JSJ decomposition

E888035

3-manifold decomposition tool in 3-manifold topology topological construction

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.

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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 ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

geometrization conjecture → relatedTo → JSJ decomposition ⓘ