## On central extensions of mapping class groups (1995)

Venue: | MATH. ANNALLEN |

Citations: | 22 - 2 self |

### BibTeX

@ARTICLE{Masbaum95oncentral,

author = {G. Masbaum and J. D. Roberts},

title = {On central extensions of mapping class groups},

journal = {MATH. ANNALLEN},

year = {1995},

volume = {302},

pages = {131--150}

}

### OpenURL

### Abstract

### Citations

632 |
Quantum field theory and the Jones polynomial
- Witten
- 1989
(Show Context)
Citation Context ... : 20 5.3 The relationship with the TQFT-functors of [6] : : : : : : : : : : 21 References 22 1 Introduction The mapping class group representations related to quantum invariants (`a la Jones, Witten =-=[35]-=-, Reshetikhin-Turaev [26], Lickorish[17, 18], Blanchet et al. [5], Wenzl [34], Turaev-Wenzl [31], and probably many more) are naturally only projective representations, as argued by Atiyah in [3]. A p... |

282 |
Invariants of 3-manifolds via link polynomials and quantum groups
- Turaev, Reshetikhin
- 1991
(Show Context)
Citation Context ...p with the TQFT-functors of [6] : : : : : : : : : : 21 References 22 1 Introduction The mapping class group representations related to quantum invariants (`a la Jones, Witten [35], Reshetikhin-Turaev =-=[26]-=-, Lickorish[17, 18], Blanchet et al. [5], Wenzl [34], Turaev-Wenzl [31], and probably many more) are naturally only projective representations, as argued by Atiyah in [3]. A particularly nice way to o... |

172 |
States models and the Jones polynomial, Topology 26
- Kauffman
- 1987
(Show Context)
Citation Context ...tral extension by a cyclic group 1The skein module S(M ) of a compact oriented 3-manifold M is the Z[A; A\Gamma 1]\Gamma module generated by isotopy classes of banded links in M , modulo the Kauffman =-=[14]-=- bracket relations (see for example [18] where S(T ) is denoted by A.) The Kauffman relations imply that S(S3) ss Z[A; A \Gamma 1]; the isomorphism is called the Kauffman bracket and denoted by ! ?. I... |

126 |
Topological quantum field theories
- Atiyah
- 1988
(Show Context)
Citation Context ..., the corrected action baep is quite natural from the point of view of the TQFT-functors Vp of [6]. (An introduction to the concept of Topological Quantum Field Theories (TQFT) can be found in Atiyah =-=[2]-=-.) Here is the precise relationship. Consider a closed surface \Sigmasequipped with the fixed p1-structure ,0. The TQFT-functor Vp associates to \Sigmasa module Vp(\Sigma ) (which is free of finite ra... |

126 | Racks and links in codimension two - Fenn, Rourke - 1992 |

112 |
Topological quantum field theories derived from the Kauffman bracket, Topology 34
- Blanchet, Habegger, et al.
- 1995
(Show Context)
Citation Context ...ks 20 5.1 Cocycles cohomologous to the signature cocycle : : : : : : : : : : 20 5.2 Three-manifold invariants without Kirby's theorem : : : : : : : : 20 5.3 The relationship with the TQFT-functors of =-=[6]-=- : : : : : : : : : : 21 References 22 1 Introduction The mapping class group representations related to quantum invariants (`a la Jones, Witten [35], Reshetikhin-Turaev [26], Lickorish[17, 18], Blanch... |

95 |
A presentation for the mapping class group of a closed orientable surface, Topology 19
- Hatcher, Thurston
- 1980
(Show Context)
Citation Context ... is divisible by 4). 3.4 A presentation of the mapping class group The first explicit presentation of \Gammasfor all genus was written down by Wajnryb [32], following the work of Hatcher and Thurston =-=[13]-=- and Harer [11]. For our purposes, the following (infinite) presentation, implicit in Harer's work and easily derived from Wajnryb's (finite) presentation, is sufficient. Let Cns ae C be the set of is... |

90 |
The second homology group of the mapping class group of an oriented surface
- Harer
- 1983
(Show Context)
Citation Context ...f the original geometric action to Dehn twists defines an extension e\Gamma 1 which is of index two in e\Gamma 2. The presentation of e\Gamma 1 induced by the geometric action looks just like Harer's =-=[11]-=- presentation of the mapping class group except that one relator is omitted. This presentation is given in theorem 3.8. It is nice to observe that this is a straightforward generalisation of the genus... |

