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

We prove that the image of the mapping class group by the representations arising in the SU(2)-TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r ̸ = 2,3,4,6 (and r ̸ = 10 for g = 2). In particular it follows that the quotient groups Mg/N (tr) by the normalizer of the r-th power of a Dehn twist t are infinite if g ≥ 3 and r ̸ = 2,3,4,6,8,12. 1. Introduction. Witten [50] constructed a TQFT in dimension 3 using path integrals and afterwards several rigorous constructions arose, like those using the quantum group approach ([39, 25]), the Temperley-Lieb algebra ([30, 31]), the theory based on the Kauffman bracket ([4, 5]) or that obtained from the mapping

Abstract. We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus 3 and p = 5, we still give an explicit basis. 1.

We prove that the 2-groupoid of transformations of rigid structures on surfaces has a finite presentation, establishing a result first conjectured by Moore and Seiberg. We also show that a finite dimensional, unitary, cyclic topological quantum field theory gives rise to a representation of this 2-groupoid.

Dedicated to Rob Kirby on his sixtieth birthday Abstract. Suppose an oriented 3-manifold ˜ M has a free Zd- action with orbit space M. Suppose d is an odd prime. Let r be relatively prime to d. We consider certain Witten-Reshetikhin-Turaev SU(2) invariants wr(M) in Z [ 1

Abstract. We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan’s property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of SL(2, Z) on L 2 (R). As a consequence we obtain a curious analytic fact about the Fourier transform on R which may not have been previously observed.

Abstract. We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of links in L(p, q) as a function of the level r − 2. They are given by 1 √ r times one of p Laurent polynomials evaluated at e 2πi 4pr. The congruence class of r modulo p determines which polynomial is applicable. If p ≡ 0 (mod 4), the meridian of L(p,q) is non-trivial in the skein module but has trivial Witten-Reshetikhin-Turaev SU(2) invariants. On the other hand, we show that one may recover the element in the Kauffman bracket skein module of L(p, q) represented by a link from the collection of the WRT invariants at all levels if p is a prime or twice an odd prime. By a more delicate argument, this is also shown to be true for p = 9. This version: 2/16/98, First Version:1/28/98 We may consider the Witten-Reshetikhin-Turaev [W,RT] SU(2) invariants

Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f: weak f-congruence, f-congruence, and strong f-congruence. If f is odd, weak f-congruence preserves the ring structure on cohomology with Zf-coefficients. We show that strong f-congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum SU(2) invariants are well-behaved under this congruence. We strengthen this result and extend it to the SO(3) quantum invariants. We also obtain some corresponding results for the coarser equivalence relations, and for quantum invariants associated to more general modular categories. We compare S 3, the Poincaré homology sphere, the Brieskorn homology sphere Σ(2, 3, 7) and their mirror images up to strong f-congruence. We distinguish the weak f-congruence classes of some manifolds with the same Zf-cohomology ring structure. 57M99; 57R56

We show that central extensions of the mapping class group Mg (or M 1 g) by Z are residually finite. Further we give rough estimates of the largest N = Ng such that homomorphisms from Mg to SU(N) have finite image. In particular, homomorphisms of Mg into PGL ( √ g + 1, C) have finite image, for g ≥ 3. Both results come from properties of quantum representations of mapping class groups.

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