Centrally extended mapping class groups from quantum Teichmuller theory
aa r X i v : . [ m a t h . G T ] O c t Centrally extended mapping class groups from quantumTeichm¨uller theory ∗ Louis Funar Rinat M. Kashaev
Institut Fourier BP 74, UMR 5582 Section de Math´ematiques, Case Postale 64University of Grenoble I Universit´e de Gen`eve38402 Saint-Martin-d’H`eres cedex, France 2-4, rue du Li`evre, 1211 Geneve 4, Suissee-mail: [email protected] e-mail: [email protected]
July 31, 2018
Abstract
The central extension of the mapping class groups of punctured surfaces of finite type that arises inquantum Teichm¨uller theory is 12 times the Meyer class plus the Euler classes of the punctures. This isanalogous to the result obtained in [12] for the Thompson groups.2000 MSC Classification: 57M07, 20F36, 20F38, 57N05.Keywords: Mapping class group, Ptolemy groupoid, quantization, Teichm¨uller space, Meyer class, Eulerclass.
Introduction
The quantum theory of Teichm¨uller spaces of punctured surfaces of finite type, originally constructed in [6, 18]and subsequently generalized to higher rank Lie groups and cluster algebras in [10, 11], leads to one parameterfamilies of projective unitary representations of Ptolemy modular groupoids associated to ideal triangulationsof punctured surfaces. We will call such representations (quantum) dilogarithmic representations, since themain ingredient in the theory is Faddeev’s quantum dilogarithm function first introduced in the context ofquantum integrable systems by L.D. Faddeev in [7].These representations are infinite dimensional so that a priori it is not clear if they come from suitable 2 + 1-dimensional topological quantum field theories (TQFT). Nonetheless, it is expected that in the singular limit,when the deformation parameter tends to a root of unity , the ”renormalized” theory corresponds to a finitedimensional TQFT first defined in [16, 17] by using the cyclic representations of the Borel Hopf sub-algebra BU q ( sl (2)), and subsequently developed and generalized in [2]. One can get the same finite dimensionalrepresentations of Ptolemy modular groupoids directly from compact representations of quantum Teichm¨ullertheory at roots of unity [5, 3, 18].Projective representations of a group are well known to be equivalent to representations of central extensionsof the same group by means of the following procedure. To a group G , a C -vector space V and a grouphomomorphism h : G → P GL ( V ) ≃ GL ( V ) / C ∗ , where C ∗ is identified with a (normal) subgroup of GL ( V )through the embedding z z id V , one can associate a central extension e G of G by a sub-group A of C ∗ together with a representation e h : e G → GL ( V ) such that the following diagram is commutative and hasexact rows: 1 / / C ∗ / / GL ( V ) / / P GL ( V ) / / / / A / / ?(cid:31) O O e G / / e h O O G / / h O O ∗ This version: February 2010. L.F. was partially supported by the ANR-06-BLAN-0311 Repsurf and ANR 2011 BS 01 02001 ModGroup. R.M.K. is partially supported by Swiss National Science Foundation. One should distinguish between two different limits, depending on whether log( q )2 πi tends to a positive or a negative rationalnumber. In the case when this limit is a positive rational number, the limit of the representation is non-singular and so it staysinfinite dimensional. e G of the central extension GL ( V ) → P GL ( V ) under the homomorphism G → P GL ( V ), which is canonically defined. However it is possible to find also smaller extensions associatedto proper sub-groups A ⊂ C ∗ . The central extension e G associated to the smallest possible sub-group A ⊂ C ∗ for which there exists a linear representation as in the diagram above resolving the projective representationof G will be called the minimal reduction of e G .In this light, quantum Teichm¨uller theory gives rise to representations of certain central extensions of thesurface mapping class groups which are the vertex groups of the Ptolemy modular groupoids. The maingoal of this paper is to identify the isomorphism classes of those central extensions. Namely, by using thequantization approach of [18], we extend the analysis of the particular case of a once punctured genus threesurface performed in [19] to arbitrary punctured surfaces of finite type.Let a group G with a given presentation be identified as the quotient group F/R , where F is a free groupand R , the normal subgroup generated by the relations. Then, a central extension of G can be obtainedfrom a homomorphism h : F → GL ( V ) with the property h ( R ) ⊂ C ∗ so that it induces a homomorphism h : G → P GL ( V ). In this case, the homomorphism h will be called an almost linear representation of G , inorder to distinguish it from a projective representation.In quantum Teichm¨uller theory, central extensions of surface mapping class groups appear through almostlinear representations. Specifically, let Γ sg,r be the mapping class group of a surface Σ sg,r of genus g with r boundary components and s punctures. These are mapping classes of homeomorphisms which fix the bound-ary point-wise and fix the set of punctures (not necessarily point-wise). Denoting Γ sg = Γ sg, , the projectiverepresentations of Γ sg for (2 g − s ) s >
0, constructed in [18, 19], are almost linear representations cor-responding to certain central extensions f Γ sg . By considering embeddings Σ sg,r ⊂ Σ th, , the central extensions f Γ sg can be used to define central extensions for the mapping class groups Γ sg,r for s ≥
1. According to [24],any embedding Σ sg,r ⊂ Σ th, , for which Σ th \ Σ sg,r contains no disk, punctured disk or cylinder components,induces an embedding of the corresponding mapping class groups. Using this fact, we can define the centralextension g Γ sg,r as the pull-back of the central extension f Γ th by the injective homomorphism Γ sg,r ֒ → Γ th inducedby an embedding of the corresponding surfaces. A priori, it is not clear whether such definition depends ona particular choice of the embedding, but our main result below shows that this is indeed the case.Central extensions by an Abelian group A of a given group G are known to be classified, up to isomorphism,by elements of the 2-cohomology group H ( G ; A ). In the case of surface mapping class groups Γ sg,r , thelatter was first computed by Harer in [15] for g ≥ g ≥ H (Γ sg,r ) = Z s +1 , if g ≥ , where the generators are given by (one fourth of) the Meyer signature class χ (it is the only generator forthe groups H (Γ g ) ∼ = H (Γ g, ) ≃ Z , see [23, 15, 21] for its definition) and s Euler classes e i associated with s punctures. In the case when g = 3, the group H (Γ s ,r ) still contains the sub-group Z s +1 generated by theabove mentioned classes, but it is not known whether there are other (2-torsion) classes. When g = 2 wewill show that H (Γ s ,r ) contains the subgroup Z / Z ⊕ Z s , whose torsion part is generated by χ and whosefree part is generated by the Euler classes. The Universal Coefficients Theorem permits then to compute H ( G ; A ) for every Abelian group A .Denote as above by g Γ sg , r the canonical central extension of Γ sg,r by C ∗ which is obtained as the pull-back ofthe canonical central extension GL ( H ) → P GL ( H ) under the quantum projective representation associatedto a semi-symmetric T in the Hilbert space H (see the next section). Quantum representations depend onsome parameter ζ ∈ C ∗ . Our main result is the following theorem. Theorem 0.1.
The central extension g Γ sg , r can be reduced to a minimal central extension g Γ sg,r of Γ sg,r by acyclic Abelian A ⊂ C ∗ , where A is the subgroup of C ∗ generated by ζ − . Moreover its cohomology class is c g Γ sg,r = 12 χ + s X i =1 e i ∈ H (Γ sg,r ; A ) if g ≥ and s ≥ . Here χ and e i are one fourth of the Meyer signature class and respectively the i -th Eulerclass with A coefficients. There is a geometric interpretation of this extension. 2 orollary 0.2.
Let us consider the extension \ Γ g,r + s of class χ . Then there is an exact sequence: → A s − → \ Γ g,r + s → g Γ sg,r → In some sense the quantum representations of punctured mapping class groups can be lifted to the mappingclass groups of surfaces with boundary obtained by blowing up the punctures.
Corollary 0.3.
The cohomology class of the central extension g Γ sg , r is c g Γ sg , r = 12 χ + s X i =1 e i ∈ H (Γ sg,r ; C ∗ ) if g ≥ and s ≥ . The same formula holds also when g = 2 but the class χ vanishes in H (Γ sg,r ; C ∗ ) . Here χ and e i are one fourth of the Meyer signature class and respectively the i -th Euler class with C ∗ coefficients.Remark . The central extension arising from SU (2)-TQFT with p -structures was computed in [13, 22]for Γ g and it equals 12 χ . It can be shown that their computations extend to the case of punctured surfacesand the associated class for Γ sg,r is 12 χ + P si =1 e i . Our result shows that this extension coincides with thecentral extension arising from quantum Teichm¨uller theory.The organization of the paper is as follows. In Section 1, we review the quantization of the Teichm¨uller spaceof a punctured surface and define the associated quantum representations of the decorated Ptolemy groupoidwhich correspond to linear representations of a central extension of the decorated Ptolemy groupoid. Then,in Section 2, we prove Theorem 0.1 by finding the pull-back of this central extension to the mapping classgroup of the surface. The key idea is to use a Grothendieck type principle. Namely, one can identify a centralextension of the mapping class group of some surface, if one understands its restrictions to the mapping classgroups of sub-surfaces of bounded topological types. The core of the proof consists in computing explicitlythe lifts to the central extension of the decorated Ptolemy groupoid of the relations known to hold in themapping class groups. When properly interpreted, these lifts yield the class of the mapping class groupextension. Acknowledgements
The authors are indebted to Stephane Baseilhac and Vlad Sergiescu for useful discussions and to the refereefor his/her suggestions which helped improving the presentation.