56 |
Quantum invariants of 3-manifolds associated with classical simple Lie algebras
- Turaev, Wenzl
(Show Context)
Citation Context ...Introduction The mapping class group representations related to quantum invariants (`a la Jones, Witten [35], Reshetikhin-Turaev [26], Lickorish[17, 18], Blanchet et al. [5], Wenzl [34], Turaev-Wenzl =-=[31]-=-, and probably many more) are naturally only projective representations, as argued by Atiyah in [3]. A particularly nice way to observe this is through the approach from the skein theory of the Kauffm... |

54 |
A simple presentation for the mapping class group of an orientable surface
- Wajnryb
- 1983
(Show Context)
Citation Context ...gnature of a surface bundle over a closed surface is divisible by 4). 3.4 A presentation of the mapping class group The first explicit presentation of \Gammasfor all genus was written down by Wajnryb =-=[32]-=-, following the work of Hatcher and Thurston [13] and Harer [11]. For our purposes, the following (infinite) presentation, implicit in Harer's work and easily derived from Wajnryb's (finite) presentat... |

45 | Three-manifold invariants derived from the Kauffman bracket, Topology 31 - Blanchet, Habegger, et al. - 1992 |

37 |
Computer calculation of Wittenâ€™s 3-manifold invariant
- Freed
- 1991
(Show Context)
Citation Context ...ss ss extends to the solid torus T , and its geometric action on the skein module S(T ), hence on Vp, is indeed the identity.) Note: This renormalised action of P Sl(2; Z) was used by Freed and Gompf =-=[8]-=- to compute the Witten [35] invariant for certain Seifert manifolds. Here is the precise relationship: Assume p = 2k + 4 is even. Put i2p = \Gamma e2ssi=2p and ^p = e2ssic=24 where c = 3k=(k + 2) is t... |

36 | On Witten's 3-Manifold Invariants, (preprint - Walker - 1991 |

28 |
Topological invariants for 3-manifolds using representations of mapping class groups I, Kyushu Univ
- Kohno
- 1990
(Show Context)
Citation Context ...heory of the Kauffman bracket, where a simple existence proof of projective actions was given in Roberts [27]. (Related projective actions had previously been constructed by other authors, e.g. Kohno =-=[16]-=-, using other methods.) The aim of this paper is to describe the central extensions of the mapping class group generated by these projective actions, and to see how the signature cocycle arises. Fix a... |

26 |
The cohomology of the moduli space of curves
- Harer
(Show Context)
Citation Context ...the signature cocycle. Remark 3.11 It is known that the group H2(\Gamma ; Z) is cyclic in all genus: in fact H2(\Gamma ; Z) is Z in genus at least three, Z=12 in genus one, and Z=10 in genus two, see =-=[11, 12]-=-. The fact that the signature cocycle represents four times a generator (in all genus) is well known (Meyer [23], p. 240.) Proof of theorem 3.10. The first statement follows from theorem 3.3 by standa... |

20 |
On framings of 3-manifolds. Topology 29
- Atiyah
- 1990
(Show Context)
Citation Context ...ten [35], Reshetikhin-Turaev [26], Lickorish[17, 18], Blanchet et al. [5], Wenzl [34], Turaev-Wenzl [31], and probably many more) are naturally only projective representations, as argued by Atiyah in =-=[3]-=-. A particularly nice way to observe this is through the approach from the skein theory of the Kauffman bracket, where a simple existence proof of projective actions was given in Roberts [27]. (Relate... |

17 |
Invariants of links and 3-manifolds from skein theory and from quantum groups
- Morton
- 1993
(Show Context)
Citation Context ...lement in \Sigmas\ThetasI obtained by cabling ff(\Upsilon ) by !p. Remark 2.5 This result is well known. The skein element !p is up to scalar multiples the one used by many authors (e.g. [17, 18] [5] =-=[24]-=-) to construct 3-manifold invariants. The precise normalisation of !p is as in [6], section 2. Formula (7) in genus one is an equation which determines !p completely (not as a skein element, but as an... |

15 |
Skeins and mapping class groups
- Roberts
- 1994
(Show Context)
Citation Context ...Atiyah in [3]. A particularly nice way to observe this is through the approach from the skein theory of the Kauffman bracket, where a simple existence proof of projective actions was given in Roberts =-=[27]-=-. (Related projective actions had previously been constructed by other authors, e.g. Kohno [16], using other methods.) The aim of this paper is to describe the central extensions of the mapping class ... |

