Mapping class group actions from Hopf monoids and ribbon graphs
MMapping class group actions from Hopf monoids andribbon graphs
Catherine Meusburger Thomas Voß Department MathematikFriedrich-Alexander-Universität Erlangen-NürnbergCauerstraße 11, 91058 Erlangen, GermanyFebruary 10, 2020
Abstract
We show that any pivotal Hopf monoid H in a symmetric monoidal category C gives riseto actions of mapping class groups of oriented surfaces of genus g ≥ n ≥ H . They are associated withedge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give aconcrete description of these mapping class group actions in terms of generating Dehn twistsand defining relations. For the case where C is finitely complete and cocomplete, we also obtainactions of mapping class groups of closed surfaces by imposing invariance and coinvariance underthe Yetter-Drinfeld module structure. Mapping class group representations arising from Hopf algebras have been investigated extensivelyin the context of topological quantum field theories, Chern-Simons theories and conformal fieldtheories and in the quantisation of moduli spaces of flat connections.In [Ly95a, Ly95b, Ly96] Lyubashenko constructed projective representations of surface mappingclass groups from Hopf algebras in certain abelian ribbon categories. At the same time, projectivemapping class group representations were obtained from the quantisation of Chern-Simons gaugetheories, via Reshetikhin-Turaev TQFTs [RT] and via the combinatorial quantisation formalism dueto Alekseev, Grosse and Schomerus [AGS95, AGS96, AS] and Buffenoir and Roche [BR95, BR96].As shown in [AS], the mapping class group representations from the latter are equivalent to theones obtained from [RT]. Recently, they were revisited and generalised by Faitg [Fa18a, Fa18b],who related them to Lyubashenko’s representations. Brochier, Ben-Zvi and Jordan placed them inthe context of factorisation homology [BBJ18].A construction based on similar representation theoretical data was employed by Fuchs, Schweigertand Stigner in [FS, FSS12, FSS14] to construct mapping class group invariants in the contextof conformal field theories. Recently, Fuchs, Schaumann and Schweigert [FSS] constructed amore general modular functor and associated mapping class group representations from bimodulecategories over certain tensor categories.The conditions on the Hopf algebras and the underlying tensor categories in these works are lessrestrictive than the ones for TQFTs. However, the underlying monoidal category is still requiredto have duals, to be linear over a field and abelian. This excludes interesting examples of Hopfmonoids such as group objects in cartesian monoidal categories (including groups, strict 2-groups [email protected] [email protected] a r X i v : . [ m a t h . QA ] F e b nd simplicial groups) or combinatorial Hopf monoids such as the ones in [AM]. It also does notdirectly include geometric examples, such as mapping class group actions on the moduli spaceHom( π (Σ) , G ) /G of flat G -connections on a surface Σ or on Teichmüller space.In this article, we construct mapping class group actions from more general data, namely a pivotalHopf monoid H in a symmetric monoidal category C . We neither require that C is linear oversome field nor that it is abelian, has a zero object or is equipped with duals. In particular, theconstruction can be applied to cartesian monoidal categories such as Set, Top, Cat or the categoryPSh( A ) = Set A op , where Hopf monoids correspond to group objects.We show that pivotality, which generalises the notion of a pivotal Hopf algebra in C = Vect F , issufficient to obtain actions of mapping class groups of surfaces with boundary. Already in the case C = Vect F this assumption is much weaker than involutivity, semisimplicity or the requirement that H is ribbon and allows for additional examples. We also emphasise that pivotality is a choice ofstructure. Even for involutive Hopf monoids, where the square of the antipode is the identity, onemay choose a non-trivial pivotal structure and thus modify the mapping class group actions.If the symmetric monoidal category C is finitely complete and cocomplete and the pivotal structureinvolutive, our construction also yields actions of mapping class groups of closed surfaces. These areconcretely in terms of generating Dehn twists, and the description is simple enough to verify itsrelations by explicit computations.More specifically, we consider the mapping class group Map(Σ) of an oriented surface Σ of genus g ≥ H in a symmetric monoidal category C .To the graph Γ with edge set E , we associate the E -fold tensor product H ⊗ E in C . Edge orientationreversal corresponds to an involution constructed from the pivotal structure and the antipode. Everyedge of Γ is associated with two H -module and two H -comodule structures on H ⊗ E , and every pairof a vertex and an adjacent face in Γ defines a H -Yetter-Drinfeld module structure on H ⊗ E .The mapping class group actions are obtained from edge slides that generalise the chord slides in[B, ABP]. They transform the graph Γ by sliding an edge end in Γ along an adjacent edge. Toeach edge slide we assign an automorphism of H ⊗ E constructed from the H -module and comodulestructures for these edges. From the presentation of the mapping class group in [B] we then obtaina mapping class group action by automorphisms of Yetter-Drinfeld modules. Theorem 1:
Let Σ be an oriented surface of genus g ≥ ρ : Map(Σ) → Aut
Y D ( H ⊗ g ).We then generalise this result to surfaces Σ of genus g ≥ n + 1 ≥ π (Σ) and n additional edges that extend to the boundary. To each generatingDehn twist in [G01] we assign an automorphism of the object H ⊗ n + g ) . These are automorphismsof Yetter-Drinfeld modules with respect to a certain Yetter-Drinfeld module structure on H ⊗ n + g ) and satisfy the defining relations of the mapping class group Map(Σ). Theorem 2:
Let Σ be an oriented surface of genus g ≥ n + 1 ≥ ρ : Map(Σ) → Aut
Y D ( H ⊗ n + g ) ).If C is finitely complete and cocomplete, one can define (co)invariants of H -(co)modules and H -modules as (co)equalisers of their (co)action morphisms and associate to each Yetter-Drinfeld module M over H an object M inv in C that is both, invariant and coinvariant. If the pivotal structure isinvolutive, this yields mapping class group actions for closed surfaces by automorphisms of M inv .2 heorem 3: Let C be a symmetric monoidal category that is finitely complete and cocompleteand H a pivotal Hopf monoid in C with an involutive pivotal structure. Then the mapping classgroup action of Theorem 2 induces a group homomorphism ρ : Map(Σ ) → Aut( H ⊗ n + g ) inv ), whereΣ is obtained by attaching a disc to a boundary component of Σ.For n = 0, this defines actions of mapping class groups of closed surfaces. In particular, we obtainthe mapping class group action on the moduli space Hom( π (Σ) , G ) /G for a group G and a surfaceΣ of genus g ≥
1. We can also modify these actions with different (involutive) pivotal structures.The article is structured as follows. Section 2 contains the required background on Hopf monoids insymmetric monoidal categories and on their modules and comodules. Section 3 summarises thebackground on ribbon graphs and Section 4 the descriptions of mapping class groups from [G01, B].In Section 5 we introduce the H -module, H -comodule and H -Yetter-Drinfeld module structures on H ⊗ E associated with an embedded graph Γ. This is inspired by Kitaev’s quantum double models[Ki, BMCA], but no background on these models is required.Section 6 introduces graph transformations by edge slides and the associated automorphisms. Itshows that these edge slides satisfy the defining relations in [B] and thus define mapping class groupactions for oriented surfaces of genus g ≥ B on H ⊗ and an action of the modular group SL(2 , Z ) on H ⊗ inv .In Section 8 we generalise edge slides to slides along certain closed paths in the underlying graphand introduce Dehn twists. These are the technical prerequisites for Section 9, where we consider agenerating set of Dehn twists and prove that they satisfy the relations for the mapping class groupof a surface Σ of genus g ≥ n + 1 ≥ C and aninvolutive pivotal structure and is dedicated to the proof of Theorem 3. We illustrate this with theexample for the mapping class group of a surface of genus 2 with one or no boundary components. Throughout the article we consider a symmetric monoidal category C with unit object e and crossingor braiding morphisms τ X,Y : X ⊗ Y → Y ⊗ X . We denote by 1 X the identity morphism on anobject X . We use the usual diagrammatic notation for symmetric monoidal categories. Diagramsare read from left to right for tensor products and from top to bottom for the composition ofmorphisms. In formulas, we suppress associators and left and right unit constraints of C . Definition 2.1.
