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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(2) these representations (denoted V A(Y )) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4rth 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 (Y ) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of (Y ). (Note that (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 (Y ) there is an r0(h) such that if r r0(h) and A is a primitive 4rth root of unity then h acts projectively nontrivially on V A(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 hid Bn admits a cabling c = c1,,cn so that ρN(c(h))id, N = c1 + + cn.

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