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

Freedman, Michael H¹ ; Walker, Kevin¹ ; Wang, Zhenghan²

¹ Microsoft Research, Redmond, Washington 98052, USA
² Department of Mathematics, Indiana University, Bloomington, Indiana 47045, USA

Geometry & topology, Tome 6 (2002) no. 2, pp. 523-539.

Résumé

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 $G=SU\left(2\right)$ these representations (denoted $V_A\left(Y\right)$) 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 $\mathcal{ℳ}\left(Y\right)$ faithfully. However, taken together, the quantum $SU\left(2\right)$ representations are faithful on non-central elements of $\mathcal{ℳ}\left(Y\right)$. (Note that $\mathcal{ℳ}\left(Y\right)$ 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 $h\in \mathcal{ℳ}\left(Y\right)$ there is an $r_0\left(h\right)$ such that if $r\ge r_0\left(h\right)$ and $A$ is a primitive $4r$th root of unity then $h$ acts projectively nontrivially on $V_A\left(Y\right)$. Jones’ original representation ${\rho }_n$ of the braid groups $B_n$, sometimes called the generic $q$–analog–$SU\left(2\right)$–representation, is not known to be faithful. However, we show that any braid $h\ne id\in B_n$ admits a cabling $c=c_1,\dots ,c_n$ so that ${\rho }_N\left(c\left(h\right)\right)\ne id$, $N=c_1+\dots +c_n$.

Détail

DOI : 10.2140/gt.2002.6.523

Keywords: quantum invariants, Jones–Witten theory, mapping class groups

Bibliographie

[1] J E Andersen, Asymptotic faithfulness of the quantum $\mathrm{SU}(n)$ representations of the mapping class groups, Ann. of Math. $(2)$ 163 (2006) 347

[2] M Atiyah, On framings of 3–manifolds, Topology 29 (1990) 1

[3] S J Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471

[4] S J Bigelow, R D Budney, The mapping class group of a genus two surface is linear, Algebr. Geom. Topol. 1 (2001) 699

[5] C Blanchet, N Habegger, G Masbaum, P Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995) 883

[6] E K Grossman, On the residual finiteness of certain mapping class groups, J. London Math. Soc. $(2)$ 9 (1974/75) 160

[7] A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221

[8] N Ivanov, Mapping class groups, online notes at Michigan State University

[9] V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. $(2)$ 126 (1987) 335

[10] P Köerbe, Kontaktprobleme der Konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.–Phys. Kl. 88 (1936) 141

[11] L H Kauffman, S L Lins, Temperley–Lieb recoupling theory and invariants of 3–manifolds, Annals of Mathematics Studies 134, Princeton University Press (1994)

[12] M Korkmaz, Low-dimensional homology groups of mapping class groups: a survey, Turkish J. Math. 26 (2002) 101

[13] W B R Lickorish, A finite set of generators for the homeotopy group of a 2–manifold, Proc. Cambridge Philos. Soc. 60 (1964) 769

[14] V G Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter Co. (1994)

[15] B Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983) 157

[16] K Walker, On Witten's 3–manifold invariants, preprint (1991)

[17] E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351