JSJ decomposition

In the mathematical field of 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. Each piece is either atoroidal (cannot be cut along an embedded torus in an interesting way) or 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 orientable compact 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible 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, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently.

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