A diagrammatic presentation of the category 3Cob (Q6561357)

The authors consider the category \(3\)Cob of \(3\)-dimensional cobordisms (objects are closed oriented surfaces and \(3\)-dimensional cobordisms are morphisms in \(3\)Cob), and give a presentation of \(3\)Cob via planar diagrams. This presentation is complete in the sense that: (1) every morphism is presented by a diagram; (2) an algorithm for deciding whether two diagrams present the same morphism is defined; and (3) a rule for attaching two diagrams that correspond to composition of morphisms is established. The correctness of the presentation is proved in the paper.\N\NThe diagrams consist of knotted and linked circles and wedges of circles. In order to achieve property (2) for their presentation the authors develop a calculus of Kirby type by defining a set of moves which transform one diagram to another. The rule (3) is stated as the main topic of the paper. It relies on the surgery theory of manifolds.\N\NThe paper abounds with colourful illustrations and examples, which make it easier and more pleasant to read.