13 |
Calculations with the Temperley-Lieb algebra
- Lickorish
- 1992
(Show Context)
Citation Context ...-functors of [6] : : : : : : : : : : 21 References 22 1 Introduction The mapping class group representations related to quantum invariants (`a la Jones, Witten [35], Reshetikhin-Turaev [26], Lickorish=-=[17, 18]-=-, Blanchet et al. [5], Wenzl [34], Turaev-Wenzl [31], and probably many more) are naturally only projective representations, as argued by Atiyah in [3]. A particularly nice way to observe this is thro... |

10 |
Three-manifolds and the Temperley-Lieb algebra
- Lickorish
- 1991
(Show Context)
Citation Context ...-functors of [6] : : : : : : : : : : 21 References 22 1 Introduction The mapping class group representations related to quantum invariants (`a la Jones, Witten [35], Reshetikhin-Turaev [26], Lickorish=-=[17, 18]-=-, Blanchet et al. [5], Wenzl [34], Turaev-Wenzl [31], and probably many more) are naturally only projective representations, as argued by Atiyah in [3]. A particularly nice way to observe this is thro... |

6 | Verlinde Formulae for Surfaces with Spin Structure - Masbaum, Vogel - 1994 |

4 |
A simple proof of the fundamental theorem of Kirby calculus of links
- Lu
- 1992
(Show Context)
Citation Context ...n of Ip(M ), and apply the methods of [27] to prove its topological invariance, without using Kirby's theorem. (One needs to use a presentation of \Gamma , and the Suzuki generators [30] (see also Lu =-=[20, 21]-=-) of the handlebody groups). Note: While aep is defined over Q(i2p), the corrected action baep : e\Gamma 4 ! Gl(Vp(g)) depends on a choice of sign for ^3p. (Only ^6p, which is a power of the skein var... |

2 |
The Logarithm of the Dedekind j-Function
- Atiyah
- 1987
(Show Context)
Citation Context ...case of interest here. (Compare this with the case of genus one, where the signature cocycle is the coboundary of a unique (rational) cochain which is almost the Rademacher OE-function (see e.g. [23] =-=[1]-=- [4] [15]).) 5.2 Three-manifold invariants without Kirby's theorem Let f 2 \Gammasbe represented by a word wf 2 F ree(D), and let M = Mf have Heegaard splitting (\Sigma ; f ) as in subsection 2.3. In ... |

2 |
Dedekind sums, mu invariants and the signature cocycle, preprint
- Kirby, Melvin
- 1992
(Show Context)
Citation Context ...interest here. (Compare this with the case of genus one, where the signature cocycle is the coboundary of a unique (rational) cochain which is almost the Rademacher OE-function (see e.g. [23] [1] [4] =-=[15]-=-).) 5.2 Three-manifold invariants without Kirby's theorem Let f 2 \Gammasbe represented by a word wf 2 F ree(D), and let M = Mf have Heegaard splitting (\Sigma ; f ) as in subsection 2.3. In [27], the... |

1 |
Presentations and extensions of mapping class groups, Nantes Preprint
- Th`ese
- 1994
(Show Context)
Citation Context ...ts existence shows that \Gammasis perfect in genus * 3, and (hence) has a universal central extension. According to Harer [11], p. 238, the latter is presented as ! Dns j Rconj [ fdg ?. Note. Gervais =-=[9, 10]-=- has shown that one may replace Rconj by the subset of those conjugation relators rff;fi where the curves ff and fi intersect in at most one point. 3.5 Proof of proposition 3.4 (ii) Here is the proof ... |

1 |
Homeomorphisms of a handlebody and Heegaard splittings of the 3\Gamma sphere
- Lu
- 1988
(Show Context)
Citation Context ...n of Ip(M ), and apply the methods of [27] to prove its topological invariance, without using Kirby's theorem. (One needs to use a presentation of \Gamma , and the Suzuki generators [30] (see also Lu =-=[20, 21]-=-) of the handlebody groups). Note: While aep is defined over Q(i2p), the corrected action baep : e\Gamma 4 ! Gl(Vp(g)) depends on a choice of sign for ^3p. (Only ^6p, which is a power of the skein var... |

1 |
Die Signatur von Fl"achenb"undeln
- Meyer
- 1973
(Show Context)
Citation Context ...section 3.5, where we will make use of the presentation of the mapping class group which we have so far avoided. Note. It is possible to prove (ii) without using a presentation (in fact using Meyer's =-=[23]-=- result that the signature of a surface bundle over a closed surface is divisible by 4). 3.4 A presentation of the mapping class group The first explicit presentation of \Gammasfor all genus was writt... |

1 | Satellites and surgery invariants, in "Knots 90 - Morton, Strickland - 1992 |

1 | On homeomorphisms of a 3\Gamma dimensional handlebody, Canad - Thesis, Cambridge - 1994 |