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Jaco-Shalen-Johannson torus decompositionThe Jaco-Shalen-Johannson torus decomposition is a topological construct defined as follows:"Irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either atoroidal or Seifert-fibered" See Thurston's conjecture for relevance. 
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Jaco-Shalen-Johannson torus decomposition".