Let C be a symmetric monoidal category.1. A monoid in C is a triple ( A, m, η ) of an object A in C and morphisms m : A ⊗ A → A and η : e → A such that, up to coherence data, m ◦ (1 A ⊗ m ) = m ◦ ( m ⊗ A ) m ◦ (1 A ⊗ η ) = m ◦ ( η ⊗ A ) = 1 A .
2. A comonoid in C is a triple ( C, ∆ , (cid:15) ) of an object C in C and morphisms ∆ : C → C ⊗ C and (cid:15) : C → e such that, up to coherence data, (∆ ⊗ C ) ◦ ∆ = (1 C ⊗ ∆) ◦ ∆ ( (cid:15) ⊗ C ) ◦ ∆ = (1 C ⊗ (cid:15) ) ◦ ∆ = 1 C . . A bimonoid in C is a pentuple ( H, m, η, ∆ , (cid:15) ) such that ( H, m, η ) is a monoid, ( H, ∆ , (cid:15) ) acomonoid in C and, up to coherence data ∆ ◦ m = ( m ⊗ m ) ◦ (1 H ⊗ τ H,H ⊗ H ) ◦ (∆ ⊗ ∆) ∆ ◦ η = η ⊗ η(cid:15) ◦ m = (cid:15) ⊗ (cid:15) (cid:15) ◦ η = 1 e .
4. A bimonoid ( H, m, η, ∆ , (cid:15) ) in C is a Hopf monoid if there is a morphism S : H → H with m ◦ ( S ⊗ H ) ◦ ∆ = m ◦ (1 H ⊗ S ) ◦ ∆ = η ◦ (cid:15). If a bimonoid in C has an antipode, then this antipode is unique. The antipode is an anti-monoidand anti-comonoid morphism: m ◦ ( S ⊗ S ) ◦ τ H,H = S ◦ m, S ◦ η = η, τ H,H ◦ ( S ⊗ S ) ◦ ∆ = ∆ ◦ S, (cid:15) ◦ S = (cid:15). The multiplication m : H → H , unit η : e → H , comultiplication ∆ : H → H ⊗ H , counit (cid:15) : H → e and the antipode S : H → H are described by the following diagrams, respectively , , , , . (1)The condition that H is a monoid in C reads= , = = , (2)and the condition that H is a comonoid in C is= , = = . (3)Using (2) and (3), we sometimes write m n : H ⊗ ( n +1) → H and ∆ n : H → H ⊗ ( n +1) for the n -foldcomposites of m and ∆. We denote them by diagrams analogous to (1) but with n + 1 lines meetingin a single point or on a horizontal line.The compatibility conditions between the monoid and comonoid structure correspond to the diagrams= , = , = , = , and the defining conditions on the antipode read= = . (4)That the antipode is an anti-monoid and anti-comonoid morphism is expressed in the diagrams= , = , = , = . (5)4here this simplifies the presentation, we also use Sweedler notation. For C = Vect F this is the usualSweedler notation for a Hopf algebra over a field F . For a general symmetric monoidal category C itis to be interpreted as a shorthand notation for a diagram in C .In the following, we consider Hopf monoids with additional structure, which generalise pivotal Hopfalgebras over a field F . The concept of a pivotal Hopf algebra in C = Vect F was introduced in [BaW,Def 3.1] as a Hopf algebra H over F together with a choice of a grouplike element p ∈ H , the pivotalelement , such that h = pS ( h ) p − for all h ∈ H . It is clear from its definition that the pivotalelement p is unique up to multiplication with central grouplike elements in H .This definition easily generalises to a Hopf monoid in a symmetric monoidal category C . In thiscase, a grouplike element p ∈ H is replaced by a morphism p : e → H satisfying ∆ ◦ p = p ⊗ p and (cid:15) ◦ p = 1 e . The second condition generalises the requirement p = 0 for a grouplike element p ∈ H .Note also that these conditions imply together with (4) m ◦ ( S ⊗ H ) ◦ ( p ⊗ p ) = η = m ◦ (1 H ⊗ S ) ◦ ( p ⊗ p ) , which replaces the identity S ( p ) = p − for grouplike elements. The condition h = pS ( h ) p − for all h ∈ H can then be formulated as a condition on the morphism p : e → H . This yields the followingdefinition, which reduces to the notion of pivotality in [BaW] for C = Vect F . Definition 2.2.
Let C be a symmetric monoidal category. A pivotal Hopf monoid in C is a pair ( H, p ) of a Hopf monoid H in C and a morphism p : e → H satisfying the identities ∆ ◦ p = p ⊗ p (cid:15) ◦ p = 1 e m ◦ (1 H ⊗ S ⊗ S ) ◦ ( p ⊗ H ⊗ p ) = 1 H . A Hopf monoid H in C is called involutive if ( H, η ) is a pivotal Hopf monoid. We denote the morphism p : e → H by a diagram similar to the one for the unit morphism, butwith a small black circle labeled p instead of a small white circle. The conditions in Definition 2.2then correspond to the diagrams p = p p , p = , p p = . (6)Note that the third condition implies that the antipode S : H → H is an isomorphism with inverse S − = m ◦ (1 H ⊗ S ⊗ S ) ◦ ( p ⊗ H ⊗ p ) : H → H . In diagrams we denote S − by a grey circle.Identities (4) and (5) then imply identities analogous to (5) and the following counterpart of (4)= = . The condition that a Hopf monoid H in C has a pivotal structure is much less restrictive thaninvolutivity. Many standard examples of Hopf algebras are pivotal Hopf monoids in C = Vect F . Example 2.3.
