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

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 $G = S U (2)$ these representations (denoted $V_{A} (Y)$ ) have a particularly simple description in terms of the Kauffman skein modules with parameter $A$ a primitive $4 r$ 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 $S U (2)$ 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 $r \geq r_{0} (h)$ and $A$ is a primitive $4 r$ th root of unity then $h$ acts projectively nontrivially on $V_{A} (Y)$ . Jones’ original representation $ρ_{n}$ of the braid groups $B_{n}$ , sometimes called the generic $q$ –analog– $S U (2)$ –representation, is not known to be faithful. However, we show that any braid $h \neq id \in B_{n}$ admits a cabling $c = c_{1}, \dots, c_{n}$ so that $ρ_{N} (c (h)) \neq id$ , $N = c_{1} + \dots + c_{n}$ .

Keywords

quantum invariants, Jones–Witten theory, mapping class groups

Mathematical Subject Classification 2000

Primary: 57R56, 57M27

Secondary: 14N35, 22E46, 53D45

References

Forward citations

Publication

Received: 14 September 2002

Accepted: 19 November 2002

Published: 23 November 2002

Corrected: 17 July 2003 (minor corrections)

Proposed: Robion Kirby

Seconded: Joan Birman, Vaughan Jones

Authors

Michael H Freedman
Microsoft Research Redmond Washington 98052 USA

Kevin Walker
Microsoft Research Redmond Washington 98052 USA

Zhenghan Wang
Department of Mathematics Indiana University Bloomington Indiana 47045 USA

Volume 6, issue 2 (2002)

Michael H Freedman, Kevin Walker and Zhenghan Wang