JSJ decomposition
Encyclopedia
In mathematics
, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
The acronym JSJ is for William Jaco
, Peter Shalen
, and Klaus Johannson. The first two worked together, and the third worked independently.
The submanifold Σ with the smallest number of boundary tori is called the characteristic submanifold of M; it is unique (up to isotopy).
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
incompressible
tori
such that closure of each component of the 3-manifold obtained by cutting along the tori is either atoroidal
or Seifert-fibered.
Warning: Cutting the manifold along the tori bounding the characteristic submanifold is also sometimes called the JSJ decomposition, though it may have more tori than the JSJ decomposition defined in the introduction.
Warning: 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
- IrreducibleIrreducible (mathematics)In mathematics, the concept of irreducibility is used in several ways.* In abstract algebra, irreducible can be an abbreviation for irreducible element; for example an irreducible polynomial....
orientableOrientabilityIn mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
closed (i.e., compact and without boundary) 3-manifold3-manifoldIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s have a unique (up to isotopyHomotopyIn topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
) minimal collection of disjointly embeddedEmbeddingIn mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
incompressibleIncompressible surfaceIn mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
toriTorusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
such that each component of the 3-manifold obtained by cutting along the tori is either atoroidalAtoroidalIn mathematics, there are three definitions for atoroidal as applied to 3-manifolds:*A 3-manifold is atoroidal if it does not contain an embedded, non-boundary parallel, incompressible torus....
or Seifert-fibered.
The acronym JSJ is for William Jaco
William Jaco
William "Bus" H. Jaco is the Grace B. Kerr Professor of mathematics at Oklahoma State University. He works on 3-manifolds and discovered the JSJ decomposition....
, Peter Shalen
Peter Shalen
Peter B. Shalen 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 College in 1966 and his Ph.D. from Harvard University in 1972...
, 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 Σ that is a Seifert manifold (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).
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
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
) minimal collection of disjointly embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
incompressible
Incompressible surface
In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
tori
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
such that closure of each component of the 3-manifold obtained by cutting along the tori is either atoroidal
Atoroidal
In mathematics, there are three definitions for atoroidal as applied to 3-manifolds:*A 3-manifold is atoroidal if it does not contain an embedded, non-boundary parallel, incompressible torus....
or Seifert-fibered.
Warning: Cutting the manifold along the tori bounding the characteristic submanifold is also sometimes called the JSJ decomposition, though it may have more tori than the JSJ decomposition defined in the introduction.
Warning: the JSJ decomposition is not quite the same as the decomposition in the geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. For example, the mapping torus
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...
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.
External links
- Allen Hatcher, Notes on Basic 3-Manifold Topology.
- William Jaco JSJ Decomposition of 3-manifolds 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.