to low-dimensional topology and knot theory.
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E.
Wirtinger presentation of knot groups
The Wirtinger presentation of knot groups is a classical method in knot theory that describes the fundamental group of a knot complement using generators and relations derived from a knot diagram.
- 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: Dehn–Lickorish theorem Target entity description: The Dehn–Lickorish theorem is a fundamental result in low-dimensional topology stating that the mapping class group of a closed, orientable surface is generated by finitely many Dehn twists.
-
A.
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.
-
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 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.
Lickorish
Lickorish is a mathematician known for his influential contributions to low-dimensional topology and knot theory.
-
E.
Wirtinger presentation of knot groups
The Wirtinger presentation of knot groups is a classical method in knot theory that describes the fundamental group of a knot complement using generators and relations derived from a knot diagram.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in low-dimensional topology ⓘ |
| about |
Dehn twists
NERFINISHED
ⓘ
closed orientable surfaces ⓘ mapping class group of a surface ⓘ |
| appliesTo |
mapping class group of a closed orientable surface
ⓘ
mapping class group of a compact connected orientable surface ⓘ |
| assertsExistenceOf | finite generating set of Dehn twists for the mapping class group ⓘ |
| concerns |
finite generation of mapping class groups
ⓘ
generation of mapping class groups ⓘ homeomorphisms of surfaces up to isotopy ⓘ |
| context |
compact connected orientable 2-manifolds without boundary
ⓘ
orientation-preserving homeomorphisms of surfaces ⓘ |
| field |
geometric topology
ⓘ
low-dimensional topology ⓘ topology ⓘ |
| hasConsequence |
any mapping class can be expressed as a product of right and left Dehn twists
ⓘ
mapping class group is generated by Dehn twists about nonseparating curves and some separating curves ⓘ |
| historicalContributor |
Max Dehn
NERFINISHED
ⓘ
W. B. R. Lickorish NERFINISHED ⓘ |
| holdsFor |
closed orientable surface of genus at least 1
ⓘ
mapping class group of a surface of genus g ≥ 1 ⓘ |
| implies |
every element of the mapping class group of a closed orientable surface can be written as a product of Dehn twists
ⓘ
the mapping class group of a closed orientable surface is finitely generated ⓘ |
| isFundamentalResultIn |
surface topology
ⓘ
theory of mapping class groups ⓘ |
| isUsedIn |
3-manifold topology via Heegaard splittings
ⓘ
construction of presentations of mapping class groups ⓘ study of moduli space of Riemann surfaces ⓘ symplectic topology of surfaces ⓘ theory of Lefschetz fibrations NERFINISHED ⓘ |
| namedAfter |
Max Dehn
NERFINISHED
ⓘ
William Bernard Raymond Lickorish NERFINISHED ⓘ |
| relatedTo |
Dehn twist factorization of mapping classes
ⓘ
Humphries generators for the mapping class group ⓘ Nielsen–Thurston classification NERFINISHED ⓘ |
| statesThat | the mapping class group of a closed orientable surface is generated by finitely many Dehn twists ⓘ |
| typicalProofUses |
cutting a surface along simple closed curves
ⓘ
decomposition of homeomorphisms into twists ⓘ surgery on curves on surfaces ⓘ |
| usesConcept |
Dehn twist
NERFINISHED
ⓘ
isotopy class of homeomorphisms ⓘ mapping class group NERFINISHED ⓘ simple closed curve on a surface ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Dehn–Lickorish theorem Description of subject: The Dehn–Lickorish theorem is a fundamental result in low-dimensional topology stating that the mapping class group of a closed, orientable surface is generated by finitely many Dehn twists.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.