1. If H is an involutive Hopf algebra over a field F , it is trivially pivotal with p = 1 ∈ H . Pivotalstructures on H are in bijection with central grouplike elements p ∈ H .2. If H is a ribbon Hopf algebra over F with Drinfeld element u = S ( R (2) ) R (1) and ribbon element ν satisfying uS ( u ) = ν and ∆( ν ) = ( ν ⊗ ν )( R R ) − , then H is pivotal with p = u − ν . The term pivotal Hopf algebra was subsequently adopted in other publications. The grouplike element p in thisarticle corresponds to the inverse of the grouplike element in [BaW, Def 3.1]. . If H is a Hopf algebra over F with an invertible antipode, one can adjoin a grouplike elementto form a pivotal Hopf algebra as follows. Let G ∼ = Z be the free group generated by the singleelement p . Then the tensor product H ⊗ F [ G ] is a Hopf algebra with ( h ⊗ p m ) · ( k ⊗ p n ) = hS − m ( k ) ⊗ p m + n η (1) = 1 ⊗ h ⊗ p m ) = ( h (1) ⊗ p m ) ⊗ ( h (2) ⊗ p m ) (cid:15) ( h ⊗ p m ) = (cid:15) ( h ) S ( h ⊗ p m ) = S m ( h ) ⊗ p − m , and the element ⊗ p ∈ H ⊗ F [ G ] is pivotal. This shows that any Hopf algebra with invertibleantipode is embedded in a pivotal Hopf algebra. If H is finite-dimensional with dim F ( H ) = d ,then adding the additional relation p d = 1 yields a finite-dimensional pivotal Hopf algebracontaining H . This finite-dimensional example is from [So].4. For any complex semisimple finite-dimensional Lie algebra g , the Drinfeld-Jimbo algebra U q ( g ) generated by elements E i , F i , K i , K − i as in [KlS] is a pivotal Hopf algebra. The pivotal elementis the inverse of the grouplike element K ρ , where ρ is the half-sum of the positive roots. Aproof is given in [KlS, Ch. 6, Prop. 6].5. It is clear from Example 4. that U q ( g ) and U q ( g ) res for q an n th root of unity with n > arealso pivotal Hopf algebras. The same holds for their Hopf-subalgebras U q ( b + ) and U q ( b − ) .6. In a cartesian monoidal category C , where the tensor product is given by a product, Hopfmonoids H in C are precisely the group objects in C . This includes groups for C = Set ,topological groups for C = Top , abelian groups for C = Grp or C = Ab and strict 2-groups for C = Cat or C = Grpd . If C = G − Set for a group G , group objects in C are semidirect products H (cid:111) φ G . If C is concrete, pivotal structures on H correspond to central elements of H .7. If A is a small category and B a symmetric monoidal category, then the category B A offunctors from A to B and natural transformations between them is a symmetric monoidalcategory. In particular, this applies to the cartesian monoidal category B = Set and thecategory C = PSh( A ) = Set A op of presheaves on A . Hopf monoids in PSh( A ) correspond tofunctors F : A op → Grp , and these are involutive. In particular, Hopf monoids in the category C = sSet = PSh(∆) of simplicial sets are simplicial groups. The reason why we require a pivotal structure on a Hopf monoid H is that it defines an involutiveautomorphism of H . This automorphism is denoted by a double circle in diagrams and given by T = m ◦ ( p ⊗ S ) = m ◦ ( S − ⊗ p ) : H → H, (7):= p = p . Lemma 2.4.
Let ( H, p ) be a pivotal Hopf monoid in a symmetric monoidal category C .1. The morphism T from (7) is an involution: T ◦ T = 1 H .2. It is an anti-comonoid morphism = , = . (8)
3. It satisfies the diagrammatic identities = = , = p , = p = . (9)6 roof. The identities follow by direct diagrammatic computations. That T is an involution isobtained from the defining diagram for T in (7), the second identity in (5) and the last identity in(6). The identities in (8) follow from the defining diagram for T in (7), the first and last identitiesin (5) and the first diagram in (6). The first two identities in (9) follow from (7), the second andthird identity in (5) and the associativity of the multiplication. The last two identities in (9) followsfrom (7), the associativity of the multiplication and the defining diagrams (4) for the antipode. In this section, we summarise background and notation for (co)modules over (co)monoids in asymmetric monoidal category.
Definition 2.5.
Let C be a symmetric monoidal category, ( A, m, η ) a monoid in C and ( C, ∆ , (cid:15) ) acomonoid in C .1. A (left) module over A is an object M ∈ Ob C together with a morphism (cid:66) : A ⊗ M → M such that (cid:66) ◦ ( m ⊗ M ) = (cid:66) ◦ (1 A ⊗ (cid:66) ) and (cid:66) ◦ ( η ⊗ M ) = 1 M .2. A morphism of A -modules from ( M, (cid:66) ) to ( M , (cid:66) ) is a morphism f : M → M such that (cid:66) ◦ (1 A ⊗ f ) = f ◦ (cid:66) .3. A (left) comodule over C is an object M ∈ Ob C together with a morphism δ : M → C ⊗ M such that (∆ ⊗ M ) ◦ δ = (1 C ⊗ δ ) ◦ δ and ( (cid:15) ⊗ M ) ◦ δ = 1 M .4. A morphism of C -comodules from ( M, δ ) to ( M , δ ) is a morphism f : M → M with δ ◦ f = (1 C ⊗ f ) ◦ δ . In the diagrams we denote (co)modules by fat coloured vertical lines and the action and coactionmorphisms by vertices on these lines. The defining conditions for left modules and left comodulesthen take the form= , = , = , = . (10)The defining properties of left module and left comodule morphisms are given by the diagrams MM f = MM f , MM f = MM f . (11)Right (co)modules over a (co)monoid in C , morphisms of right (co)modules and their diagrams aredefined analogously.Note also that if H is a pivotal Hopf monoid in C , the involution T : H → H from (7) relates H -left and right comodule structures. As T is an anti-comonoid morphism by Lemma 2.4, for any H -left-comodule ( M, δ ), we obtain a H -right comodule ( M, δ ) by setting δ = τ H,M ◦ ( T ⊗ M ) ◦ δ : M → M ⊗ H. (12)Bimodules over a monoid A are defined as triples ( M, (cid:66) , (cid:67) ) such that ( M, (cid:66) ) is a left module,( M, (cid:67) ) a right module over A and (cid:67) ◦ ( (cid:66) ⊗ A ) = (cid:66) ◦ (1 A ⊗ (cid:67) ). Bicomodules over a comonoid C as7riples ( M, δ L , δ R ) such that ( M, δ L ) is a left-comodule over C , ( M, δ R ) a right-comodule over C and( δ L ⊗ C ) ◦ δ R = (1 C ⊗ δ R ) ◦ δ L . Morphisms of bimodules or of bicomodules are morphisms that areboth left- and right module morphisms or left- and right comodule morphisms. The compatibilityconditions between left and right actions and left and right coactions are given by the diagrams= , = . In the following, we will need to consider (co)invariants of (co)modules over Hopf monoids in C . Aswe do not restrict attention to abelian categories or even categories with zero objects, we impose thatthe category C has all coequalisers and equalisers and define them as coequalisers and equalisers. Definition 2.6.
Let C be a symmetric monoidal category that has all equalisers and coequalisersand H a Hopf monoid in C .1. The invariants of an H -left module ( M, (cid:66) ) are the coequaliser ( M H , π ) of (cid:66) and (cid:15) ◦ M : H ⊗ M (cid:15) ⊗ M (cid:47) (cid:47) (cid:66) (cid:47) (cid:47) M π (cid:47) (cid:47) M H .