Let Σ = Σ sg,r be an oriented closed surface of genus g with r boundary components and s ≥ r > boundary punctures. When weneed to single out the s punctures lying in the interior we will call them interior punctures. In this paper wewill only consider the situation when each boundary component has exactly one boundary puncture, so thatthere is a total of s + r punctures among which r are boundary punctures. The triangulations of Σ sg,r whosevertices are the s + r punctures will be called ideal triangulations . Then Σ is large if and only if N s > N = 4 g − s + 3 r is the number of triangles in an ideal triangulation. Definition 1.1. A decorated ideal triangulation of Σ is an ideal triangulation τ up to isotopy fixing theboundary, where all triangles are provided with a marked corner, and a bijective ordering map ¯ τ : { , . . . , N } ∋ j ¯ τ j ∈ T ( τ ) is fixed. Here T ( τ ) is the set of all triangles of τ . Graphically, the marked corner of a triangle is indicated by an asterisk and the corresponding number is putinside the triangle. The set of all decorated ideal triangulations of Σ is denoted ∆ Σ .Recall that if a group G freely acts on a set X , then there is an associated groupoid defined as follows. Theobjects are the G -orbits in X , while morphisms are G -orbits in X × X with respect to the diagonal action.Denote by [ x ] the object represented by an element x ∈ X and [ x, y ] the morphism represented by a pair of3lements ( x, y ) ∈ X × X . Two morphisms [ x, y ] and [ u, v ], are composable if and only if [ y ] = [ u ] and theircomposition is [ x, y ][ u, v ] = [ x, gv ], where g ∈ G is the unique element sending u to y . The inverse and theidentity morphisms are given respectively by [ x, y ] − = [ y, x ] and id [ x ] = [ x, x ]. In what follows, products ofthe form [ x , x ][ x , x ] · · · [ x n − , x n ] will be shortened as [ x , x , x , . . . , x n − , x n ].The mapping class group Γ sg,r of Σ acts freely on ∆ Σ . In this case, we let G Σ denote the correspondinggroupoid, called the groupoid of decorated ideal triangulations , or decorated Ptolemy groupoid . This groupoidfirst considered in [18] is an enhanced version of the usual Ptolemy groupoid introduced and studied byPenner in [25] (see also [26]), which arises in the Fock-Goncharov quantization ([10, 11]) of the Teichm¨ullerspace. There is a presentation for G Σ with three types of generators and four types of relations.The generators are of the form [ τ, τ σ ], [ τ, ρ i τ ], and [ τ, ω i,j τ ], where τ σ is obtained from τ by replacing theordering map ¯ τ by the map ¯ τ ◦ σ , where σ ∈ S N is a permutation of the set { , . . . , N } , ρ i τ is obtained from τ by changing the marked corner of triangle ¯ τ i as in Figure 1, and ω i,j τ is obtained from τ by applying theflip transformation in the quadrilateral composed of triangles ¯ τ i and ¯ τ j as in Figure 2. (cid:0)(cid:0)❅❅ r r r ∗ i (cid:0)(cid:0)❅❅ r r r ∗ i ρ i −→ Figure 1: The transformation ρ i . ❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅ rr r r i j ∗ ∗ ❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅ rr r r ij ∗ ∗ ω i,j −→ Figure 2: The transformation ω i,j .There are two sets of relations satisfied by these generators. The first set is as follows:[ τ, τ α , ( τ α ) β ] = [ τ, τ αβ ] , α, β ∈ S N , (1)[ τ, ρ i τ, ρ i ρ i τ, ρ i ρ i ρ i τ ] = id [ τ ] , (2)[ τ, ω ij τ, ω ik ω ij τ, ω jk ω ik ω ij τ ] = [ τ, ω jk τ, ω ij ω jk τ ] , (3)[ τ, ω ij τ, ρ i ω ij τ, ω ji ρ i ω ij τ ] = [ τ, τ ( ij ) , ρ j τ ( ij ) , ρ i ρ j τ ( ij ) ] , (4)where the first two relations are evident, while the other two are shown graphically in Figures 3, 4. ❇❇❇❇❇ ✂✂✂✂✂★★★★ ❝❝❝❝ r rr rr ∗ ∗ ∗ i j k ❇❇❇❇❇ ✂✂✂✂✂★★★★ ❝❝❝❝❩❩❩❩❩❩ r rr rr ∗ ∗ ∗ ij k ❇❇❇❇❇ ✂✂✂✂✂★★★★ ❝❝❝❝❩❩❩❩❩❩ r rr rr ∗ ∗ ∗ ij k ❇❇❇❇❇ ✂✂✂✂✂★★★★ ❝❝❝❝✚✚✚✚✚✚ r rr rr ∗ ∗ ∗ i j k ❇❇❇❇❇ ✂✂✂✂✂★★★★ ❝❝❝❝✚✚✚✚✚✚ r rr rr ∗ ∗ ∗ ij k ց ωj,k ւ ωj,k ω i,j −→ ω i,j −→ ω i,k −→ Figure 3: The Pentagon relation (3).The following commutation relations fulfill the second set of relations:[ τ, ρ i τ, ( ρ i τ ) σ ] = [ τ, τ σ , ρ σ − ( i ) τ σ ] , (5)[ τ, ω ij τ, ( ω ij τ ) σ ] = [ τ, τ σ , ω σ − ( i ) σ − ( i ) τ σ ] , (6)[ τ, ρ j τ, ρ i ρ j τ ] = [ τ, ρ i τ, ρ j ρ i τ ] , (7)[ τ, ρ i τ, ω jk ρ i τ ] = [ τ, ω jk τ, ρ i ω jk τ ] , i
6∈ { j, k } , (8)[ τ, ω ij τ, ω kl ω ij τ ] = [ τ, ω kl τ, ω ij ω kl τ ] , { i, j } ∩ { k, l } = ∅ . (9)4 ❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅ rr r r i j ∗ ∗ ❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅ rr r r ij ∗ ∗ ω i,j −→↓ ( i, j ) ◦ ρi × ρj ρi ↓ ❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅ rr r r j i ∗ ∗ ❅❅ (cid:0)(cid:0)(cid:0)(cid:0) ❅❅ rr r r ij ∗∗ ω j,i ←− Figure 4: The Inversion relation (4).Consider now an embedding of Σ sg,r into Σ th,v sending all punctures (both interior and boundary) to punc-tures. Of course boundary punctures are sent into interior punctures unless the respective boundary circleis also a boundary of the larger surface.
Lemma 1.1.
Assume that each component of Σ th,v \ int(Σ sg,r ) is large. Then there is a natural embeddingof G Σ sg,r into G Σ th,v .Proof. Let τ ext be a fixed decorated triangulation of Σ th,v \ int(Σ sg,r ). If τ is a decorated triangulation of Σ sg,r we denote by τ ∪ τ ext the result of gluing the two triangulations along their corresponding boundary circleswith the induced decoration. The isotopy class of the resulting triangulation is unique up to the action ofDehn twists along boundary components of Σ sg,r . This induces an injective map between the set of objectsof the two groupoids. Then, the map which associates to the class [ τ , τ ] of decorated triangulations of Σ sg,r the class [ τ ∪ τ ext , τ ∪ τ ext ] is well-defined. Since the restriction of a homeomorphism of Σ th,v preservingthe isotopy class of the decorated triangulation τ ext to Σ th,v \ int(Σ sg,r ) is isotopic to identity by Alexander’strick, the map defined above is injective. Remark . When r > G Σ sg,r depends on the choiceof the set of boundary punctures, which might have more than r elements, in general. In what follows, we work with Hilbert spaces
H ≡ L ( R ) , H ⊗ n ≡ L ( R n ) . Any two self-adjoint operators p and q , acting in H and satisfying the Heisenberg commutation relation pq − qp = (2 π i ) − id H , (10)can be realized as differentiation and multiplication operators. Such ”coordinate” realization in Dirac’sbra-ket notation has the form h x | p = 12 π i ∂∂x h x | , h x | q = x h x | . (11)Formally, the set of ”vectors” {| x i} x ∈ R forms a generalized basis of H with the following orthogonality andcompleteness properties: h x | y i = δ ( x − y ) , Z R | x i dx h x | = id H . For any 1 ≤ i ≤ m we shall use the following notation ι i : End H ∋ a a i = 1 ⊗ · · · ⊗ | {z } i − ⊗ a ⊗ ⊗ · · · ⊗ ∈ End H ⊗ m . Besides that, if u ∈ End H ⊗ k for some 1 ≤ k ≤ m and { i , i , . . . , i k } ⊂ { , , . . . , m } , then we shall write u i i ...i ≡ ι i ⊗ ι i ⊗ · · · ⊗ ι i k ( u ) . The symmetric group S m naturally acts in H ⊗ m : P σ ( x ⊗ · · · ⊗ x i ⊗ · · · ⊗ x m ) = x σ − (1) ⊗ · · · ⊗ x σ − ( i ) ⊗ . . . ⊗ x σ − ( m ) , σ ∈ S m . (12)5 .3 Semi-symmetric T -matrices We define now the algebraic structure needed for constructing representations of the decorated Ptolemygroupoid G Σ . Definition 1.2. A semi-symmetric T -matrix consists of two operators A ∈ End( H ) and T ∈ End( H ⊗ ) satisfying the equations: A = 1 , (13) T T T = T T , (14) T A T = ζ A A P (12) , (15) where ζ ∈ C ∗ and the permutation operator P (12) is defined by equation (12) , for σ denoting the transposition (12) . Examples of semi-symmetric T -matrices could be obtained as follows. Fix some self-conjugate operators p , q satisfying the Heisenberg commutation relation(10). Choose a parameter b satisfying the condition:(1 − | b | )Im b = 0 , and define then two unitary operators by the following formulas: A ≡ e − i π/ e i π q e i π ( p + q ) ∈ End( H ) , (16) T ≡ e i π p q ϕ b ( q + p − q ) ∈ End( H ⊗ ) . (17)They satisfy the defining relations for a semi-symmetric T -matrix, where ζ = e i πc b / , c b = i b + b − ) , (18)and ϕ b is Faddeev’s quantum dilogarithm defined on { z ∈ C ; | Im(z) | < | Im(c b ) |} by means of ϕ b ( z ) = exp (cid:18) − Z ∞−∞ exp( − izx ) d x sinh( xb ) sinh( x/b ) x (cid:19) (19)Faddeev’s quantum dilogarithm is closely related to the double gamma and double sine functions ([1, 27])and was used by Baxter ([4]) and Faddeev (see [7, 8]). Its main feature is the following functional equation(see [7, 8]) it satisfies: ϕ b ( q ) ϕ b ( p ) = ϕ b ( p ) ϕ b ( p + q ) ϕ b ( q )whenever pq − qp = πi .Remark that the operator A is characterized (up to a normalization factor) by the equations: AqA − = p − q , ApA − = − q . Note that equations (13)—(15) correspond to relations (2)—(4).Let us introduce now some notation which will be useful in the sequel. For any operator a ∈ End H we set: a ˆ k ≡ A k a k A − k , a ˇ k ≡ A − k a k A k . (20)It is evident that a ˇˆ k = a ˆˇ k = a k , a ˆˆ k = a ˇ k , a ˇˇ k = a ˆ k , where the last two equations follow from equation (13). In particular, we have p ˆ k = − q k , q ˆ k = p k − q k , (21) p ˇ k = q k − p k , q ˇ k = − p k . (22)Besides that, it will be also useful to use the notation P ( kl...m ˆ k ) ≡ A k P ( kl...m ) , P ( kl...m ˇ k ) ≡ A − k P ( kl...m ) , (23)where ( kl . . . m ) is the cyclic permutation( kl . . . m ) : k l . . . m k. Equation (15) in this notation takes a rather compact form T T = ζP (12ˆ1) . (24)6 emark . Notice that the Pentagon relation (14) can be applied whenever any of the indices k ∈ { , } arising among subscripts is replaced everywhere by either ˆ k or else ˇ k . Remark . A T -matrix has the following symmetry property: T = T ˆ2ˇ1 . This can be obtained using twicerelation (24): T = T T T − = ζP (12ˆ1) T − = T − ζP (12ˆ1) = T − ζP (ˆ1ˆ2ˇ1) = T ˆ2ˇ1 (25) The quantization of the Teichm¨uller space of a punctured surface Σ with boundary induced by a semi-symmetric T -matrix is defined by means of a quantum functor : F : G Σ → End( H ⊗ N ) , Its meaning is that we have an operator valued function: F : ∆ Σ × ∆ Σ → End( H ⊗ N ) , satisfying the following equations: F ( τ, τ ) = id H ⊗ N , F ( τ, τ ′ ) F ( τ ′ , τ ′′ ) F ( τ ′′ , τ ) ∈ C \ { } , ∀ τ, τ ′ , τ ′′ ∈ ∆ Σ , (26) F ( f ( τ ) , f ( τ ′ )) = F ( τ, τ ′ ) , ∀ f ∈ M Σ , (27) F ( τ, ρ i τ ) ≡ A i , (28) F ( τ, ω i,j τ ) ≡ T ij , (29) F ( τ, τ σ ) ≡ P σ , ∀ σ ∈ S N , (30)where operator P σ is defined by equation (12). Consistency of these equations is ensured by the consistencyof equations (13)—(15) with relations (2)—(4).A particular case of equation (26) corresponds to τ ′′ = τ : F ( τ, τ ′ ) F ( τ ′ , τ ) ∈ C \ { } . (31)As an example, we can calculate the operator F ( τ, ω − i,j ( τ )). Denoting τ ′ ≡ ω − i,j ( τ ) and using equation (31),as well as definition (29), we obtain F ( τ, ω − i,j ( τ )) = F ( ω i,j ( τ ′ ) , τ ′ ) ≃ ( F ( τ ′ , ω i,j ( τ ′ ))) − = T − ij , (32)where ≃ means equality up to a numerical multiplicative factor.The operations ˆ and ˇ at the indices level have the following geometric interpretation. If the distinguishedcorners of the decorated ideal triangulation are precisely those from Figure 2 then the quantum functorassigns to the flip on that edge the endomorphism T − ij . Now, changing the distinguished corner in thetriangle labeled i amounts of changing i into ˆ i or ˇ i (and similarly for j ) in the expression of the quantumfunctor endomorphism. This rules will be intensively used when we compute the expressions of Dehn twistsin terms of the generators of the decorated Ptolemy groupoid in the next section.The quantum functor induces a unitary projective representation of the mapping class group Γ sg of Σ asfollows: Γ sg ∋ f F ( τ, f ( τ )) ∈ End( H ⊗ N ) . Indeed, we have the following relation (up to a non-zero scalar): F ( τ, f ( τ )) F ( τ, h ( τ )) = F ( τ, f ( τ )) F ( f ( τ ) , f ( h ( τ ))) ≃ F ( τ, f h ( τ )) . The main question addressed in this present paper is to identify the central extension of the mapping classgroup corresponding to this projective representation. Observe that the projective factor lies in the sub-groupof C ∗ generated by ζ .In [18, 19] one considered only punctured surfaces without boundary. However, the construction extendswithout essential modifications to the case when Σ is a surface with boundary Σ sg,r when s ≥ g Γ sg,r by using the decorated Ptolemy groupoid of the punctured surface with boundary, withoutreference to a larger surface without boundary. 7 Presentation of g Γ sg,r We start with a number of notations and definitions. Our setup consists of an embedding Σ sg,r ⊂ Σ th, sendingpunctures into punctures. We assume that each component of Σ th, \ int(Σ sg,r ) is large, namely it admits idealtriangulations whose vertices are those punctures of Σ th, which are not interior punctures of Σ sg,r (henceboundary punctures of Σ sg,r being allowed). In particular, if we discard the boundary punctures of Σ sg,r thecomplement Σ th, \ int(Σ sg,r ) contains no disk, punctured disk or cylinder components. According to [24]the surface embedding induces an embedding between the corresponding mapping class groups Γ sg,r ֒ → Γ th, .The pull-back of the central extension f Γ th to Γ sg,r is a central extension g Γ sg,r . Our main concern is to studythis central extension. The central extension obtained by the present construction is isomorphic to thecentral extension obtained by the direct quantization of the Teichm¨uller space associated to Σ sg,r followingthe procedure of section 1.4. This follows from the fact that the map between the mapping class groupsΓ sg,r ֒ → Γ th, is covered by an injective map between the decorated Ptolemy groupoids according to Lemma1.1.Since the restriction of the Euler class corresponding to the ( s + 1)-th puncture to Γ sg,r vanishes, it is enoughto consider t = s below. Our strategy is to compute explicit lifts to g Γ sg,r of a set of relations arising in a grouppresentation of Γ sg,r by expressing (lifts of) the generators as elements of the decorated Ptolemy groupoid ofthe larger punctured surface Σ sh, . The independence on the particular embedding of the subsurface Σ sg,r ,under the assumptions of the main theorem is a consequence of the so-called Grothendieck principle. In theform proved by Gervais in [13] it states that all relations in Γ sg,r are determined by an explicit set of relationsamong mapping classes supported on small subsurfaces, namely Σ , , Σ , and Σ , , where Σ g,r = Σ g,r . Weexpress then these relations in terms of elements of the decorated Ptolemy groupoids of the surfaces Σ , ,Σ , and Σ , , respectively. According to Lemma 1.1 these relations also hold in G Σ sh, , provided that s ≥ a is a simple closed curve on Σ sg,r we denote by D a ∈ Γ sg,r the right Dehn twist along a . Definition 2.1. A chain relation C on the surface Σ sg,r is given by an embedding Σ , ⊂ Σ sg,r and thestandard chain relation on this 2-holed torus, namely ( D a D b D c ) = D e D f where a, b, c, d, e, f are the following curves of the embedded 2-holed torus: ace fb Definition 2.2. A lantern relation L on the surface Σ sg,r is given by an embedding Σ , ⊂ Σ sg,r and thestandard lantern relation on this 4-holed sphere, namely D a D a D a D − a D − a D − a D − a = 1 (33) where a , a , a , a , a , a , a are the following curves of the embedded 4-holed sphere: a aa a a a
12 13 23 a Definition 2.3.
Consider an embedding Σ , ⊂ Σ sg,r such that the boundary components a , a , a of Σ , are non-separating curves. Let then a , a , a be embedded curves on Σ , so that a jk bounds a pair of ants Σ , ⊂ Σ , along with a j and a k , for all ≤ j = k ≤ . Then the puncture relation P (supported atthe puncture of Σ , ) on the surface Σ sg,r is: D a D a D a D − a D − a D − a = 1 (34) Remark . The puncture relation is, in fact, a consequence of the lantern relation and the fact that theDehn twist along a small loop encircling a puncture is trivial.The first step in proving Theorem 0.1 is to find an explicit presentation for the central extension g Γ sg,r .Specifically, by using Gervais’ presentation [13], we have the following description. Proposition 2.1.
Suppose that g ≥ and s ≥ . Then the group g Γ sg,r has the following presentation.1. Generators:(a) With each non-separating simple closed curve a in Σ sg,r is associated a generator e D a ;(b) One (central) element z .2. Relations:(a) Centrality: z e D a = e D a z (35) for any non-separating simple closed curve a on Σ sg,r ;(b) Braid type -relations: e D a e D b = e D b e D a (36) for each pair of disjoint non-separating simple closed curves a and b ;(c) Braid type -relations: e D a e D b e D a = e D b e D a e D b (37) for each pair of non-separating simple closed curves a and b which intersect transversely at onepoint;(d) One lantern relation on a -holed sphere subsurface with non-separating boundary curves: e D a e D a e D a e D a = e D a e D a e D a (38) (e) One chain relation on a -holed torus subsurface with non-separating boundary curves: ( e D a e D b e D c ) = z e D e e D f (39) (f ) Puncture relations: e D a i ) e D a ( i ) e D a ( i ) = z e D a ( i ) e D a ( i ) e D a ( i ) (40) for each puncture p i of Σ sg,r , i ∈ { , , . . . , s } .(g) Scalar equation: z N = 1 (41) where N is the order of ζ − , in the case where ζ ∈ C ∗ is a root of unity. Lemma 2.1.
For any lifts e D a of the Dehn twists D a we have e D a e D b = e D b e D a , for any two disjoint simpleclosed curves a and b , and thus the braid-type 0-relations (b) are satisfied.Proof. The commutativity relations are satisfied for particular lifts coming from a semi-symmetric T -matrix.If we change the lifts by multiplying each lift by some central element the commutativity is still valid. Thus,the commutativity holds for any lifts. Lemma 2.2.
