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JSJ decomposition
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In [[mathematics]], the '''JSJ decomposition''', also known as the '''toral decomposition''', is a [[topological]] construct given by the following theorem: :[[Irreducible (mathematics)|Irreducible]] [[Orientability|orientable]] closed (i.e., compact and without boundary) [[3-manifold]]s have a unique (up to [[homotopy|isotopy]]) minimal collection of disjointly [[Embedding|embedded]] [[incompressible surface|incompressible]] [[torus|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. ==The characteristic submanifold== An alternative version of the JSJ decomposition states: :A closed irreducible orientable 3-manifold ''M'' has a submanifold Ξ£ that is a [[Seifert manifold]] (possibly disconnected and with boundary) whose complement is atoroidal (and possibly disconnected). The submanifold Ξ£ 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 Ξ£ 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 [[homotopy|isotopy]]) minimal collection of disjointly [[Embedding|embedded]] [[incompressible surface|incompressible]] [[torus|tori]] such that ''closure'' of each component of the 3-manifold obtained by cutting along the tori is either [[atoroidal]] or [[Seifert-fibered]]. The JSJ decomposition is not quite the same as the decomposition in the [[geometrization conjecture]], because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. For example, the [[mapping torus]] of an [[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]] *[[Manifold decomposition]] *[[Satellite knot]] ==References== *{{citation|last1=Jaco|first1= William H.|authorlink1=William Jaco|last2= Shalen|first2= Peter B|authorlink2=Peter Shalen|title= Seifert fibered spaces in 3-manifolds|journal=[[Memoirs of the American Mathematical Society]] |volume= 21 |year=1979|issue= 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]], [http://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html ''Notes on Basic 3-Manifold Topology'']. *William Jaco, [https://web.archive.org/web/20141020175022/https://www.mathdept.okstate.edu/~jaco/Documents/LectureVB.pdf An Algorithm to Construct the JSJ Decomposition of a 3-manifold]. An algorithm is given for constructing the JSJ-decomposition of a 3-manifold and deriving the Seifert invariants of the Characteristic submanifold. [[Category:3-manifolds]]
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