2. The coinvariants of an H -left comodule ( M, δ ) are the equaliser ( M coH , ι ) of δ and η ⊗ M : M coH ι (cid:47) (cid:47) M η ⊗ M (cid:47) (cid:47) δ (cid:47) (cid:47) H ⊗ M. (Co)invariants of H -right (co)modules are defined analogously. Note that for C = Vect F Definition 2.6of coinvariants coincides with the usual definition as the subset M coH = { m ∈ M | δ ( m ) = 1 ⊗ m } .This is not the case for the definition of invariants in terms of the coequaliser, which yields a quotientof the H -module M , not a linear subspace.It follows directly from Definition 2.6 that morphisms of H -(co)modules induce morphisms betweentheir (co)invariants. In the following we say that two morphisms f : M → M of H -(co)modules agree on the (co)invariants if the induced morphisms between the (co)invariants agree. Lemma 2.7.
Let C be a symmetric monoidal category that has all equalisers and coequalisers and H a Hopf monoid in C .1. For every H -module morphism f : ( M, (cid:66) ) → ( M , (cid:66) ) there is a unique morphism f H : M H → M H with f H ◦ π = π ◦ f .2. For every H -comodule morphism f : ( M, δ ) → ( M , δ ) there is a unique morphism f coH : M coH → M coH with ι ◦ f coH = f ◦ ι .Proof. If f : ( M, δ ) → ( M , δ ) is a morphism of H -comodules, then δ ◦ f = (1 H ⊗ f ) ◦ δ and δ ◦ ( f ◦ ι ) = (1 H ⊗ f ) ◦ δ ◦ ι = (1 H ⊗ f ) ◦ ( η ⊗ M ) ◦ ι = ( η ⊗ M ) ◦ ( f ◦ ι ) . By the universal property of the equaliser M coH , there is a unique morphism f coH : M coH → M coH with ι ◦ f coH = f ◦ ι . The proof for invariants is analogous.In the following, we also consider morphisms that are both, module and comodule morphisms, andimpose both, invariance and coinvariance. As we do not restrict attention to categories with a zeroobject, we implement this with the (non-abelian) notion of an image, see for instance [Mi, I.10].An image of a morphism f : C → C in C is a pair ( P, I ) of a monomorphism I : im( f ) → C anda morphism P : C → im( f ) with I ◦ P = f such that for any pair ( Q, J ) of a monomorphism8 : X → C and a morphism Q : C → X with J ◦ Q = f there is a unique morphism v : im( f ) → X with I = J ◦ v . Note that if C has all equalisers, then the morphism P : C → im( f ) is an epimorphism[Mi, Sec I.10]. If C is finitely complete and cocomplete, images are obtained as equalisers of cokernelpairs [KaS, Def 5.1.1], and their existence is guaranteed. In particular, this holds for the categoriesSet, Top, Ab, Grp, Cat, the category G − Set for a group G and the category PSh( A ) for a smallcategory A from Example 2.3, 6. and 7. Note also that if C is abelian, this coincides with the usualnotion of an image in an abelian category. Definition 2.8.
Let C be a symmetric monoidal category that is finitely complete and cocompleteand H a Hopf monoid in C . The biinvariants of an object M in C that is both an H -left moduleand H -left comodule are the image of the morphism π ◦ ι : M coH → M H M coH P (cid:40) (cid:40) ι (cid:47) (cid:47) M π (cid:47) (cid:47) M H M inv := im( π ◦ ι ) . I (cid:55) (cid:55) (13)By Lemma 2.7 morphisms of H -(co)modules induce morphisms between their (co)invariants. Asimilar statement holds for isomorphisms that respect both, the H -module and H -comodule structure,and for the associated biinvariants. Lemma 2.9.
Let H be a Hopf monoid in a finitely complete and cocomplete symmetric monoidalcategory C . Then for any isomorphism φ : M → M of H -modules and H -comodules there is aunique morphism φ inv : M inv → M inv with π ◦ φ ◦ ι = I ◦ φ inv ◦ P , and φ inv is an isomorphism.Proof. By Lemma 2.7 the isomorphism φ : M → M induces isomorphisms φ coH : M coH → M coH and φ H : M H → M H such that the following diagram commutes M coHφ coH (cid:15) (cid:15) ι (cid:47) (cid:47) M φ (cid:15) (cid:15) π (cid:47) (cid:47) M Hφ H (cid:15) (cid:15) M coH ι (cid:47) (cid:47) M π (cid:47) (cid:47) M H . (14)Let ( M inv , I, P ) and ( M inv , I , P ) be the biinvariants for M and M from (13). Then by (13) and(14) the morphisms J = ( φ H ) − ◦ I : M inv → M H and Q = P ◦ φ coH : M coH → M inv satisfy J ◦ Q = π ◦ ι , and there is a unique morphism φ inv : M inv → M inv with I = J ◦ φ inv = ( φ H ) − ◦ I ◦ φ inv .This condition implies I ◦ φ inv ◦ P = π ◦ φ ◦ ι by (13) and (14). As P is an epimorphism and I amonomorphism, the last equation determines φ inv uniquely. By applying this to the inverse of φ ,one finds that φ inv is an isomorphism. Example 2.10.
1. A Hopf monoid H in Set is a group, H -modules are H -sets and an H -comodule is a set M together with a map f : M → H , whose graph is the morphism δ : M → H × M . • The invariants of an H -set M are the orbit space M H = M/H with the canonicalsurjection π : M → M/H . • The coinvariants of an H -comodule ( M, f ) are the subset M coH = { m ∈ M | f ( m ) = 1 } with the inclusion ι : M coH → M . • The biinvariants of an H -module and comodule M are the set M inv = π ( M coH ) togetherwith the inclusion I : M inv → M/H and the surjection P : M coH → M inv , m π ( m ) .2. If C is abelian, the invariants of an H -module ( M, (cid:66) ) are the cokernel of the morphism (cid:66) − (cid:15) ⊗ M , the coinvariants of an H -comodule ( M, δ ) are the kernel of the morphism δ − η ⊗ M , and the biinvariants are the image of π ◦ ι : M coH → M H in C . . If H is a finite-dimensional semisimple Hopf algebra in C = Vect F with char( F ) = 0 , then • the invariants of an H -module ( M, (cid:66) ) are M H = { m ∈ M | h (cid:66) m = (cid:15) ( h ) m ∀ h ∈ H } with π : M → M H , m ‘ (cid:66) m , where ‘ is the normalised Haar integral of H , • the coinvariants of an H -comodule ( M, δ ) are M coH = { m ∈ M | δ ( m ) = 1 ⊗ m } withthe inclusion ι : M coH → M , • the biinvariants of an H -module and H -comodule M are M inv = M H ∩ M coH with theinclusion I : M inv → M H and the surjection P : M coH → M inv , m ‘ (cid:66) m . For a bimonoid H in C an object M in C that has both a H -module structure (cid:66) : H ⊗ M → M anda H -comodule structure δ : M → H ⊗ M one can impose certain compatibility conditions between (cid:66) and δ . This leads to the notions of Hopf modules [LS] and Yetter-Drinfeld modules [Y]. Definition 2.11.
