Math 2.0

Manifolds and cell complexes Three-manifolds Theorem

JSJ decomposition

Builds on 3-manifoldOrientabilityIncompressible surfaceSeifert fiber spaceSatellite knot

Statement

Let MM be a closed connected orientable irreducible 3-manifold, where irreducible means that every embedded 22-sphere bounds a 33-ball. A compact 33-manifold NN is atoroidal if every incompressible torus in NN is parallel to a boundary torus of NN. Then MM contains a finite collection 𝒯\mathcal{T} of disjoint embedded incompressible tori with the following property. Each component of MM cut along 𝒯\mathcal{T} is atoroidal or a Seifert fiber space. A minimal collection 𝒯\mathcal{T} with this property is unique up to isotopy.

Remarks

The theorem is due to Jaco and Shalen, and independently to Johannson. Existence rests on Haken's finiteness theorem, which bounds the number of disjoint, pairwise non-parallel incompressible surfaces. Uniqueness comes from the study of how incompressible tori meet Seifert fibered pieces. A version of the theorem holds for compact manifolds whose boundary is a union of tori. For example, take a satellite knot in S3S^3. The boundary of a neighborhood of the companion knot is an incompressible torus in the knot complement, so the complement is not atoroidal. If MM is itself Seifert fibered or atoroidal, the minimal collection is empty.

Going deeper

For a student

The JSJ decomposition is the second step of geometrization, after the prime decomposition. By the geometrization theorem, the atoroidal pieces that are not Seifert fibered have hyperbolic interiors. So every closed orientable irreducible 33-manifold is assembled from Seifert fibered and hyperbolic pieces glued along tori.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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3-manifold, concept, no evidence level3-manifoldconceptIncompressible surface, concept, no evidence levelIncompressible surfaceconceptOrientability, concept, no evidence levelOrientabilityconceptSatellite knot, concept, no evidence levelSatellite knotconceptSeifert fiber space, concept, no evidence levelSeifert fiber spaceconceptJSJ decomposition, statement, no evidence levelJSJ decompositionstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. 3-manifold concept
  2. Incompressible surface concept
  3. Orientability concept
  4. Satellite knot concept
  5. Seifert fiber space concept
  6. JSJ decomposition statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/JSJ_decomposition
About this page
TagTX4MSZ0SRP
URIm2://TX4MSZ0SRP
Kindstatement, theorem
WritingW0 stub
MSC57K30
StatusThis copy of the site has no build data. Run m2 status TX4MSZ0SRP.
About3-manifold
Uses3-manifold, Orientability, Incompressible surface, Seifert fiber space, Satellite knot
Same as
Changed2 November 2026

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