JSJ decomposition
E888035
The JSJ decomposition is a fundamental tool in 3-manifold topology that splits a 3-manifold along tori into simpler, canonical pieces that are either Seifert fibered or atoroidal, forming a key step toward its geometric classification.
All labels observed (1)
| Label | Occurrences |
|---|---|
| JSJ decomposition canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10807795 — 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: JSJ decomposition Context triple: [geometrization conjecture, relatedTo, JSJ decomposition]
-
A.
Dehn algorithm
The Dehn algorithm is a decision procedure in combinatorial group theory that solves the word problem for certain groups by systematically reducing words using defining relations.
-
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.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
-
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’s decision problems in group theory
Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
- 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: JSJ decomposition Target entity description: The JSJ decomposition is a fundamental tool in 3-manifold topology that splits a 3-manifold along tori into simpler, canonical pieces that are either Seifert fibered or atoroidal, forming a key step toward its geometric classification.
-
A.
Dehn algorithm
The Dehn algorithm is a decision procedure in combinatorial group theory that solves the word problem for certain groups by systematically reducing words using defining relations.
-
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.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
-
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’s decision problems in group theory
Dehn’s decision problems in group theory are foundational early 20th-century problems that introduced algorithmic questions about the solvability of word, conjugacy, and isomorphism problems in finitely presented groups, helping launch the field of algorithmic group theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
3-manifold decomposition
ⓘ
tool in 3-manifold topology ⓘ topological construction ⓘ |
| alsoKnownAs | Jaco–Shalen–Johannson decomposition NERFINISHED ⓘ |
| appliesTo |
Haken 3-manifolds
NERFINISHED
ⓘ
compact 3-manifolds ⓘ irreducible 3-manifolds ⓘ orientable 3-manifolds ⓘ |
| assumes |
manifold is irreducible
ⓘ
manifold is sufficiently large in the Haken sense ⓘ |
| characterizedBy | uniqueness of the maximal Seifert fibered submanifold ⓘ |
| componentType |
Seifert fibered 3-manifold
ⓘ
acylindrical 3-manifold ⓘ atoroidal 3-manifold ⓘ |
| decomposesAlong |
embedded tori
ⓘ
incompressible tori ⓘ |
| field |
3-manifold topology
ⓘ
geometric topology ⓘ |
| framework | Haken hierarchy ⓘ |
| generalizationOf | torus decomposition of 3-manifolds ⓘ |
| goal | split a 3-manifold into canonical geometric pieces ⓘ |
| groupTheoreticAnalogue | JSJ decomposition of groups ⓘ |
| historicalDevelopment | independently developed by Jaco–Shalen and Johannson in the late 1970s ⓘ |
| implies | each atoroidal piece admits at most one hyperbolic structure in many cases ⓘ |
| influenced | modern 3-manifold theory ⓘ |
| involves |
JSJ tori
ⓘ
maximal family of disjoint incompressible tori ⓘ |
| namedAfter |
Klaus Johannson
NERFINISHED
ⓘ
Peter Shalen NERFINISHED ⓘ William Jaco NERFINISHED ⓘ |
| precedes | geometric decomposition into Thurston model geometries ⓘ |
| producesPieces |
Seifert fibered components
ⓘ
atoroidal components ⓘ |
| property |
canonical up to isotopy
ⓘ
unique up to isotopy of the tori ⓘ |
| refines | prime decomposition of 3-manifolds ⓘ |
| relatedTo |
Perelman’s proof of geometrization
ⓘ
Thurston geometrization conjecture NERFINISHED ⓘ prime decomposition of 3-manifolds ⓘ |
| requires |
incompressible surface theory
ⓘ
normal surface theory ⓘ |
| studiedIn | low-dimensional topology ⓘ |
| typicalOutput | graph of groups decomposition of the fundamental group ⓘ |
| usedFor |
geometric classification of 3-manifolds
ⓘ
identifying Seifert fibered pieces in a 3-manifold ⓘ isolating hyperbolic pieces in a 3-manifold ⓘ preparing 3-manifolds for Thurston geometrization ⓘ understanding the structure of Haken 3-manifolds ⓘ |
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: JSJ decomposition Description of subject: The JSJ decomposition is a fundamental tool in 3-manifold topology that splits a 3-manifold along tori into simpler, canonical pieces that are either Seifert fibered or atoroidal, forming a key step toward its geometric classification.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.