Let H be a bimonoid in a symmetric monoidal category C . A left-left-Hopfmodule over H is a triple ( M, (cid:66) , δ ) such that ( M, (cid:66) ) is a left module over H , ( M, δ ) is a left-comodule over H and the following compatibility condition is satisfied = (15) A morphism of left-left- H -Hopf modules from ( M, (cid:66) , δ ) to ( M , (cid:66) , δ ) is a morphism f : M → M that is a morphism of H -left modules and of H -left comodules. There are analogous definitions of left-right H -Hopf modules, right-left H -Hopf modules and aright-right H -Hopf modules, where the left action of H is replaced by a right action or the leftcoaction by a right coaction. In these cases the diagrams that replace (15) read= , = , = . Analogously, a Hopf bimodule over a Hopf monoid H in C is defined as a pentuple ( M, (cid:66) , (cid:67) , δ L , δ R )such that ( M, (cid:66) , (cid:67) ) is a H -bimodule, ( M, δ L , δ R ) is a H -bicomodule and ( M, (cid:66) , δ L ), ( M, (cid:66) , δ R ),( M, (cid:67) , δ L ), ( M, (cid:67) , δ R ) are left-left, left-right, right-left and right-right- H -Hopf modules, respectively. Example 2.12.
Let H be a Hopf monoid in C .1. H is a Hopf bimodule over itself with (cid:66) = (cid:67) = m : H ⊗ H → H , δ L = δ R = ∆ : H → H ⊗ H .2. For any object M in C , the object H ⊗ M is a Hopf bimodule with (cid:66) = m ⊗ M : H ⊗ H ⊗ M → H ⊗ M (cid:67) = ( m ⊗ M ) ◦ (1 H ⊗ τ ) : H ⊗ M ⊗ H → H ⊗ Mδ L = ∆ ⊗ M : H ⊗ M → H ⊗ H ⊗ M, δ R = (1 H ⊗ τ ) ◦ (∆ ⊗ M ) : H ⊗ M → H ⊗ M ⊗ H. Hopf bimodules and the associated Hopf modules of this type are called trivial.
The main examples of Hopf (bi)modules considered in this article are the trivial ones from Example2.12. By the fundamental theorem of Hopf modules [LS, Prop. 1], any Hopf module M over a Hopfmonoid H in C = Vect F is isomorphic to the trivial Hopf module H ⊗ M coH . For a Hopf algebra inan abelian rigid braided monoidal category an analogous result was shown in [Ly95b, Thm 1.2].10ote also that for finite-dimensional Hopf algebras in C = Vect F , the notion of a Hopf modulecoincides with the notion of a module over the Heisenberg double of H , which exists in differentversions corresponding to left-left, left-right, right-left and right-right Hopf modules.Just as Hopf modules can be viewed as categorical analogues of modules over Heisenberg doubles,there is also an analogue of modules over the Drinfeld double D ( H ), namely Yetter-Drinfeld modules.Just as Hopf modules, they come in four variants, for left- and right module and comodule structures.We restrict attention to left-module and left-comodule structures. Definition 2.13.
Let H be a bimonoid in a symmetric monoidal category C . A left-left Yetter-Drinfeld module over H is a triple ( M, (cid:66) , δ ) such that ( M, (cid:66) ) is a left module over H , ( M, δ ) aleft-comodule over H and the following compatibility condition is satisfied = . (16) A morphism of H -left-left Yetter-Drinfeld modules from ( M, (cid:66) , δ ) to ( M , (cid:66) , δ ) is a mor-phism f : M → M that is a morphism of H -left modules and of H -left comodules. In this section, we summarise the required background on embedded graphs or ribbon graphs . Formore details, see for instance [LZ]. All graphs considered in this article are finite, but we allowloops, multiple edges and univalent vertices. Paths in a graph are most easily described by orienting the edges of Γ and considering the freegroupoid generated by the resulting directed graph. Note that different choices of orientation yieldisomorphic groupoids.
Definition 3.1.
The path groupoid G Γ of a graph Γ is the free groupoid generated by Γ . A path in Γ from a vertex v to a vertex w is a morphism γ : v → w in G Γ . The objects of G Γ are the vertices of Γ. A morphism from v to w in G Γ is a finite sequence p = α (cid:15) ◦ ... ◦ α (cid:15) n n , (cid:15) i ∈ {± } of oriented edges α i and their inverses such that the starting vertex ofthe first edge α (cid:15) n n is v , the target vertex of the last edge α (cid:15) is w , and the starting vertex of eachedge in the sequence is the target vertex of the preceding edge. These sequences are taken withthe relations α ◦ α − = 1 t ( α ) and α − ◦ α = 1 s ( e ) , where α − denotes the edge α with the reversedorientation, s ( α ) the starting and t ( α ) the target vertex of α and we set s ( α ± ) = t ( α ∓ ).An edge α ∈ E with s ( α ) = t ( α ) is called a loop , and a path ρ ∈ G Γ is called closed if it is anautomorphism of a vertex. We call a path ρ ∈ G Γ a subpath of a path γ ∈ G Γ if the expression for γ as a reduced word in E is of the form γ = γ ◦ ρ ◦ γ with (possibly empty) reduced words γ , γ . Wecall it a proper subpath of γ if γ , γ are not both empty. We say that two paths ρ, γ ∈ G Γ overlap ifthere is an edge in Γ that is traversed by both ρ and γ , and by both in the same direction. The graphs we consider have additional structure. They are called ribbon graphs , fat graphs or embedded graphs and give a combinatorial description of oriented surfaces with or without boundary.11 23456 123 74 65Figure 1: A ciliated vertex and a ciliated face in a directed ribbon graph Definition 3.2. A ribbon graph is a graph together with a cyclic ordering of the edge ends ateach vertex. A ciliated vertex in a ribbon graph is a vertex together with a linear ordering of theincident edge ends that is compatible with their cyclic ordering. A ciliated vertex in a ribbon graph is obtained by selecting one of its incident edge ends as thestarting end of the linear ordering. We indicate this in figures by assuming the counterclockwisecyclic ordering in the plane and inserting a line, the cilium , that separates the edges of minimal andmaximal order, as shown in Figure 1. We say that an edge end β at a ciliated vertex v is between two edge ends α and γ incident at v if α < β < γ or γ < β < α . We denote by s ( α ) the startingend and by t ( α ) the target end of a directed edge α .The cyclic ordering of the edge ends at each vertex allows one to thicken the edges of a ribbon graphto strips or ribbons and its vertices to polygons. It also equips the ribbon graph with the notionof a face. One says that a path in a ribbon graph Γ turns maximally left at a vertex v if it enters v through an edge end α and leaves it through an edge end β that comes directly before α withrespect to the cyclic ordering at v . Definition 3.3.
Let Γ be a ribbon graph.1. A face path in Γ is a path that turns maximally left at each vertex in the path and traverseseach edge at most once in each direction.2. A ciliated face in Γ is a closed face path whose cyclic permutations are also face paths.3. A face of Γ is an equivalence class of ciliated faces under cyclic permutations. Examples of faces and face paths are shown in Figures 1 and 3. Each face defines a cyclic ordering ofthe edges and their inverses in the face. A ciliated face is a face together with the choice of a startingvertex and induces a linear ordering of these edges. These orderings are taken counterclockwise, asshown in Figure 1.A graph Γ embedded into an oriented surface Σ inherits a cyclic ordering of the edge ends at eachvertex from the orientation of Σ and hence a ribbon graph structure. Conversely, every ribbongraph Γ defines a compact oriented surface Σ Γ that is obtained by attaching a disc at each face.For a graph Γ embedded into an oriented surface Σ, the surfaces Σ Γ and Σ are homeomorphic ifand only if Σ \ Γ is a disjoint union of discs. An oriented surface with a boundary is obtained froma ribbon graph Γ by attaching annuli instead of discs to some of its faces.