There are lifts e D a of the Dehn twists D a , for each non-separating simple closed curve a suchthat we have e D a e D b e D a = e D b e D a e D b for any simple closed curves a, b with one intersection point, and thusthe braid type -relations (c) are satisfied. Moreover, the choice of lifts of all e D x , with x non-separating,satisfying these requirements is uniquely defined by fixing the lift e D a of one particular Dehn twist. roof. Consider an arbitrary lift of one braid type 1-relation (to be called the fundamental one), which hasthe form e D a e D b e D a = z k e D b e D a e D b . Change then the lift e D b into z k e D b . With the new lift the relation abovebecomes e D a e D b e D a = e D b e D a e D b .Choose now an arbitrary braid type 1-relation of Γ sg,r , say D x D y D x = D y D x D y . There exists a 1-holedtorus Σ , ⊂ Σ sg,r containing x, y , namely a neighborhood of x ∪ y . Let T be the similar torus containing a, b .Since a, b and x, y are non-separating there exists a homeomorphism ϕ : Σ sg,r → Σ sg,r such that ϕ ( a ) = x and ϕ ( b ) = y . We have then D x = ϕD a ϕ − , D y = ϕD b ϕ − . Let us consider now an arbitrary lift e ϕ of ϕ , which is well-defined only up to a central element, and set e D x = e ϕ e D a e ϕ − , e D y = e ϕ e D b e ϕ − . These lifts are well-defined since they do not depend on the choice of e ϕ (the central elements coming from e ϕ and e ϕ − mutually cancel). Moreover, we have then e D x e D y e D x = e D y e D x e D y and so the braid type 1-relations (c) are all satisfied.For the second part of the lemma observe that the choice of e D a fixes the choice of e D b . If x is a non-separatingsimple closed curve on Σ sg,r , then there exists another non-separating curve y which intersects it in one point.Thus, by the argument which was used above to prove the existence of the lifts the choice of e D x is unique. Lemma 2.3.
One can choose the lifts of Dehn twists in g Γ sg,r so that all braid type relations are satisfied andthe lift of the lantern relation (d) is trivial, namely e D a e D a e D a e D a = e D a e D a e D a for the non-separating curves on an embedded Σ , ⊂ Σ sg,r .Proof. An arbitrary lift of that lantern relation is of the form e D a e D a e D a e D a = z k e D a e D a e D a . In thiscase, we change the lift e D a into z − k e D a and adjust the lifts of all other Dehn twists along non-separatingcurves the way that all braid type 1-relations are satisfied. Then, the required form of the lantern relationis satisfied.We say that the lifts of the Dehn twists are normalized if all braid type relations and one lantern relationare lifted in a trivial way. Lemma 2.4.
Assume that s ≥ . Then a normalized Dehn twist in quantum Teichm¨uller theory is conjugatedto the inverse T -matrix times ζ − i.e. e D α = F ( τ, D α τ ) = ζ − U α T − kl U − α . As the computations involved in the proof are rather laborious we postpone it after the proof of Lemma 2.6.We will suppose henceforth that the lifts of Dehn twists are normalized.
Lemma 2.5.
Let a, b, c, e, f be the five curves appearing in the chain relation ( D a D b D c ) = D e D f on anembedded 2-holed torus sitting inside Σ sg,r . If s ≥ , then the lifts of Dehn twists in g Γ sg,r satisfy the relation ( e D a e D b e D c ) = ζ − e D e e D f Proof. If s ≥ g ≥
2, then there is an embedding Σ , ⊂ Σ sg,r .We consider a surface S homeomorphic to Σ , , i.e. a torus with two holes and two punctures drawn in theleft picture of Figure 5 where the opposite sides of the rectangle are identified. Notice that the two puncturesare located on the two boundary components.The central picture of Figure 5 specifies five simple closed curves a, b, c, e, f in S , the Dehn twists alongwhich enter the chain relation.We also choose a particular decorated ideal triangulation τ of S given by the right picture of Figure 5,where the ideal arcs are drawn in black and the positions of the numbers in ideal triangles correspond to10 cbe f
12 3 45 6
Figure 5the marked corners. Notice that our choice is manifestly symmetric with respect to the exchange of the leftand the right halves of the rectangle accompanied with relabeling (1 , , ↔ (4 , , D α along a given simple closedcurve α is to use a specific decorated ideal triangulation where the contour α intersects only two ideal arcs,so that the annular neighborhood of α is given by only two ideal triangles. With respect to such (decorated)ideal triangulation the quantum operator realizing D α is given by a single T -operator. Let us work out thisprocedure in the case of the curves a, b, c, e, f .For any simple closed curve α , we denote ¯ F α = e D − α ≃ F ( D α τ, τ ). To derive the operator representing theDehn twist D a , we apply the following change of triangulation:
12 3 45 6 a T ˇ23 / / a where the operator above the arrow realizes the corresponding element of the groupoid of decorated idealtriangulations within the quantum Teichm¨uller theory. Thus, ζ − ¯ F a = Ad( T ˇ23 )( T ) = T ˇ23 T ¯ T ˇ23 = T T , where in the last equality, we have applied once the Pentagon relation, and we use the notation ¯ T = T − .Here, we use the normalization where the braid-type and the lantern relations are satisfied without projectivefactors. By the above mentioned left-right symmetry (1 , , ↔ (4 , , D c : ζ − ¯ F c = T T . To calculate the quantum realization of D b we use a two-step chain of transformations of τ :
12 3 45 6 b ¯ T / /
12 3 456 b T ¯ T / /
12 3 456 b Thus, we have the following sequence of equalities: ζ − ¯ F b = Ad( ¯ T T ¯ T )( T ) = ¯ T T ¯ T T T ¯ T T = ¯ T T T T ¯ T T = T T T ¯ T T = T T T T , where in each step the underlined fragment is transformed by using the Pentagon relation.11o calculate the realization of D e , we consider the following sequence of ideal triangulations:
12 3 45 6 e T (cid:15) (cid:15) e
12 3 4 5 6 e T ¯ T / / e T ˇ45 / / e T ˇ56 O O Thus, we have ζ − ¯ F e = Ad( T T ¯ T T ˇ45 T ˇ56 )( T ˇ2ˆ6 ) = T T ¯ T T ˇ45 T ˆ6ˆ5 T ˇ2ˆ6 ¯ T ˆ6ˆ5 ¯ T ˇ45 T ¯ T ¯ T = T T ¯ T T ˆ5ˆ4 T ˇ2ˆ6 T ˇ2ˆ5 ¯ T ˆ5ˆ4 T ¯ T ¯ T = T T ˆ4ˇ1 ¯ T T ˇ2ˆ6 T ˇ2ˆ5 T ˇ2ˆ4 T ¯ T ˆ4ˇ1 ¯ T = T ¯ T ˆ3ˇ6 T ˇ6ˆ2 T ˇ2ˆ5 T ˇ2ˆ4 T ˇ2ˇ1 T ˆ3ˇ6 ¯ T = T T ˇ23 T ˇ6ˆ2 T ˇ2ˆ5 T ˇ2ˆ4 T ˇ2ˇ1 ¯ T = T ˇ23 T ˇ24 T ˆ4ˇ3 T ˇ6ˆ2 T ˇ2ˆ5 T ˇ2ˆ4 T ˇ2ˇ1 ¯ T ˆ4ˇ3 = T ˇ23 T ˇ24 T ˇ2ˆ6 T ˇ2ˆ5 T ˇ2ˆ4 T ˇ2ˇ3 T ˇ2ˇ1 , where, as before, in each step the underlined fragment is transformed by applying the Pentagon relation.We use throughout these computations the fact that T ij and T kl commute if { i, j } ∩ { k, l } = ∅ . Again, usingthe symmetry (1 , , ↔ (4 , , ζ − ¯ F f = T ˇ56 T ˇ51 T ˇ5ˆ3 T ˇ5ˆ2 T ˇ5ˆ1 T ˇ5ˇ6 T ˇ5ˇ4 . In order to check the Chain relation, we first calculate the following product: ζ − ¯ F c ¯ F b ¯ F a = T T T T T T T T = T T T T T ζP (31ˆ3) T = ζT T T T T T ˆ3ˆ2 P (31ˆ3) , where we have applied the Inversion relation to the underlined fragment. Next, we calculate ζ − (¯ F c ¯ F b ¯ F a ) = ζ T T T T T T ˆ3ˆ2 P (31ˆ3) T T T T T T ˆ3ˆ2 P (31ˆ3) = ζ T T T T T T ˆ3ˆ2 T T T T T T ˆ1ˆ2 P (3ˆ3)(1ˆ1) = ζ T T T T T ˆ3ˆ2 T ˆ6ˆ5 T T T T ˆ1ˆ2 P (64ˆ6) P (3ˆ3)(1ˆ1) = ζ T T T ˆ6ˆ5 T ˇ51 T T ˆ3ˆ2 T ˇ24 T T T T T ˆ1ˆ2 P (64ˆ6) P (3ˆ3)(1ˆ1) = ζ T ˆ6ˆ5 T T ˇ51 T ˆ3ˆ2 T ˇ24 P (61ˆ6) P (34ˆ3) T T ˆ1ˆ2 P (64ˆ6) P (3ˆ3)(1ˆ1) = ζ T ˆ6ˆ5 T T ˇ51 T ˇ23 T ˇ24 T ˆ3ˆ1 T ˇ6ˆ2 P (1ˇ63ˆ4ˇ1) , where each equality is obtained by transforming the underlined fragment by applying the Pentagon relation(twice in the forth and once in the fifth equalities), the Inversion relation (once in the third and twice in thefifth equalities), and the extended symmetric group action (in the second, the third, and the sixth equalities).Finally, taking the square of the obtained identity, we have ζ − (¯ F c ¯ F b ¯ F a ) = ζ T ˆ6ˆ5 T T ˇ51 T ˇ23 T ˇ24 T ˆ3ˆ1 T ˇ6ˆ2 P (1ˇ63ˆ4ˇ1) T ˆ6ˆ5 T T ˇ51 T ˇ23 T ˇ24 T ˆ3ˆ1 T ˇ6ˆ2 P (1ˇ63ˆ4ˇ1) = ζ T ˇ56 T T ˇ51 T ˇ23 T ˇ24 T ˆ3ˆ1 T ˇ6ˆ2 T ˇ5ˆ3 T ˆ1ˇ3 T ˇ5ˇ6 T ˇ2ˆ4 T ˇ2ˆ1 T ˇ46 T P (13ˇ1) P (46ˇ4) = ζ T ˇ56 T T ˇ51 T ˇ23 T ˇ24 T ˇ6ˆ2 T ˇ5ˆ3 T ˇ5ˆ1 T ˆ3ˆ1 T ˆ1ˇ3 T ˇ5ˇ6 T ˇ2ˆ4 T ˇ2ˆ1 T ˇ46 T P (13ˇ1) P (46ˇ4) = ζ T ˇ56 T T ˇ51 T ˇ23 T ˇ24 T ˇ6ˆ2 T ˇ5ˆ3 T ˇ5ˆ1 T ˇ5ˇ6 T ˇ2ˆ4 T ˇ2ˇ3 T ˇ46 T P (46ˇ4) = ζ T ˇ56 T ˇ51 T ˇ23 T ˇ24 T ˇ2ˆ6 T T ˇ6ˆ2 T ˇ5ˆ3 T ˇ5ˆ1 T ˇ5ˇ6 T ˇ2ˆ4 T ˇ46 T ˇ2ˇ3 T P (46ˇ4) = ζ T ˇ56 T ˇ51 T ˇ23 T ˇ24 T ˇ2ˆ6 T ˇ5ˆ3 T ˇ5ˆ1 T ˇ2ˆ4 T T ˇ5ˇ6 T ˇ46 T ˇ2ˇ3 T P (46ˇ4) = ζ T ˇ56 T ˇ51 T ˇ23 T ˇ24 T ˇ2ˆ6 T ˇ5ˆ3 T ˇ5ˆ1 T ˇ2ˆ4 T ˇ5ˇ6 T ˇ5ˇ4 T T ˇ46 T ˇ2ˇ3 T P (46ˇ4) = ζ T ˇ56 T ˇ51 T ˇ23 T ˇ24 T ˇ2ˆ6 T ˇ5ˆ3 T ˇ5ˆ1 T ˇ2ˆ4 T ˇ5ˇ6 T ˇ5ˇ4 T ˇ2ˇ3 T = ζ T ˇ56 T ˇ51 T ˇ5ˆ3 T ˇ5ˆ2 T ˇ23 T ˇ24 T ˇ5ˆ1 T ˇ5ˇ6 T ˇ5ˆ2 T ˇ2ˆ6 T ˇ5ˇ4 T ˇ5ˆ2 T ˇ2ˆ4 T ˇ2ˇ3 T = ζ T ˇ56 T ˇ51 T ˇ5ˆ3 T ˇ5ˆ2 T ˇ23 T ˇ5ˆ1 T ˇ5ˇ6 T ˇ5ˇ4 T ˇ24 T ˇ2ˆ6 T ˇ5ˆ2 T ˇ2ˆ4 T ˇ2ˇ3 T = ¯ F f ¯ F e , F f (respectively ¯ F e ) Lemma 2.6.
