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We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for the S ±1 -matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projective SL(2, Z)-action on the center of U q (sl 2 ) for q an l = 2m + 1-st root of unity. It appears that the 3m + 1-dimensional representation decomposes into an m + 1-dimensional finite representation and a 2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of SL(2, Z) and the finite, m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category of U q (sl 2 ) .

We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachments. 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 groups, 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 categorical 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. Academic PressContents

Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object Int H is invertible. The fully braided version of Radford's formula for the fourth power of the antipode is obtained. Connections of integration with cross-product and transmutation are studied.