Dehn lemma
E265412
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dehn lemma canonical | 3 |
| "On Dehn's lemma and the asphericity of knots" | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2416868 — 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 lemma Context triple: [Max Dehn, notableWork, Dehn lemma]
-
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.
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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C.
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.
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D.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
-
E.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
- 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 lemma Target entity description: 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.
-
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.
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.
-
C.
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.
-
D.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
-
E.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in 3-manifold topology
ⓘ
theorem in topology ⓘ |
| appliesTo |
compact 3-manifolds
ⓘ
orientable 3-manifolds ⓘ |
| assumption |
loop has a map of a disk into the manifold with boundary that loop
ⓘ
loop on the boundary is null-homotopic in the 3-manifold ⓘ |
| category | low-dimensional topology theorem ⓘ |
| concerns |
3-manifolds
ⓘ
embedded disks ⓘ loops on the boundary of 3-manifolds ⓘ null-homotopic curves on the boundary ⓘ |
| conclusion | there exists an embedded disk in the manifold with the same boundary loop ⓘ |
| field |
3-manifold topology
ⓘ
geometric topology ⓘ topology ⓘ |
| gapRepairedBy | Christos Papakyriakopoulos ⓘ |
| givesConditionFor | when a loop on the boundary of a 3-manifold bounds an embedded disk ⓘ |
| implies | existence of an embedded disk with given boundary under certain conditions ⓘ |
| influenced |
development of 3-manifold topology in the mid-20th century
ⓘ
work of Friedhelm Waldhausen ⓘ work of John Stallings ⓘ |
| involvesConcept |
boundary of a manifold
ⓘ
embedded surfaces ⓘ immersed disks ⓘ null-homotopy ⓘ |
| isPartOf | classical results in low-dimensional topology ⓘ |
| modernProofBy | Christos Papakyriakopoulos ⓘ |
| namedAfter | Max Dehn ⓘ |
| oftenStatedWith | loop theorem ⓘ |
| originallyProvedBy | Max Dehn ⓘ |
| originalProof | contained a gap ⓘ |
| proofMethod |
use of group-theoretic techniques in topology
ⓘ
use of towers of covering spaces ⓘ |
| publishedIn | Annals of Mathematics ⓘ |
| relatedTo |
Haken manifolds
ⓘ
Poincaré conjecture ⓘ incompressible surfaces ⓘ loop theorem ⓘ sphere theorem ⓘ |
| strengthenedBy | loop theorem ⓘ |
| typeOfResult | existence theorem ⓘ |
| usedIn |
3-manifold decomposition theory
ⓘ
construction of incompressible surfaces ⓘ knot theory ⓘ proofs of the loop theorem ⓘ study of fundamental groups of 3-manifolds ⓘ |
| yearOfCorrectProof | 1957 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Dehn lemma Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Dehn lemma