Suppose that s ≥ . Then the lift of each puncture relation is ζ .Proof. Observe first that the central element P i which is the lift of the puncture relation at the puncture p i is independent of the particular subsurface S , . If we consider another subsurface, there exists a homeomor-phism ϕ : S sg,r → S sg,r fixing the puncture p i and sending it to the initial subsurface, because the boundarycomponents are non-separating. The new puncture relation is then conjugate of P i by e ϕ and hence theycoincide, as they are elements of the center.If s ≥ S , ⊂ S sg,r , such that each boundary component of S , has a punctureon it. Consider first the following decomposition τ of the punctured pair of pants into triangles. The positionof the label of each triangle indicates also the marked corner. (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) Then we can express easily the action of each Dehn twist D a j on the triangulation τ as a composition offlips. If we set F a j = F ( τ, D a j ( τ )) then we have: F a = T − , F a = T − , F a = T − Further we use the sequence of transformations below, in order to change the triangulation τ into a trian-gulation which intersects the curve a in only two points.13 (cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)
34 5 67 21 5 65 6655 6 3 5 61212 32 1 312 332 1 47 47 47477 4
T 4 7^ ^ T 2 4T 4 1^ ^T3 4 ^ ^3 2^T ^^
Here and in the pictures below we marked by a dot the edges where a flip occurs, in order to help the readervisualise the sequence of transformations. Then the method outlined above permits to compute the Dehntwist F a = F ( τ, D a ( τ )) as follows: F a = Ad( T ˆ4ˆ7 T ˇ24 T ˆ4ˆ1 T ˆ3ˆ4 T ˆ3ˇ2 )( T − )Let us first simplify the formula for F a . We have¯ F a = T ˇ74 T ˇ24 T ˆ4ˆ1 T ˆ3ˆ4 T ˆ3ˇ2 T ˇ7ˆ3 ¯ T ˆ3ˇ2 ¯ T ˆ3ˆ4 ¯ T ˆ4ˆ1 ¯ T ˇ24 ¯ T ˇ74 = T ˇ74 T ˇ24 T ˆ4ˆ1 T ˆ3ˆ4 T ˇ7ˆ3 T ˇ7ˇ2 ¯ T ˆ3ˆ4 ¯ T ˆ4ˆ1 ¯ T ˇ24 ¯ T ˇ74 = T ˇ74 T ˇ24 T ˆ4ˆ1 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˇ2 ¯ T ˆ4ˆ1 ¯ T ˇ24 ¯ T ˇ74 = T ˇ74 T ˇ24 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˆ1 T ˇ7ˇ2 ¯ T ˇ24 ¯ T ˇ74 = T ˇ74 T ˇ24 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˆ1 ¯ T ˇ24 T ˇ7ˇ2 = T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˆ2 T ˇ7ˆ1 T ˇ7ˇ2 where in each step the underlined fragment is transformed by using the Pentagon equation, and in the lastequality it is also combined with the symmetry relation T ˇ24 = T ˆ4ˆ2 .Our triangulation is invariant under the following simultaneous cyclic permutations π : P P P P , ˇ6 , ˆ5 ˇ4 , ˇ7 , so that the contours a j and a kl are transformed as follows: π : a a a a , a a a a . Thus, it suffices to know the explicit formula for F a in order to write out the other two without any furthercalculation: ¯ F a = π (¯ F a ) = π ( T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˆ2 T ˇ7ˆ1 T ˇ7ˇ2 ) = T ˆ7ˆ2 T ˆ7ˆ1 T ˆ7ˇ2 T ˆ7ˇ5 T ˆ76 T ˆ75 , F a = π (¯ F a ) = π ( T ˆ7ˆ2 T ˆ7ˆ1 T ˆ7ˇ2 T ˆ7ˇ5 T ˆ76 T ˆ75 ) = T T T T T T . Now, we have¯ F a ¯ F a ¯ F a = T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˆ2 T ˇ7ˆ1 T ˇ7ˇ2 T ˆ7ˆ2 T ˆ7ˆ1 T ˆ7ˇ2 T ˆ7ˇ5 T ˆ76 T ˆ75 T T T T T T = T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˆ2 T ˇ7ˆ1 ζP (2ˆ7ˆ2) T ˆ7ˆ1 T ˆ7ˇ2 T ˆ7ˇ5 T ˆ76 ζP (ˆ57ˇ5) T T T T T = ζ T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˇ7ˆ2 T ˇ7ˆ1 T ˆ2ˆ1 T ˆ27 T ˆ2ˇ5 T ˆ26 T ˇ56 T ˇ5ˇ2 T ˇ54 T ˇ5ˆ3 T ˇ5ˆ4 P (2ˆ75ˇ2) = ζ T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˆ2ˆ1 T ˇ7ˆ2 T ˆ27 T ˇ56 T ˆ2ˇ5 T ˇ5ˇ2 T ˇ54 T ˇ5ˆ3 T ˇ5ˆ4 P (2ˆ75ˇ2) = ζ T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˆ2ˆ1 ζP (ˇ7ˆ27) T ˇ56 ζP (ˆ2ˇ5ˇ2) T ˇ54 T ˇ5ˆ3 T ˇ5ˆ4 P (2ˆ75ˇ2) = ζ T ˇ74 T ˇ7ˆ3 T ˇ7ˆ4 T ˆ2ˆ1 T ˇ56 T ˆ74 T ˆ7ˆ3 T ˆ7ˆ4 P (7ˆ7) = ζ T ˇ74 T ˇ7ˆ3 T ˆ2ˆ1 T ˇ56 ζP (ˇ4ˆ74) T ˆ7ˆ3 T ˆ7ˆ4 P (7ˆ7) = ζ T ˇ74 T ˇ7ˆ3 T ˆ2ˆ1 T ˇ56 T T P (74ˇ7) = ζ T ˆ2ˆ1 T ˇ56 T T ˇ74 T P (74ˇ7) = ζ T ˆ2ˆ1 T ˇ56 T ζP (ˇ747) P (74ˇ7) = ζ T ˆ2ˆ1 T ˇ56 T = ζ ¯ F a ¯ F a ¯ F a where in the underlined fragments the Pentagon equation is used twice in the forth and once in the ninthequalities, the Inversion relation is used twice in the second and the fifth, and once in the seventh andthe tenth equalities, while in the third, sixth, eighth, and eleventh equalities the permutation operators aremoved to the right and the powers of ζ , to the left. Proof of Lemma 2.4.
