Graph manifold
Encyclopedia
In topology
, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold
which is obtained by gluing some circle bundles. They were invented and classified by the German topologist Friedhelm Waldhausen
in 1967. This definition allows a very convenient combinatorial description as a graph whose vertices are the fundamental parts and (decorated) edges stand for the description of the gluing, hence the name.
A very important class of examples is given by the Seifert bundles. This leads to a more modern definition: a graph manifold is a manifold whose prime summands
have only Seifert pieces in their JSJ decomposition
. Waldhausen's article can be seen as the first breakthrough towards the discovery of JSJ decomposition.
One of the numerous consequences of the Thurston-Perelman geometrization theorem
is that graph manifolds are precisely the 3-manifolds whose Gromov norm
vanishes.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold
3-manifold
In 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...
which is obtained by gluing some circle bundles. They were invented and classified by the German topologist Friedhelm Waldhausen
Friedhelm Waldhausen
Friedhelm Waldhausen is a German mathematician known for his work in algebraic topology.-Academic life:...
in 1967. This definition allows a very convenient combinatorial description as a graph whose vertices are the fundamental parts and (decorated) edges stand for the description of the gluing, hence the name.
A very important class of examples is given by the Seifert bundles. This leads to a more modern definition: a graph manifold is a manifold whose prime summands
Prime decomposition (3-manifold)
In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique collection of prime 3-manifolds....
have only Seifert pieces in their JSJ decomposition
JSJ decomposition
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...
. Waldhausen's article can be seen as the first breakthrough towards the discovery of JSJ decomposition.
One of the numerous consequences of the Thurston-Perelman geometrization theorem
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...
is that graph manifolds are precisely the 3-manifolds whose Gromov norm
Gromov norm
In mathematics, the Gromov norm of a compact oriented n-manifold is a norm on the homology given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle...
vanishes.