# JSJ decomposition

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In the mathematical field of [topology](/source/Topology), the **JSJ decomposition**, also known as the **toral decomposition**, is a decomposition of a 3-manifold into a finite number of simpler pieces by cutting along a finite number of embedded [tori](/source/Torus). Each piece is either [atoroidal](/source/Atoroidal) (cannot be cut along an embedded torus in an interesting way) or [Seifert-fibered](/source/Seifert-fibered) (can be decomposed into a disjoint union of circles in a nice way).

The statement of the JSJ decomposition is as follows:

- [Irreducible](/source/Irreducible_(mathematics)) [orientable](/source/Orientability) compact [3-manifolds](/source/3-manifold) have a unique (up to [isotopy](/source/Homotopy)) minimal collection of disjointly [embedded](/source/Embedding) [incompressible](/source/Incompressible_surface) tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.

The acronym JSJ is for [William Jaco](/source/William_Jaco), [Peter Shalen](/source/Peter_Shalen), and [Klaus Johannson](/source/Klaus_Johannson). The first two worked together, and the third worked independently.

## The characteristic submanifold

An alternative version of the JSJ decomposition states:

- A closed irreducible orientable 3-manifold *M* has a submanifold Σ that is a [Seifert manifold](/source/Seifert_manifold) (possibly disconnected and with boundary) whose complement is atoroidal (and possibly disconnected).

The submanifold Σ with the smallest number of boundary tori is called the **characteristic submanifold** of *M*; it is unique (up to isotopy). Cutting the manifold along the tori bounding the characteristic submanifold is also sometimes called a JSJ decomposition, though it may have more tori than the standard JSJ decomposition.

The boundary of the characteristic submanifold Σ is a union of tori that are almost the same as the tori appearing in the JSJ decomposition. However there is a subtle difference: if one of the tori in the JSJ decomposition is "non-separating", then the boundary of the characteristic submanifold has two parallel copies of it (and the region between them is a Seifert manifold isomorphic to the product of a torus and a unit interval). The set of tori bounding the characteristic submanifold can be characterised as the unique (up to [isotopy](/source/Homotopy)) minimal collection of disjointly [embedded](/source/Embedding) [incompressible](/source/Incompressible_surface) [tori](/source/Torus) such that *closure* of each component of the 3-manifold obtained by cutting along the tori is either [atoroidal](/source/Atoroidal) or [Seifert-fibered](/source/Seifert-fibered).

The JSJ decomposition is not quite the same as the decomposition in the [geometrization conjecture](/source/Geometrization_conjecture), because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. For example, the [mapping torus](/source/Mapping_torus) of an [Anosov map](/source/Anosov_map) of a torus has a finite volume sol structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.

## See also

- [Geometrization conjecture](/source/Geometrization_conjecture)
- [Manifold decomposition](/source/Manifold_decomposition)
- [Satellite knot](/source/Satellite_knot)

## References

- Jaco, William H. & Shalen, Peter B (1979), "Seifert fibered spaces in 3-manifolds", *[Memoirs of the American Mathematical Society](/source/Memoirs_of_the_American_Mathematical_Society)*. **21** (220).
- Jaco, William; Shalen, Peter B. *Seifert fibered spaces in 3-manifolds. Geometric topology* (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 91–99, Academic Press, New York-London, 1979.
- Jaco, William; Shalen, Peter B. *A new decomposition theorem for irreducible sufficiently-large 3-manifolds.* Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 71–84, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.
- Johannson, Klaus, *Homotopy equivalences of 3-manifolds with boundaries.* Lecture Notes in Mathematics, 761. Springer, Berlin, 1979. ISBN 3-540-09714-7

## External links

- [Allen Hatcher](/source/Allen_Hatcher), [*Notes on Basic 3-Manifold Topology*](https://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html).
- William Jaco, [An Algorithm to Construct the JSJ Decomposition of a 3-manifold](https://web.archive.org/web/20141020175022/https://www.mathdept.okstate.edu/~jaco/Documents/LectureVB.pdf). An algorithm is given for constructing the JSJ-decomposition of a 3-manifold and deriving the Seifert invariants of the Characteristic submanifold.

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