The idea of the proof is to calculate the lift of the lantern relation. Consider thefollowing decorated triangulation τ of the 4-holed disk with 4 punctures: (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) a a a a The trick used in [18, 19] for computing D a is to use a sequence of flips to change the triangulation into onewhich intersects some curve isotopic to a into two points. Then the Dehn twist along a can be expressedas the flip of one of the two edges of the latter triangulation intersecting a . This recipe generalizes to thecase where the curve a intersects several edges of the triangulation, if a is a boundary component withone puncture on it. Specifically, let e , . . . , e s be the edges issued from the puncture, in counterclockwiseorder. Then the Dehn twist D a can be expressed as the result of composing the flips of e , e , . . . , e s − . Weillustrate this procedure with the case of the left Dehn twist D − a on the triangulation τ above:15 (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)
12 345 8 76 12 45 8 63712 4 83675
T5 3 T 3 8T3 7T3 6
12 345 7 6812 45 3 8 67
In particular, we find the following expression for the right Dehn twist along a :¯ F a = ¯ F ( τ, D a τ ) = T T T T (42)We used above the symmetry property of the T -matrix T = T (see Remark 1.3 equation (25)). The samerecipe for the remaining Dehn twists along boundary components gives us:¯ F a = ¯ F ( τ, D a τ ) = T T T T (43)¯ F a = ¯ F ( τ, D a τ ) = T T T T (44)¯ F a = ¯ F ( τ, D a τ ) = T ˇ8ˆ5 T ˇ8ˇ4 T ˇ8ˇ1 T ˇ8ˇ7 (45)In order to compute F a we need to transform the triangulation τ into one which intersects a curve isotopicto a into precisely two points. This can be done as follows:16 (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)
12 345 8 76
T5 3 TTT
12 345 7 682 678 31 452 8145 7
632 678 315 4 25 678 31 4
Therefore we have: ¯ F a = ¯ F ( τ, D a τ ) = Ad( T T T T T )( T ˇ67 ) (46)The following sequence of transformations 17 (cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) T TTT 4 5T
12 345 67 81 3457 8 6 21 37 8 6 2451 37 8 65 4 27 12 345 8 76 can be used to compute: ¯ F a = ¯ F ( τ, D a τ ) = Ad( T ˆ8ˆ7 T ˇ2ˇ6 T T T ˇ5ˆ8 )( T ˇ4ˇ5 ) (47)Eventually use the transformations 18 (cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) T TT
28 76 1 34 5
25 8 76 1 43 1 72 3 645 8 15 8 72 3 642 3 65 8 7 1 4 2 3 645 8 7 112 345 8 76 1 345 8 762 in order to obtain: ¯ F a = ¯ F ( τ, D a τ ) = Ad( T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T ˇ3ˇ1 T ˇ4ˆ5 )( T ) (48)The next step is to simplify the expression of the last three Dehn twist, as follows:¯ F a = T T T T T T ˇ67 ¯ T ¯ T ¯ T ¯ T ¯ T = T T T T T T ˇ67 ¯ T ¯ T ¯ T ¯ T ¯ T == T T T T T ˇ67 ¯ T ¯ T ¯ T ¯ T ¯ T = T T T T T ˇ67 ¯ T T ¯ T ¯ T = T T T T ˇ67 T ˇ65 T ¯ T ¯ T == T T ˇ67 T T T ˇ65 ¯ T T ¯ T = T T ˇ67 T T ¯ T T T ˇ65 ¯ T T ¯ T = T T ˇ67 T T T ˇ65 T T ¯ T The first equality above corresponds to the commutativity of T ij and T kl in the case when the two sets ofindices are disjoint, for each one of the underlined fragments. We further also made use of the symmetryproperty from (Remark 1.3, relation (25)) in order to be able to use the Pentagon relation, as in the lastequality above. Specifically, the rightmost reduction consists of the the following steps: T T ˇ65 ¯ T = T T ˇ65 ¯ T = T ˇ65 T T ¯ T = T ˇ65 T T ¯ T = T ˇ65 T (49)Similar simplifications lead to: 19 F a = T ˆ8ˆ7 T ˇ2ˇ6 T T T ˇ5ˆ8 T ˇ4ˇ5 ¯ T ˇ5ˆ8 ¯ T ¯ T ¯ T ˇ2ˇ6 ¯ T ˆ8ˆ7 = T ˆ8ˆ7 T ˇ2ˇ6 T T ˇ5ˆ8 T T ˇ4ˇ5 ¯ T ¯ T ˇ5ˆ8 ¯ T ¯ T ˇ2ˇ6 ¯ T ˆ8ˆ7 == T ˆ8ˆ7 T ˇ2ˇ6 T T ˇ5ˆ8 T ˇ4ˇ5 T ¯ T ˇ5ˆ8 ¯ T ¯ T ˇ2ˇ6 ¯ T ˆ8ˆ7 = ζT ˆ8ˆ7 T ˇ2ˇ6 T ˇ5ˆ8 T P (45ˆ4) ¯ T ¯ T ˇ5ˆ8 ¯ T ¯ T ˇ2ˇ6 ¯ T ˆ8ˆ7 == ζT ˆ8ˆ7 T ˇ2ˇ6 T ˇ5ˆ8 T ¯ T ˆ4ˇ2 ¯ T ¯ T ¯ T ˇ2ˇ6 ¯ T ˆ8ˆ7 P (45ˆ4) = ζT ˆ8ˆ7 T ˇ2ˇ6 T ˇ5ˆ8 T T ¯ T ¯ T ˇ2ˇ6 ¯ T ˆ8ˆ7 P (45ˆ4) == ζT ˆ8ˆ7 T ˇ5ˆ8 T ˇ2ˇ6 T T ¯ T ˇ2ˇ6 ¯ T ¯ T ˆ8ˆ7 P (45ˆ4) = ζT ˆ8ˆ7 T ˇ5ˆ8 T ˇ2ˇ6 T ¯ T ˇ2ˇ6 T ˇ2ˇ6 T ¯ T ˇ2ˇ6 ¯ T ¯ T ˆ8ˆ7 P (45ˆ4) == ζT ˆ8ˆ7 T ˇ5ˆ8 T T ˆ4ˇ6 T T ˇ8ˇ6 ¯ T ¯ T ˆ8ˆ7 P (45ˆ4) ¯ F a = T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T ˇ3ˇ1 T ˇ4ˆ5 T ¯ T ˇ4ˆ5 ¯ T ˇ3ˇ1 ¯ T ˇ4ˇ1 ¯ T ˇ1ˇ7 ¯ T ˇ5ˆ8 ¯ T ˇ36 ¯ T == T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T ˇ4ˆ5 T ˇ3ˇ1 T ¯ T ˇ3ˇ1 ¯ T ˇ4ˆ5 ¯ T ˇ4ˇ1 ¯ T ˇ1ˇ7 ¯ T ˇ5ˆ8 ¯ T ˇ36 ¯ T == T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T ˇ4ˆ5 T T ¯ T ˇ4ˆ5 ¯ T ˇ4ˇ1 ¯ T ˇ1ˇ7 ¯ T ˇ5ˆ8 ¯ T ˇ36 ¯ T = T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T ˇ4ˆ5 T ¯ T ˇ4ˆ5 T ¯ T ˇ1ˇ7 ¯ T ˇ5ˆ8 ¯ T ˇ36 ¯ T == T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T ˇ4ˆ5 T T ¯ T ˇ4ˆ5 ¯ T ˇ4ˇ1 ¯ T ˇ1ˇ7 ¯ T ˇ5ˆ8 ¯ T ˇ36 ¯ T = T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T T T ¯ T ˇ1ˇ7 ¯ T ˇ5ˆ8 ¯ T ˇ36 ¯ T Putting all these together we obtain:¯ F a ¯ F a ¯ F a = ζT T ˇ67 T T T ˇ65 T T ¯ T T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T T T ¯ T ˇ1ˇ7 ¯ T ˇ5ˆ8 ¯ T ˇ36 ¯ T T ˆ8ˆ7 T ˇ5ˆ8 T T ˆ4ˇ6 T T ˇ8ˇ6 ×× ¯ T ¯ T ˆ8ˆ7 P (45ˆ4) = ζT T ˇ67 T T T ˇ65 T T ¯ T T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T T T ¯ T ˇ1ˇ7 ¯ T ˇ36 T ˇ5ˆ7 T ˆ8ˆ7 T ˆ4ˇ6 T T ˇ8ˇ6 ¯ T ×× ¯ T ˆ8ˆ7 P (45ˆ4) = ζ T T ˇ67 T T T ˇ65 T T ¯ T T T ˇ36 T ˇ5ˆ8 T ˇ1ˇ7 T ˇ4ˇ1 T T ¯ T ˇ1ˇ7 T P (5ˇ7ˆ5) ¯ T ˇ36 T ˆ8ˆ7 T ˆ4ˇ6 T T ˇ8ˇ6 ¯ T ×× ¯ T ˆ8ˆ7 P (45ˆ4) = ζ T T ˇ67 T T T ˇ65 T T ¯ T T T ˇ36 T ˇ5ˆ8 T ˇ4ˇ1 T ˇ4ˇ7 T T T ¯ T ˇ36 T ˆ4ˇ6 T ˆ85 T T ˇ8ˇ6 ¯ T ˆ85 ¯ T ˇ7ˆ4 ×× P (5ˇ7ˆ5) P (45ˆ4) = ζ T T ˇ67 T T T ˇ65 T T T T T ˇ36 T ˇ5ˆ8 T ˇ4ˇ7 T T T ¯ T ˇ36 T ˆ4ˇ6 T ˆ85 T T ˇ8ˇ6 ¯ T ˆ85 ¯ T ˇ7ˆ4 ×× P (5ˇ7ˆ5) P (45ˆ4) = ζ T T ˇ67 T T T ˇ65 T T T T P (47ˆ4) T ˇ36 T ˇ5ˆ8 T T T ¯ T ˇ36 T ˆ4ˇ6 T ˆ85 T T ˇ8ˇ6 ¯ T ˆ85 ¯ T ˇ7ˆ4 ×× P (5ˇ7ˆ5) P (45ˆ4) = ζ T T T T ¯ T ¯ T T ˇ67 T ¯ T T T T ˇ65 T T T P (47ˆ4) T ˇ36 T ˇ5ˆ8 T T T ¯ T ˇ36 T ˆ4ˇ6 T ˆ85 T ×× T ˇ8ˇ6 ¯ T ˆ85 ¯ T ˇ7ˆ4 P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ T ¯ T ˇ67 ¯ T T ¯ T T T ¯ T T T T ˇ65 T T P (47ˆ4) T ˇ36 T ˇ5ˆ8 T T T ¯ T ˇ36 T ˆ4ˇ6 ×× T ˆ85 T T ˇ8ˇ6 ¯ T ˆ85 ¯ T ˇ7ˆ4 P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a T ˇ67 ¯ T T T T T T T ˇ65 T T T ˇ36 P (47ˆ4) T ˇ5ˆ8 T T T ¯ T ˇ36 T ˆ4ˇ6 ×× T ˆ85 T T ˇ8ˇ6 ¯ T ˆ85 ¯ T ˇ7ˆ4 P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a T T T T ¯ T T ˇ67 ¯ T T T T T ˇ65 P (ˇ6ˇ36) P (47ˆ4) T ˇ5ˆ8 T T T ¯ T ˇ36 T ˆ4ˇ6 ×× T ˆ85 T T ˇ8ˇ6 ¯ T ˆ85 ¯ T ˇ7ˆ4 P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a T ˇ67 ¯ T ¯ T T T T ˇ65 T ˇ5ˆ8 T ˆ6ˇ5 T T ¯ T T ˆ7ˇ3 T ˆ85 T T ˇ8ˇ3 ¯ T ˆ85 ¯ T ×× P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a T ˇ67 ¯ T ¯ T T T T ˇ5ˆ8 T ˇ6ˆ8 T ¯ T ˆ53 T ˇ5ˆ7 T ˆ86 T T ˇ8ˇ7 ¯ T ˆ86 ¯ T ×× P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a T ˇ67 ¯ T T T T ˇ5ˆ8 T T ˇ6ˆ8 ¯ T ˆ53 T ˇ5ˆ7 T ˆ86 T T ˇ8ˇ7 ¯ T ˆ86 ¯ T ×× P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a T ˇ67 ¯ T T T T ˇ5ˆ8 T ¯ T ˆ53 T ˇ5ˆ7 P (ˇ6ˆ86) T ˆ8 T ˇ8ˇ7 ¯ T ˆ86 ¯ T ×× P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a T ˇ67 ¯ T T T T ˇ5ˆ8 T ¯ T ˆ53 T ˇ5ˆ7 T T ˆ6ˇ7 ¯ T ¯ T ×× P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a T T T ˇ5ˆ8 T ¯ T ˆ53 T ˇ5ˆ7 T ˇ6ˆ5 P (ˇ676) ¯ T ¯ T ×× P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ T ˇ8ˇ7 ¯ T ˇ8ˇ1 ¯ T ˇ8ˇ4 ¯ T ˇ8ˆ5 T T T ˇ5ˆ8 T ¯ T ˆ53 T ˇ5ˆ7 T ˇ6ˆ5 ¯ T ˆ7ˇ8 ¯ T ×× P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ T ˇ8ˇ7 ¯ T ˇ8ˇ1 ¯ T ˇ8ˇ4 T T T T ¯ T ˆ53 T ˇ5ˆ7 T ˇ6ˆ5 ¯ T ˆ7ˇ8 ¯ T ×× P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ T ˇ8ˇ7 T T ¯ T ˆ53 T ˇ5ˆ7 T ˇ6ˆ5 ¯ T ˆ7ˇ8 ¯ T ×× P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) In the previous lines we used both the Pentagon relation coupled with the symmetry property several timesand the commutativity relations corresponding to the underlined fragments. Sometimes several simplifi-cations are recorded in the same line, as in the first equality above where the underlined factors ¯ T and T commute with T ˆ8ˆ7 T ˇ5ˆ8 and therefore cancel each other, so that along with the first underlined factor weobtain a subproduct ¯ T ˇ5ˆ8 T ˆ8ˆ7 T ˇ5ˆ8 and the Pentagon relation can be applied.Use now the identity: ¯ T ˇ8ˇ7 T ˇ5ˆ7 ¯ T ˆ7ˇ8 = T ˇ5ˆ8 T ˇ5ˆ7 ¯ T ˇ8ˇ7 ¯ T ˆ7ˇ8 = ζ − T ˇ5ˆ8 T ˇ5ˆ7 P (ˇ7ˇ8ˆ7) F a ¯ F a ¯ F a = ζ ¯ F a ¯ F a ¯ F a T T ¯ T ˆ53 T ˇ5ˆ8 T ˇ5ˆ7 T ˇ6ˆ5 ¯ T ×× P (ˇ7ˇ8ˆ7) P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ F a ¯ T ¯ T ¯ T ¯ T T T ¯ T ˆ53 T ˇ5ˆ8 T ˇ5ˆ7 T ˇ6ˆ5 ¯ T ×× P (ˇ7ˇ8ˆ7) P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ F a ¯ T ¯ T ¯ T T ¯ T ¯ T ˆ53 T ˇ5ˆ8 T ˇ5ˆ7 T ˇ6ˆ5 ¯ T ×× P (ˇ7ˇ8ˆ7) P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ F a ¯ T ¯ T ¯ T T P (ˇ53ˆ5) T ˇ5ˆ8 T ˇ5ˆ7 T ˇ6ˆ5 ¯ T ×× P (ˇ7ˇ8ˆ7) P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ F a ¯ T ¯ T ¯ T T T T T ˇ6ˇ3 ¯ T ×× P (ˇ53ˆ5) P (ˇ7ˇ8ˆ7) P (ˇ676) P (ˇ6ˆ86) P (ˇ656) P (ˆ3ˆ7ˇ3) P (ˇ6ˇ36) P (47ˆ4) P (5ˇ7ˆ5) P (45ˆ4) = ζ ¯ F a ¯ F a ¯ F a ¯ F a Thus the lift of the lantern relation is ζ . Therefore we have to renormalize each right Dehn twist by taking f D α = ζ − F α , as claimed.The following lemma is a simple consequence of a deep result of Gervais from ([13]): Lemma 2.7.
Let g ≥ and s ≥ . Then the group Γ sg,r is presented as follows:1. Generators are all Dehn twists D a along the non-separating simple closed curves a on Σ sg,r .2. Relations:(a) Braid type 0-relations: D a D b = D b D a for each pair of disjoint non-separating simple closed curves a and b ;(b) Braid type 1-relations: D a D b D a = D b D a D b for each pair of non-separating simple closed curves a and b which intersect transversely in onepoint;(c) One lantern relation for a -hold sphere embedded in Σ sg,r so that all boundary curves are non-separating;(d) One chain relation for a 2-holed torus embedded in Σ sg,r so that all boundary curves are non-separating;(e) A puncture relation for each puncture.Proof. According to ([13], Theorem B) we have a presentation of Γ g,s + r with the generators above and allbut the puncture relations. Now, the kernel of Γ g,s + r → Γ sg,r is the free Abelian group generated by theDehn twists along the boundary curves to be pinched to punctures. Such a Dehn twist is expressed (usingthe lantern relation) by the left hand side of the puncture relation. This proves the claim. Proof of Proposition 2.1 . According to the normalization coming from the braid relations and the lanternrelations the images of the standard Dehn twist generators of the mapping class group are products of ζ and elements T ij , where i, j are the labels of the triangles (possibly with ˆ or ˇ). Thus the projective factorsthat appear belong to the subgroup A generated by ζ . The only non-trivial lift of a relation from Lemma2.7 is the chain relation which lifts to ζ − . Set z for the element ζ − of g Γ sg,r . Then the presentation of thecentral extension g Γ sg,r is given by the claimed relations. Recall from ([21], Corollary 4.4) that the 2-cohomology classes χ and e i are defined for any g ≥ , s, r ≥ Z s +1 ⊂ H (Γ sg,r ). This inclusion is actually an isomorphism when g ≥ d Γ sg,r the group defined by the presentation given in Proposition 2.1, for all values of s, g, r .Thus, according to Proposition 2.1 the extension d Γ sg,r is isomorphic to g Γ sg,r if s ≥ g ≥ Lemma 2.8. If g ≥ , then we have c d Γ g,r = 12 χ ∈ H (Γ g,r ; A ) . roof. Consider first the case where ζ is not a root of unity, so that the group A is isomorphic to Z . Gervaisproved in ([13], Theorem 3.6) that d Γ g,r (namely, where s = 0) is isomorphic to the so-called p -centralextension of Γ g,r . Further in [13, 22] the authors identified the class of the p -central extension of Γ g,r tothe class 12 χ and thus c d Γ g,r = 12 χ .Here is a more direct argument. Set Γ g,r (1) for the subgroup of d Γ g,r generated by the lifts f D a of the Dehntwists and the central element u = z . Then Γ g,r (1) is the universal central extension considered by Harer(see [13, 15]) and thus c Γ g,r (1) is the generator χ of H (Γ g,r ) ∼ = Z .The cohomology class c Γ g,r (1) is represented by some explicit 2-cocycle C Γ g,r (1) : Γ g,r × Γ g,r → Z which arisesas follows. Let S : Γ g,r → Γ g,r (1) be a set-wise section. Let also i : ker(Γ g,r (1) → Γ g,r ) → Z be the groupisomorphism defined by i ( u ) = 1. It is well-known that the 2-cocycle C Γ g,r (1) ( x, y ) = i ( S ( xy ) S ( x ) − S ( y ) − ) ∈ Z represents the cohomology class c Γ g,r (1) .Let us construct now a 2-cocycle representing the extension d Γ g,r . Consider the set-wise section ι ◦ S : Γ g,r → d Γ g,r , where ι : Γ g,r (1) → d Γ g,r is the obvious inclusion. Let also j : ker( d Γ g,r → Γ g,r ) → Z be the isomorphismgiven by j ( z ) = 1. Then C d Γ g,r ( x, y ) = j (( ι ◦ S )( xy )( ι ◦ S )( x ) − ( ι ◦ S )( y ) − ) = j ( ι ( S ( xy ) S ( x ) − S ( y ) − )) ∈ Z is a 2-cocycle representing c d Γ g,r . Since j ( ι ( u )) = j ( z ) = 12 i ( u ) and S ( xy ) S ( x ) − S ( y ) − belongs to thecyclic subgroup of Γ g,r (1) generated by u , it follows that C d Γ g,r ( x, y ) = 12 C Γ g,r (1) and thus c d Γ g,r = 12 χ , where χ is one fourth of the Meyer signature class, which is a generator of H (Γ g, ) ⊂ H (Γ g ).When ζ is a root of unity of order N then the class of the extension d Γ g,r is the image of 12 χ in H (Γ g,r ; Z /N Z )by the reduction mod N .The next step is to prove a similar statement when the number s of punctures is non-zero. Definition 2.4.
For ( m , m , . . . , m s ) ∈ Z s let Γ sg,r ( m , m , . . . , m s ) be the central extension of Γ sg,r by A having the following presentation:1. Generators are the f D α , where D α are Dehn twist generators of Γ sg,r and the central element z of thesame order as ζ − ;2. Relations are as follows. For each puncture p i the lift of the corresponding puncture relation reads: ^ D a ( i ) − ^ D a ( i ) − ^ D a ( i ) − ^ D a ( i ) ^ D a ( i ) ^ D a ( i ) = z m i where f D a are lifts of Dehn twists. Furthermore the chain and lantern relations have trivial lifts. Proposition 2.2.
Suppose that g ≥ . Then c Γ sg,r ( m ,...,m s ) ∈ A n +1 ⊂ H (Γ sg,r ; A ) is the vector m e + m e + · · · + m s e s , where e i is the Euler class of the i -th puncture.Proof. This is folklore. Consider first that ζ is not a root of unity. Let Σ s − g,r +1; i denote the subsurface of Σ sg,r obtained by removing a one-punctured disk centered at the puncture p i and thus creating a new boundarycomponent b i . We have then a central extension Z → Γ s − g,r +1; i → Γ sg,r → s − g,r +1; i ֒ → Σ sg,r . It is well-known that its cohomology class is c Γ s − g,r +1; i = e i . Lemma 2.9.
