Geometry & Topology, Vol. 6 (2002) Paper no. 18, pages 523--539.

Quantum $SU(2)$ faithfully detects mapping class groups modulo center

Michael H Freedman, Kevin Walker and Zhenghan Wang

Abstract. The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted V_A(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4r-th root of unity (r=k+2). In each of these representations (as well as the general G case), Dehn twists act as transformations of finite order, so none represents the mapping class group M(Y) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of M(Y). (Note that M(Y) has non-trivial center only if Y is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus = 2.) Specifically, for a non-central h in M(Y) there is an r_0(h) such that if r>= r_0(h) and A is a primitive 4r-th root of unity then h acts projectively nontrivially on V_A(Y). Jones' [J] original representation rho_n of the braid groups B_n, sometimes called the generic q-analog-SU(2)-representation, is not known to be faithful. However, we show that any braid h not= id in B_n admits a cabling c = c_1,...,c_n so that rho_N (c(h)) not= id, N=c_1 + ... + c_n.

Keywords. Quantum invariants, Jones-Witten theory, mapping class groups

AMS subject classification. Primary: 57R56, 57M27. Secondary: 14N35, 22E46, 53D45.

DOI: 10.2140/gt.2002.6.523

E-print: arXiv:math.GT/0209150

Submitted to GT on 14 September 2002. Paper accepted 19 November 2002. Paper published 23 November 2002.
Minor correction made 17 July 2003:   Sentence added to proof of lemma 4.1, page 536, lines 7-8.

Notes on file formats

Michael H Freedman, Kevin Walker and Zhenghan Wang
Microsoft Research, Redmond, WA 98052, USA (MHF and KW)
Indiana University, Department of Mathematics
Bloomington, IN 47405, USA


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