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.
- Waldhausen, Friedhelm (1967), "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I", Inventiones Mathematicae 3 (4): 308–333, doi:10.1007/BF01402956, ISSN 0020-9910, MR 0235576
- Waldhausen, Friedhelm (1967), "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. II", Inventiones Mathematicae 4 (2): 87–117, doi:10.1007/BF01425244, ISSN 0020-9910, MR 0235576
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