In the following, we sometimes restrict attention to ribbon graphs with a single vertex. Such aribbon graph can be described equivalently by a circular chord diagram. This chord diagram (with122345678Figure 2: Ribbon graph with a single ciliated vertex and the associated chord diagram. α α α α α α − α − α α is a facepath, but not a ciliated face. α α α α α α − α − α α is not a facepath.the chords pointing outwards from the circle) is obtained by thickening the vertex of the ribbongraph to a circle. Similarly, a ribbon graph with a single ciliated vertex corresponds to a linear chord diagram that is obtained from the circular chord diagram by cutting the circle at the positionof the cilium, as shown in Figure 2.We call the resulting line segment the baseline of the chord diagram. The edges of the ribbon graphthen correspond to the chords of the diagram. The ribbon graph structure is given by the orderingof their ends on the circle or baseline, as shown in Figure 2. Two chords are called disjoint , if theydo not intersect, and two chord ends are called disjoint if they do not belong to the same chord. Inaddition to the chords, we sometimes also admit additional edges that start or end at the baselineand have a univalent vertex at their other end.For a ribbon graph with a single vertex, the path groupoid from Definition 3.1 becomes a group.In the associated chord diagram, we represent its elements by paths that start and end below thebaseline and are composed of segments along the chords and horizontal segments along the baseline,as shown in Figures 3 and 4. In this section, we review the background on mapping class groups for oriented surfaces with orwithout boundaries. Where no other references are given, we follow the presentation in [FM]. As in[FM, Sec. 2.1], we define the mapping class group of an oriented surface Σ with boundary ∂ Σ asMap(Σ) = Homeo + (Σ , ∂ Σ)Homeo (Σ , ∂ Σ) . where Homeo + (Σ , ∂ Σ) is the group of orientation preserving homeomorphisms that restrict to theidentity on the boundary and Homeo (Σ , ∂ Σ) is the subgroup of orientation preserving homeomor-phisms that fix the boundary and are homotopic to the identity.13his is not to be confused with the mapping class group of a surface with marked points or punctures.For the relation between these mapping class groups see for instance [Bi] or [FM, Sec. 2.1, 3.6] andthe references therein. In particular, Dehn twists along circles around the boundary componentsare not necessarily trivial.
A simple finite presentation of the mapping class group of a surface of genus g ≥ n ≥ | α ∩ β | = 0 if the homotopy classes of the closed simplecurves α, β have representatives that do not intersect and | α ∩ β | = 1 if they have representativeswith a single intersection point. Theorem 4.1 ([G01], Th. 1) . Let Σ be an oriented surface of genus g ≥ with n ≥ boundarycomponents such that g + n > . Then the mapping class group Map(Σ) is generated by the Dehntwists along the closed simple curves in the set G = (cid:8) α i | i = 1 , ..., g (cid:9) ∪ (cid:8) δ j | j = 1 , ..., n + 2 g − (cid:9) ∪ { γ k,l | k, l = 1 , ..., n + 2 g − , k = l (cid:9) in Figures 5 and 6, subject to the relations(i) D γ n +2 j +1 ,n +2 j = D γ n +2 j,n +2 j − for j = 1 , ..., g − and D γ ,n +2 g − = D γ n +2 g − ,n +2 g − ,(ii) D α ◦ D β = D β ◦ D α for α, β ∈ G with | α ∩ β | = 0 ,(iii) D α ◦ D β ◦ D α = D β ◦ D α ◦ D β for α, β ∈ G with | α ∩ β | = 1 ,(iv) ( D δ k ◦ D δ i ◦ D δ j ◦ D α g ) = D γ i,j ◦ D γ j,k ◦ D γ k,i for i, j, k not all equal and j ≤ i ≤ k or k ≤ j ≤ i or i ≤ k ≤ j with D γ i,i = id for ≤ i ≤ n + 2 g − . Relation (i) expresses the fact that the loops around the two legs of the j th handle for j = 1 , ..., g − γ i,j , γ j,k and γ k,i and along δ i , δ j and δ k commute by(ii), so that their relative order in (iv) is irrelevant.The essential intersection numbers of the curves in Figures 5 and 6 are given by | α i ∩ α j | = | δ i ∩ δ j | = 0 for all i = j, (17) | α i ∩ δ j | = ( i = g or j = n + 2 i | α i ∩ γ j,k | = ( j = n + 2 i or k = n + 2 i | δ i ∩ γ j,k | = ( k < i < j up to cyclic permutations0 otherwise, | γ i,j ∩ γ k,l | = j < k < l < i up to cyclic permutations0 if either j ≤ l ≤ k ≤ i, l ≤ j ≤ i ≤ k or j ≤ i ≤ l ≤ k up to cyclic permutations2 otherwise. The last relation is not stated explicitly in [G01, Th. 1] but implied by its proof and its verbal description. Notealso that our notation differs from the one in [G01]. Our curves α i are called β i , our curves δ i are called α i , and our α g is called β in [G01]. We reserve the letters α i and β i for certain generators of the fundamental group π (Σ). .. ... n − n − n n + 1 n + 2 n +2 g − n +2 g − n +2 g − n +2 g − α g − α g − α α g δ δ δ n − δ δ δ n +1 δ n − δ n δ n +2 δ n +3 δ n +2 g − δ n +2 g − δ n +2 g − δ n +2 g − Figure 5: The curves α i and δ j ... ... i − ik − k δ k jj − δ j δ i γ k,i γ i,j γ j,k α g . . . ... ... Figure 6: The curves δ i , δ j , δ k and γ i,j , γ j,k , γ k,i for j < i < k .15 .2 Presentation in terms of chord slides For an oriented surface Σ of genus g ≥ β along a chord α is defined if two ends of α and β are adjacentwith respect to the linear ordering of the chord ends at the baseline. It is given by the followingdiagram, where the holes in the circle indicate possible positions of other chord ends or the cilium. α β αβ Building on the work of Penner [P], which describes the mapping class group Map(Σ) in terms of flip moves or Whitehead moves on a triangulation, Bene [B] gives a simple presentation in terms ofchord slides.
Theorem 4.2 ([B], Th. 7.1) . Let Σ be a surface of genus g ≥ with one boundary component and Γ an embedded ribbon graph with a single ciliated vertex and face such that Σ \ Γ is an annulus.Then mapping class group Map(Σ) is presented by finite sequences of chord slides that preserve Γ upto the cilium subject to the relations1. Involutivity: the following composite of slides is the identity, α β αβ α β (18) Commutativity: chord slides of disjoint ends along disjoint chords commute, βαδγ β αδγβα δγ β α δγ (19) Triangle relation: αβ αβ αβα βα ↔ β (20)16 . Left pentagon relation: αγβ αγβ αγβαγβαγβ (21) Right pentagon relation: αγβ αγβ αγβαγβαγβ (22)Bene [B] also defines analogous chord slides and relations for oriented chords. In this case, the chordslides preserve the orientation, and each relation holds for all possible orientation of the chords. Thetriangle relation then requires an additional orientation reversal. It is also shown in [B] that therelations in Theorem 4.2 imply two further relations.
