JSJ decomposition
In mathematics, the JSJ decomposition , also known as the toral decomposition , is a topological construct given by the following theorem: Irreducible orientable closed (i.e., compact and without boundary) 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. Contents 1 The characteristic submanifold 2 See also 3 References 4 External links 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 (possibly disconnected and with boundary) whose complement is atoroidal (and possibly disconnected). The submanifold Σ with the