Graph manifold

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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.

[edit] Reference

Friedhelm Waldhausen, "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten", Invent. Math. 3, 4 (1967).