#### Volume 6, issue 2 (2002)

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

### Michael H Freedman, Kevin Walker and Zhenghan Wang

Geometry & Topology 6 (2002) 523–539
 arXiv: math.GT/0209150
##### 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 $G=SU\left(2\right)$ 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 $\mathsc{ℳ}\left(Y\right)$ faithfully. However, taken together, the quantum $SU\left(2\right)$ representations are faithful on non-central elements of $\mathsc{ℳ}\left(Y\right)$. (Note that $\mathsc{ℳ}\left(Y\right)$ 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 $h\in \mathsc{ℳ}\left(Y\right)$ 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 ${\rho }_{N}\left(c\left(h\right)\right)\ne id$, $N={c}_{1}+\dots +{c}_{n}$.

##### Keywords
quantum invariants, Jones–Witten theory, mapping class groups
##### Mathematical Subject Classification 2000
Primary: 57R56, 57M27
Secondary: 14N35, 22E46, 53D45