  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.
