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23
Bridged links and tangle presentations of cobordism categories
 Adv. Math
, 1999
"... Abstract: We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe onehandle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated ..."
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Cited by 25 (5 self)
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Abstract: We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe onehandle 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 nonsimply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3manifold by doing surgery on any other 3manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerftheoretical derivation of presentations of 2+1dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the MatveevPolyak 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 FennRourke reduction of the original
On the TQFT representations of the mapping class groups
 Pacific J. Math
"... 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 rth ..."
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Cited by 18 (5 self)
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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 rth 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 TemperleyLieb algebra ([30, 31]), the theory based on the Kauffman bracket ([4, 5]) or that obtained from the mapping
Integral bases for TQFT modules and unimodular representations of mapping class groups
"... Abstract. We construct integral bases for the SO(3)TQFTmodules 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 ..."
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Cited by 12 (9 self)
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Abstract. We construct integral bases for the SO(3)TQFTmodules 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 nonunimodular. In one such case, genus 3 and p = 5, we still give an explicit basis. 1.
On the groupoid of transformations of rigid structures on surfaces
 J. MATH. SCI. UNIV. TOKYO
, 1999
"... We prove that the 2groupoid 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 2g ..."
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Cited by 12 (6 self)
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We prove that the 2groupoid 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 2groupoid.
Quantum invariants of periodic threemanifolds, Geometry and Topology Monographs 2
, 1999
"... Dedicated to Rob Kirby on his sixtieth birthday Abstract. Suppose an oriented 3manifold ˜ 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 WittenReshetikhinTuraev SU(2) invariants wr(M) in Z [ 1 ..."
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Cited by 5 (1 self)
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Dedicated to Rob Kirby on his sixtieth birthday Abstract. Suppose an oriented 3manifold ˜ 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 WittenReshetikhinTuraev SU(2) invariants wr(M) in Z [ 1
ON THE ASYMPTOTICS OF QUANTUM SU(2) REPRESENTATIONS OF MAPPING CLASS GROUPS.
, 2004
"... 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 c ..."
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Cited by 4 (0 self)
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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.
Skein theory and WittenReshetikhinTuraev Invariants of links in lens spaces
 Comm. Math. Physics
"... Abstract. We study the behavior of the WittenReshetikhinTuraev 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 ..."
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Cited by 4 (2 self)
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Abstract. We study the behavior of the WittenReshetikhinTuraev 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 nontrivial in the skein module but has trivial WittenReshetikhinTuraev 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 WittenReshetikhinTuraev [W,RT] SU(2) invariants
Congruence and Quantum Invariants of 3manifolds
, 2007
"... Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3manifolds generated by Dehn surgery with denominator f: weak fcongruence, fcongruence, and strong fcongruence. If f is odd, weak fcongruence preserves the ring structure on cohomology with ..."
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Cited by 3 (3 self)
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Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3manifolds generated by Dehn surgery with denominator f: weak fcongruence, fcongruence, and strong fcongruence. If f is odd, weak fcongruence preserves the ring structure on cohomology with Zfcoefficients. We show that strong fcongruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum SU(2) invariants are wellbehaved 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 fcongruence. We distinguish the weak fcongruence classes of some manifolds with the same Zfcohomology ring structure. 57M99; 57R56
Integral TQFT for a oneholed torus
"... Abstract. We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the hadic expansion of the TQFTmatrix associated to a mapping class in a straightforward w ..."
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Cited by 3 (1 self)
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Abstract. We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the hadic expansion of the TQFTmatrix associated to a mapping class in a straightforward way. Truncating the hadic expansion gives an approximation of the representation by representations into finite groups. As a special case, we study the induced representations over finite fields and identify them up to isomorphism. The key technical ingredient of the paper are new bases of the Integral TQFT modules which are orthogonal with respect to the Hopf pairing. We construct these orthogonal bases in arbitrary genus, and briefly describe some other applications of them. Contents
Two questions on mapping class groups ∗
, 2010
"... We show that central extensions of the mapping class group Mg of the closed orientable surface of genus 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 SL( [ √ g + ..."
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We show that central extensions of the mapping class group Mg of the closed orientable surface of genus 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 SL( [ √ g + 1], C) have finite image. Both results come from properties of quantum representations of mapping class groups.