The extension Γ s − g,r +1; i is isomorphic to Γ sg,r (0 , . . . , , . . . , , where 1 is on the i -th position. roof. There is a natural set-wise section S i : Γ sg,r → Γ s − g,r +1; i , given by S i ( D α ) = D α , for any Dehn twist D α . In order to make sense, we might suppose that a simple closed curve α disjoint from the puncture p i isactually disjoint from b i so that it lies within Σ s − g,r +1; i .Braid, chain and lantern relations are then lifted trivially. A puncture relation at p j is lifted trivially if j = i . Consider next a puncture relation at p i in Σ sg,r , which is supported on some subsurface Σ , . Thethree boundary curves of Σ , lie within Σ s − g,r +1; i and together with b i bound a 4-holed sphere in Σ s − g,r +1; i .The lantern relation associated to this 4-holed sphere on Σ s − g,r +1; i is then the lift of the puncture relation at p i . The Dehn twist along b i is the generator z of the central factor ker(Γ s − g,r +1; i → Γ sg,r ). Thus the lift of apuncture relation at p i is the factor z . Lemma 2.10.
Let L m : Z s → Z denote the linear map L m ( n , . . . , n s ) = P si =1 m i n i , where m =( m , . . . , m s ) . Consider the central extension → Z s → Γ g,r + s → Γ sg,r → Then the map L m induces a quotient of Γ g,r + s , which is a central extension Γ sg,r ( m ) of Γ sg,r by Z which isisomorphic to Γ sg,r ( m , m , . . . , m s ) and gives rise to the following commutative diagram: → Z s → Γ g,r + s → Γ sg,r → ↓ L a ↓ π ↓ → Z → Γ sg,r ( m ) → Γ sg,r → Proof.
The class of the central extension c Γ g,r + s belongs to H (Γ sg,r ; Z s ) = ⊕ s H (Γ sg,r , Z ). By functorialitywe derive that c Γ g,r + s = ( e , e , . . . , e s ) ∈ H (Γ sg,r ; Z s ). Then the class c Γ sg,r ( a ) is the image of c Γ g,r + s into H (Γ sg,r ) by the homomorphism of coefficients rings L m : Z s → Z . There is an obvious set-wise section S defined in the same way as the S i from above. Then c Γ g,r + s is the class of the 2-cocycle L m C , where C isthe 2-cocycle associated to S and so L m C ( x, y ) = π ( S ( x ) − S ( y ) − S ( xy )) = L m (( S i ( x ) − S i ( y ) − S i ( xy ) i =1 ,s ) = s X i =1 m i C i ( x, y )where C i is the 2-cocycle associated to S i . Since the class of C i is e i it follows that the class of L m C is P si =1 m i ei .On the other hand the lifts of relations in Γ sg,r ( m ) are the same as in Γ sg,r ( m , . . . , m s ) and thus they areisomorphic. In fact the lifts of braid, chain and lantern relations to Γ g,r + s are trivial. The lift of a puncturerelation at p i is the i -th generator of the central factor Z s , according to Lemma 2.9. Therefore its imageinto Γ sg,r ( m ) is z m i , namely the lift of the puncture relation in Γ sg,r ( m , . . . , m s ).When ζ is a root of unity the extensions by Z above are replaced by extensions by Z /N Z and all argumentsgo through without essential modifications.This proves the Proposition. Proof of the Theorem.
Assume first that A is cyclic infinite. Consider the operation ⊗ (which is a push-out, or a fibered product) on central extensions defined as follows. If f i : G i → G are the projectionshomomorphisms of the central extensions G i of G by Z then G ⊗ G is the extension f ∗ G (or equivalently f ∗ G ) of G by Z . The class c G ⊗ G ∈ H ( G, Z ) is the direct sum of the classes c G i ∈ H ( G, Z ) under theidentification of H ( G, Z ) with the sum of two copies of H ( G, Z ).Let f denote the surjective homomorphism f : Γ sg,r → Γ g,r . Consider then the central extension1 → Z → f ∗ ( d Γ g,r ) ⊗ Γ sg,r (1 , , . . . , → Γ sg,r → L : Z → Z given by L ( x, y ) = x + y we find a quotient of f ∗ ( d Γ g,r ) ⊗ Γ sg,r (1 , , . . . , Z isomorphic to d Γ sg,r . In fact, there is a commutative diagram:1 → Z → f ∗ ( d Γ g,r ) ⊗ Γ sg,r (1 , , . . . , → Γ sg,r → ↓ L ↓ π ↓ → Z → d Γ sg,r → Γ sg,r → d Γ sg,r because the lifts of relations are the same.Braid and lantern relations lift trivially. Chain relations lift to z in f ∗ ( d Γ g,r ) and trivially to Γ sg,r (1 , , . . . , L (or π ) is z . Puncture relations at p i lift trivially to f ∗ ( d Γ g,r ) and to z inthe factor Γ sg,r (1 , , . . . , L (or π ) is z . As a consequence of this description the class c d Γ sg,r is the image by L of the class of f ∗ ( d Γ g,r ) ⊗ Γ sg,r (1 , , . . . , c f ∗ ( d Γ g,r ) + c Γ sg,r (1 , ,..., .On the other hand, by functoriality, the class c f ∗ ( d Γ g,r ) is f ∗ (12 χ ) = 12 χ ∈ H (Γ sg,r ), because the map f ∗ isthe standard embedding of H (Γ g,r ) = Z χ into H (Γ sg,r ). Proposition 2.2 proves the Theorem for g ≥ g = 2 one does not know the group H (Γ s ,r ), but for s = 0 and r ≤
1. Nevertheless, the classes χ and e j are still defined. It suffices to prove that: Lemma 2.11.
The subgroup of H (Γ s ,r ) generated by χ and e , . . . , e s is isomorphic to Z / Z ⊕ Z s .Proof. By the universal coefficients theorem we have1 → H (Γ s ,r ) → H (Γ s ,r ) → Hom( H (Γ s ,r ) , Z ) → H (Γ s ,r ) = Z / Z . The Meyer class χ in genus 2 is one half of theclass of Meyer’s cocycle from [23] and it generates the image of H (Γ s ,r ) into H (Γ s ,r ).Consider next the extensions Γ s ,r ( m ) for integral vectors m . According to the previous description lifts ofpuncture relations are of the form z m i . Suppose that there exists an isomorphism between the extensionsΓ s ,r ( m ) and Γ s ,r ( u ). Such an isomorphism of extensions should send f D α into z n ( α ) f D α , because it has toinduce the identity on Γ s ,r . Since lifts of braid relations are trivial in both extension groups it follows that n ( α ) = n does not depend on the non-separating curve α . But puncture relations are homogeneous, and sothey do not depend on n . This shows that m = u . In particular the classes e i span a free Z -submodule of H (Γ s ,r ).Since the class χ is of order 10 and both subgroups Z / Z (generated by χ ) and Z s (generated by e , . . . , e s )inject into H (Γ s ,r ), the claim follows.Then the arguments used above for g ≥ g = 2 and the Theorem follows. When ζ is a rootof unity the associated cohomology class is the reduction mod N of the corresponding integral cohomologyclass. Proof of Corollary 0.2 . Consider the extension \ Γ g,r + s of class 12 χ . The Corollary claims that there is anexact sequence: 1 → A s − → \ Γ g,r + s → d Γ sg,r → s blown up boundary components. Proof of Corollary 0.3 . It suffices to understand the map H (Γ sg,r ; A ) → H (Γ sg,r , C ∗ ) induced by z → ζ − .This map is injective, when g ≥ W , the following exact sequence isexact: 1 → Ext( H (Γ sg,r ) , W ) → H (Γ sg,r ; W ) → Hom( H (Γ sg,r ) , W ) → Z , W ) = 0, for any Abelian group W . This implies that H (Γ sg,r ; C ∗ ) = H (Γ sg,r ; C ∗ /A ) = 0, if g ≥
3. From the Bockstein exact sequence H (Γ sg,r ; C ∗ ) → H (Γ sg,r ; C ∗ /A ) β → H (Γ sg,r ; A ) ν → H (Γ sg,r ; C ∗ )we derive the claim.When g = 2 the Universal Coefficient Theorem shows, as above, that H (Γ s ,r ; C ∗ ) = Hom( H (Γ s ,r ) , C ∗ ) and H (Γ s ,r ; C ∗ /A ) = Hom( H (Γ s ,r ) , C ∗ /A ). Thus H (Γ s ,r ; C ∗ ) = Hom( Z / Z , C ∗ ) = U , where U is thesubgroup of roots of unity of order 10. The last isomorphism sends a homomorphism into its value on thegenerator 1. Next H (Γ s ,r ; C ∗ /A ) = Hom( Z / Z , C ∗ /A ) = U × A/ A . To explain the last isomorphism,each element f ∈ H (Γ s ,r ; C ∗ /A ) is determined by its value f (1) = As , for some s ∈ C ∗ . Here s = a n ∈ A ,where a is the generator of A . Fix some 10-th root a / ∈ C ∗ of the generator of A . Then the isomorphismabove associates to f the element ( sa − n/ , s ) ∈ U × A/ A , which is well-defined and independent of24he choice of the representative s in its A -coset. In particular the map H (Γ s ,r , C ∗ ) → H (Γ s ,r , C ∗ /A ) sends U onto the factor U of the second group.Let b f be a lift of f to b f : Z / Z = H (Γ sg,r ) → C ∗ , for instance b f ( k ) = s k , where k ∈ Z / Z . Then F ( k , k ) = b f ( k ) b f ( k ) b f ( k k ) − ∈ A is a 2-cocycle on H (Γ s ,r ) with values in A . The pull-back in H (Γ s ,r , A ) of the class of F by the map Γ s ,r → H (Γ s ,r ) is the element β ( f ). It is well-known that H ( Z / Z , A ) = A/ A is generated by the Euler class. Specifically, the cohomology class of the 2-cocycle F in H ( Z / Z , A ) is the element s ∈ A/ A , under the previous isomorphism.The Universal Coefficients Theorem shows that1 → Ext( H (Γ s ,r ) , A ) → H (Γ s ,r ; A ) → Hom( H (Γ s ,r ) , A ) → H (Γ sg,r ) , A ) = A/ A is generated by the class χ (as an A -valued cohomology class). Usingthe definition of Ext one identifies the class χ with the generator of H ( Z / Z ; A ). This implies that theimage of β is the subgroup generated by χ within H (Γ s ,r ; A ). Then Corollary 0.3 follows. References [1] E.W. Barnes,
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