Lemma 4.3 ([B], Lemma 6.1) . The relations in Theorem 4.2 imply1.
Opposite end commutativity: chord slides of different ends of a chord commute.2.
Adjacent commutativity: chord slides along different sides of a chord commute. αγβ αγβ αγβαγβ αγβ αγβ αγβαγβ (23)17
Hopf monoid labelled ribbon graphs
The mapping class group actions we derive in this article are inspired by Kitaev’s lattice models or quantum double models , which were introduced by Kitaev in [Ki] for group algebras of a finite groupand generalised in [BMCA] to finite-dimensional semisimple Hopf-*-algebras over C . We generalisesome aspects of these models to pivotal Hopf monoids in a symmetric monoidal category C . Forthe case where C = Vect C and H is a finite-dimensional semisimple Hopf ∗ -algebra, our model isequivalent to the one in [BMCA], up to minor changes in conventions.Throughout this section, let H be a pivotal Hopf monoid in a symmetric monoidal category C withpivotal morphism p : e → H and Γ a directed ribbon graph with vertex set V and edge set E .We associate to Γ the object H ⊗| E | in C with different copies of H assigned to the oriented edgesof Γ. To emphasise this assignment, we use the notation H ⊗ E . This is a direct analogue of theconstruction of the extended Hilbert space in [Ki, BMCA].Edge orientation is reversed with the involutive morphism T : H → H from (7). If Γ is obtainedfrom Γ by reversing the orientation of an edge α and f : A ⊗ H ⊗ E ⊗ B → C ⊗ H ⊗ E ⊗ D a morphismin C associated with Γ, we define the associated morphism for Γ as T α ◦ f ◦ T α , where T α applies T to the copy of H for α and is the identity morphism on all other factors of the tensor product.Note that for an involutive Hopf monoid we can choose T = S and p = η .To each oriented edge α of Γ we assign two H -left module structures (cid:66) α ± : H ⊗ H ⊗ E → H ⊗ E and two H -left comodule structures δ α ± : H ⊗ E → H ⊗ H ⊗ E . The H -left module structures areassociated with the edge ends of α and the H -left comodule structures with the edge sides of α as shown below. They are related by the involution T : H → H from (7), which corresponds toreversing the orientation of α . (cid:66) α + (cid:66) α − δ α + δ α − α Definition 5.1.
Let Γ be a directed ribbon graph and H a pivotal Hopf monoid in C . The H -leftmodule structures (cid:66) α ± : H ⊗ H ⊗ E → H ⊗ E and H -left comodule structures δ α ± : H ⊗ E → H ⊗ H ⊗ E assigned to an oriented edge α are the following morphisms in C α α = α (cid:66) α + : H ⊗ H ⊗ E → H ⊗ E , (cid:66) α − : H ⊗ H ⊗ E → H ⊗ E ,α α = αδ α + : H ⊗ E → H ⊗ H ⊗ E , δ α − : H ⊗ E → H ⊗ H ⊗ E , where the H -left and H -right (co)module structures on H in the diagrams are given by left and right(co)multiplication, as in Example 2.12. H -left module and comodule structures correspond to the triangle operators in Kitaev’slattice models. If H is a finite-dimensional semisimple Hopf algebra over C with p = η , then theformer correspond to the operators L hα ± : H ⊗ E → H ⊗ E for h ∈ H in [BMCA]. The morphisms δ α ± induce two H ∗ -right module structures on H ⊗ E . Up to antipodes that transform right H ∗ - intoleft H ∗ -modules, these define the operators T βα ± : H ⊗ E → H ⊗ E in [BMCA]. The following lemmageneralises the relations between these operators from [BMCA]. Lemma 5.2.
Let α be an edge in a directed ribbon graph Γ and H a pivotal Hopf monoid in C .1. The triples ( H ⊗ E , (cid:66) α + , δ α + ) and ( H ⊗ E , (cid:66) α − , δ α − ) are H -left-left-Hopf modules in C ,2. The two H -left module structures (cid:66) α ± and H -left comodule structures δ α ± commute: (cid:66) α − ◦ (1 H ⊗ (cid:66) α + ) = (cid:66) α + ◦ (1 H ⊗ (cid:66) α − ) ◦ ( τ H,H ⊗ H ⊗ E )(1 H ⊗ δ α − ) ◦ δ α + = ( τ H,H ⊗ H ⊗ E ) ◦ (1 H ⊗ δ α + ) ◦ δ α − . Proof.
The first claim for ( H ⊗ E , (cid:66) α + , δ α + ) follows directly from the definition of (cid:66) α ± and Example2.12, and the first claim for ( H ⊗ E , (cid:66) α − , δ α − ) follows from the one for ( H ⊗ E , (cid:66) α + , δ α + ) and the factthat T is an involution. The second claim is a direct consequence of (co)associativity for H .To every ciliated vertex v and ciliated face f in Γ we assign, respectively, an H -left module and an H -left comodule structure on H ⊗ E . These are constructed from the H -left module structures forthe edges incident at v and the H -left comodule structures for the edges traversed by f .The H -left module structure for a ciliated vertex v with n incident edge ends is obtained by applyingthe comultiplication ∆ n − : H → H ⊗ n . The different factors in the tensor product then act on thecopies of H associated to the edge ends at v , according to their ordering. The action on an edgeend is (cid:66) α + if the end of α is incoming and (cid:66) α − if it is outgoing at v .The H -left comodule structure for a ciliated face f that traverses n edges is obtained from thecomodule structures at each edge in f , where one takes the left H -coaction δ α + if the edge α istraversed parallel to its orientation and δ α − if it is traversed against its orientation. One thenapplies the multiplication m n − : H ⊗ n → H , according to the order of the edges in f . Definition 5.3.
Let Γ be a directed ribbon graph and H a pivotal Hopf monoid in C .1. The H -left module structure (cid:66) v : H ⊗ H ⊗ E → H ⊗ E for a ciliated vertex v with incident edgeends α < α < ... < α n is (cid:66) v = (cid:66) α ◦ (1 H ⊗ (cid:66) α ) ... ◦ (1 H ⊗ ( n − ⊗ (cid:66) α n ) ◦ (∆ n − ⊗ H ⊗ E ) , where (cid:66) α i = (cid:66) e ( α i )+ and (cid:66) α i = (cid:66) e ( α i ) − , respectively, if the edge end α i is incoming andoutgoing at v and e ( α i ) denotes the edge for the edge end α i .2. The H -left comodule structure δ f : H ⊗ E → H ⊗ H ⊗ E for a ciliated face f = α (cid:15) ◦ . . . ◦ α (cid:15) r r is δ f = m r − ◦ (1 H ⊗ ( r − ⊗ δ α r (cid:15) r ) ◦ . . . ◦ (1 H ⊗ δ α (cid:15) ) ◦ δ α (cid:15) . In the following we will sometimes describe the module or comodule structure associated with avertex or face with Sweedler notation on the edge labels. This is to be understood as a shorthandnotation for a diagram that describes a morphism in C , as illustrated in the following example. Example 5.4.
