## Geometry & Topology

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

#### 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 $4r$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 $h∈ℳ(Y)$ there is an $r0(h)$ such that if $r≥r0(h)$ and $A$ is a primitive $4r$th 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 $h≠id∈Bn$ 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
Accepted: 19 November 2002
First available in Project Euclid: 21 December 2017

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

#### 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

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