About: Graph manifold

In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered 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. One of the numerous consequences of the Thurston-Perelman geometrization theorem is that graph manifolds are precisely the 3-manifolds whose Gromov norm vanishes.