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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Y</mi> </math>$ indexed by a semi-simple Lie group $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math>$ and a level $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi></math>$ . In the case $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi> <mo class="MathClass-rel">=</mo> <mi>S</mi><mi>U</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ these representations (denoted $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>V</mi> </mrow><mrow><mi>A</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> </mrow><mo class="MathClass-close">)</mo></mrow></math>$ ) have a particularly simple description in terms of the Kauffman skein modules with parameter $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>$ a primitive $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn><mi>r</mi></math>$ th root of unity ( $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi> <mo class="MathClass-rel">=</mo> <mi>k</mi> <mo class="MathClass-bin">+</mo> <mn>2</mn></math>$ ). In each of these representations (as well as the general $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math>$ case), Dehn twists act as transformations of finite order, so none represents the mapping class group $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">ℳ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> </mrow><mo class="MathClass-close">)</mo></mrow></math>$ faithfully. However, taken together, the quantum $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ representations are faithful on non-central elements of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">ℳ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> </mrow><mo class="MathClass-close">)</mo></mrow></math>$ . (Note that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">ℳ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> </mrow><mo class="MathClass-close">)</mo></mrow></math>$ has non-trivial center only if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Y</mi> </math>$ is a sphere with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn><mo class="MathClass-punc">,</mo><mn>1</mn><mo class="MathClass-punc">,</mo></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>$ punctures, a torus with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn><mo class="MathClass-punc">,</mo><mn>1</mn><mo class="MathClass-punc">,</mo></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn></math>$ punctures, or the closed surface of genus $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mo class="MathClass-rel">=</mo> <mn>2</mn></math>$ .) Specifically, for a non-central $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi> <mo class="MathClass-rel">∈</mo><mi mathvariant="bold-script">ℳ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> </mrow><mo class="MathClass-close">)</mo></mrow></math>$ there is an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>h</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ such that if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi> <mo class="MathClass-rel">≥</mo> <msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>h</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>$ is a primitive $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn><mi>r</mi></math>$ th root of unity then $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math>$ acts projectively nontrivially on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>V</mi> </mrow><mrow><mi>A</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>Y</mi> </mrow><mo class="MathClass-close">)</mo></mrow></math>$ . Jones’ original representation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ of the braid groups $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ , sometimes called the generic $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>q</mi></math>$ –analog– $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ –representation, is not known to be faithful. However, we show that any braid $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi><mo class="MathClass-rel">≠</mo><mo class="qopname">id</mo> <mo class="MathClass-rel">∈</mo> <msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ admits a cabling $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi> <mo class="MathClass-rel">=</mo> <msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">,</mo><mo class="MathClass-op">…</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ so that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>N</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>c</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>h</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">)</mo></mrow><mo class="MathClass-rel">≠</mo><mo class="qopname">id</mo></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi> <mo class="MathClass-rel">=</mo> <msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub> <mo class="MathClass-bin">+</mo> <mo class="MathClass-op">…</mo> <mo class="MathClass-bin">+</mo> <msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math>$ .

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