mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the JSJ decomposition, also known as the toral decomposition, is a

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

construct given by the following theorem: :

Irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

closed (i.e., compact and without boundary)

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

s have a unique (up to isotopy) minimal collection of disjointly embedded

incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...

tori such that each component of the 3-manifold obtained by cutting along the tori is either

atoroidal In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non- boundary parallel, incompressible t ...

Seifert-fibered A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for a ...

. The acronym JSJ is for

William Jaco William "Bus" H. Jaco (born July 14, 1940 in Grafton, West Virginia) is an American mathematician who is known for his role in the Jaco–Shalen–Johannson decomposition theorem and is currently Regents Professor and Grayce B. Kerr Chair at Okl ...

Peter Shalen Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition. Life He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard Colle ...

, and

Klaus Johannson Klaus is a German, Dutch and Scandinavian given name and surname. It originated as a short form of Nikolaus, a German form of the Greek given name Nicholas. Notable persons whose family name is Klaus * Billy Klaus (1928–2006), American baseb ...

. 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 A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for a ...

(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) minimal collection of disjointly embedded

tori such that ''closure'' of each component of the 3-manifold obtained by cutting along the tori is either

. The JSJ decomposition is not quite the same as the decomposition in the

geometrization conjecture In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensi ...

, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. For example, the

mapping torus In mathematics, the mapping torus in topology of a homeomorphism ''f'' of some topological space ''X'' to itself is a particular geometric construction with ''f''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the boun ...

of an

Anosov map In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "c ...

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.

References

*. *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.

External links

Allen Hatcher Allen, Allen's or Allens may refer to: Buildings * Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee * Allen Center, a skyscraper complex in downtown Houston, Texas * Allen Fieldhouse, an indoor sports arena on the Univer ...

''Notes on Basic 3-Manifold Topology''
*

JSJ Decomposition of 3-manifolds
{Dead link, date=January 2020 , bot=InternetArchiveBot , fix-attempted=yes . This lecture gives a brief introduction to Seifert fibered 3-manifolds and provides the existence and uniqueness theorem of Jaco, Shalen, and Johannson for the JSJ decomposition of a 3-manifold. *William Jaco
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. 3-manifolds

The characteristic submanifold

See also

References

External links