Graph manifold

In topology, a graph manifold is a 3-manifold which is a connected sum of prime manifolds whose JSJ decomposition has only Seifert fiber spaces and torus bundles. Alternatively, they are the manifolds which can be obtained by gluing together some number of solid tori and 3-punctured spheres cross $S^{1}$ along their boundary.

The name "graph manifold" comes from the JSJ decomposition, which can be conveniently represented as a graph where the vertices are the Seifert-fiber pieces of the decomposition and the edges are tori in the decomposition. To completely describe the manifold, the edges need to be decorated with a rational number describing the gluing of the two pieces.

If the Geometrization conjecture is true, then graph manifolds are the 3-manifolds whose Gromov norm is zero.