Bridged Links and Tangle Presentations of Cobordism Categories

Abstract

We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical derivation of presentations of 2+1-dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the Matveev-Polyak presentations of the mapping class group, and, furthermore, find systematically surgery presentations of general mapping tori. We discuss a natural extension of the Reshetikhin Turaev invariant to the calculus of bridged links. Invariance follows now - similar as for knot invariants - from simple identifications of the elementary moves with elementary categorial relations for invariances or cointegrals, respectively. Hence, we avoid the lengthy computations and the unnatural Fenn-Rourke reduction of the original proofs. Moreover, we are able to start from a much weaker ``modularity''-condition, which implies the one of Turaev. Generalizations of the presentation to cobordisms of surfaces with boundaries are outlined.