Geometry & Topology

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

Michael H Freedman, Kevin Walker, and Zhenghan Wang

Full-text: Open access

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 VA(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4rth 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 (Y) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of (Y). (Note that (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(Y) there is an r0(h) such that if rr0(h) and A is a primitive 4rth root of unity then h acts projectively nontrivially on VA(Y). Jones’ original representation ρn of the braid groups Bn, sometimes called the generic q–analog–SU(2)–representation, is not known to be faithful. However, we show that any braid hidBn admits a cabling c=c1,,cn so that ρN(c(h))id, N=c1++cn.

Article information

Source
Geom. Topol., Volume 6, Number 2 (2002), 523-539.

Dates
Received: 14 September 2002
Accepted: 19 November 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882933

Digital Object Identifier
doi:10.2140/gt.2002.6.523

Mathematical Reviews number (MathSciNet)
MR1943758

Zentralblatt MATH identifier
1037.57024

Subjects
Primary: 57R56: Topological quantum field theories 57M27: Invariants of knots and 3-manifolds
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 22E46: Semisimple Lie groups and their representations 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]

Keywords
quantum invariants Jones–Witten theory mapping class groups

Citation

Freedman, Michael H; Walker, Kevin; Wang, Zhenghan. Quantum $SU(2)$ faithfully detects mapping class groups modulo center. Geom. Topol. 6 (2002), no. 2, 523--539. doi:10.2140/gt.2002.6.523. https://projecteuclid.org/euclid.gt/1513882933


Export citation

References

  • J Andersen, Asymptotic faithfulness of the quantum SU(n) reppresentations of the mapping class groups.
  • M Atiyah, On framings of $3$–manifolds, Topology 29 (1990) 1–7
  • Stephen J Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471–486
  • Stephen J Bigelow, Ryan D Budney, The mapping class group of a genus two surface is linear, Algebr. Geom. Topol. 1 (2001) 699–708
  • C Blanchet, N Habegger, G Masbaum, P Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995) 883–927
  • Edna K Grossman, On the residual finiteness of certain mapping class groups, J. London Math. Soc. 9 (1974/75) 160–164
  • A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221–237
  • N Ivanov, Mapping class groups, on-line notes at Michigan State University.
  • V F R Jones, Hecke algebra representations of braid groups and link polynomial, Ann. Math. 126 (1987) 335–388
  • Paul Köerbe, Kontaktprobleme der Konformen \Abbildung\, Ber. Sächs. Akad. Wiss. Leipzig, Math.–Phys. Kl. 88 (1936) 141–164
  • L Kauffmann, S Lins, Temperley–Lieb recoupling theory and invariants of 3–manifolds, Ann. Math. Studies, vol 134, Princeton Univ. Press (1994)
  • Mustafa Korkmaz, Low-dimensional homology groups of mapping class groups: a survey, Turkish J. Math. 26 (2002) 101–114
  • W B R Lickorish, A finite set of generators for the homeotopy group of a $2$–manifold, Proc. Cambridge Philos. Soc. 60 (1964) 769–778
  • V Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter Studies in Math. Vol 18 (1994)
  • B Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983) 157–174.
  • K Walker, On Witten's 3–manifold invariants, preprint (1991) available at http://messagetothefish.net/math/
  • E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351–399