The H -left module and comodule structure for the ciliated vertex and face fromFigure 1 are given as follows bcd h aS ( h (1) ) h (2) bS ( h (4) ) h (5) cS ( h (3) ) h (6) d a b c dafe bc d a (2) f (1) e (1) b (2) c (2) d (2) a (1) T ( f (2) ) T ( e (2) ) c (1) T ( d (3) ) d (1) b (1) a f e c d b The H -left (co)module structures for ciliated vertices and faces correspond to the vertex and faceoperators in Kitaev’s lattice models. For a finite-dimensional semisimple Hopf algebra H in C = Vect C ,the vertex operator A hv for an element h ∈ H from [BMCA] is A hv = h (cid:66) v − : H ⊗ E → H ⊗ E andthe face operator for an element β ∈ H ∗ by B βf = ( S ( β ) ⊗ H ⊗ E ) ◦ δ f : H ⊗ E → H ⊗ E . The H -left(co)module structures thus have properties similar to the vertex and face operators. Lemma 5.5.
Let Γ be a directed ribbon graph and H a pivotal Hopf monoid in C . The H -leftmodule and comodule structures associated with ciliated vertices and faces of Γ satisfy:1. All H -left module structures for distinct vertices v, v ∈ V commute: (cid:66) v ◦ (1 H ⊗ (cid:66) v ) = (cid:66) v ◦ (1 H ⊗ (cid:66) v ) ◦ ( τ H,H ⊗ H ⊗ E ) .
2. If two ciliated faces f, f are not related by cyclic permutations, the associated H -comodulestructures commute: (1 H ⊗ δ f ) ◦ δ f = ( τ H,H ⊗ H ⊗ E ) ◦ (1 H ⊗ δ f ) ◦ δ f .
3. If two cilia are at distinct vertices and distinct faces, the H -module structure for one of themcommutes with the H -comodule structure for the other: δ f ◦ (cid:66) v = (1 H ⊗ (cid:66) v ) ◦ ( τ H,H ⊗ H ⊗ E ) ◦ (1 H ⊗ δ f )
4. If a ciliated vertex v and a ciliated face f share a cilium, the associated H -left module and H -left comodule structure form a H -left-left Yetter-Drinfeld module: ( m ◦ τ H,H ⊗ H ⊗ E ) ◦ (1 H ⊗ δ f ) ◦ (1 H ⊗ (cid:66) v ) ◦ ( τ H,H ◦ ∆ ⊗ H ⊗ E )=( m ⊗ (cid:66) v ) ◦ (1 H ⊗ τ H,H ⊗ H ⊗ E ) ◦ (∆ ⊗ δ f )20 = = = == == =(a)(b) Figure 7: Diagrammatic proof of Lemma 5.5 Proof.
The proof of these statements is analogous to the one in [BMCA], see also [Me]. The firsttwo follow directly from the definition of the vertex and face operators and from Lemma 5.2, 2. Thethird is obvious if the vertex and face do not share an edge. If they share an edge, but not a cilium,if is sufficient to consider the constellation in case (a), and the graphical proof in Figure 7 (a). Theother cases follow by applying the involution T from (7) to reverse edge orientation.(a) (b) . . . To prove the last claim, it is sufficient to consider the constellation in (b) and to note that condition(16) for a left-left Yetter-Drinfeld module is equivalent to the condition= . The graphical proof is given in Figure 7 (b), and the identity for other edge orientations againfollows by applying the involution T . 21 β S α L S α − L α βα − β Figure 8: Isomorphism of path groupoids induced by sliding the target end of β along the left of α . In this section, we consider a generalisation of the chord slides from Section 4.2. Throughout thissection, let Γ be a directed ribbon graph with edge set E .The slide is defined in analogy to the chord slides in Section 4.2, but not restricted to ribbon graphswith a single vertex. It slides an end of an oriented edge β along another oriented edge α that sharesa vertex with this end of β and is adjacent to it with respect to the cyclic ordering. If some or allof the vertices of the ribbon graph are ciliated, we sometimes impose that edge ends do not slideover cilia. This means that the relevant end of α is required to be adjacent to β with respect to the linear ordering of the edges at this vertex.As α is an oriented edge, we distinguish slides to the start and to the target of α and slides on theright and on the left of α , viewed in the direction of its orientation. This yields the four differentcases below. We denote by S α L and S α R the slides to the target end of α , on the left and right of α ,respectively, and by S α − L and S α − R their inverses. In particular, we set S ( α − ) L = S α − R , where α − denotes the edge α with the reversed orientation. αβ S α L S α − L α β αβ S α R S α − R α β (24)The edge slides for the other orientation of β are defined analogously. The short black lines in (24)indicate other edge ends or cilia at the vertices. Their position remains unchanged under the edgeslide, just as the position of the starting end of β . Note also that the two vertices in (24) maycoincide and that the other end of β may be incident at the same vertex. The only conditions are(i) The sliding end of β shares a vertex with the starting or target end of α .(ii) It comes directly after the starting end (before the target end) of α with respect to the orderingat their common vertex for the slide S α L (for the slide S α − L ) and directly before the startingend (after the target end) of α for the slide S α R (for the slide S α − R ).(iii) The edge β is not identical to the edge α .If Γ is obtained from a directed ribbon graph Γ by an edge slide S , one has an isomorphism ofgroupoids S : G Γ → G Γ . On the vertices, it is the canonical identification of the vertices of Γ and Γ .On the morphisms, it is given by the images of the edges of Γ. If S = S α ± L or S = S α ± R slides the target end of an edge β along α , one has S ( γ ) = γ for edges γ = β and S ( β ) = α ∓ ◦ β , as shownin Figure 8. If S = S α ± L or S = S α ± R slides the starting end of an edge β along α , it is given by S ( γ ) = γ for γ = β and S ( β ) = β ◦ α ± .Given a Hopf monoid H in a symmetric monoidal category C and a directed ribbon graph Γ, weassociate to each edge slide of Γ an automorphism of the object H ⊗ E in C . It is obtained bycombining one of the H -left comodule structures for α and one of the H -left module structures for β from Definition 5.1. The choice of the module and comodule structures depends on the orientationof the edges and the direction of the slide. 22 efinition 6.1. Let Γ be a directed ribbon graph and H a pivotal Hopf monoid in C .1. If α, β are oriented edges of Γ as in (24) the left edge slide of the target end of β to thetarget end of α is the isomorphism S α L = (cid:66) β + ◦ δ α + : H ⊗ E → H ⊗ E ab S α L a (2) a (1) b β α (25) with inverse S α − L = (cid:66) β + ◦ ( S − ⊗ H ⊗ E ) ◦ δ α + : H ⊗ E → H ⊗ E a b S α − L a (2) S − ( a (1) ) b β α (26)
2. The edge slides for other edge orientations and their inverses are defined by 1. by reversingedge orientation with the involution T : H → H from (7) . That the morphism S α − L in (26) is indeed the inverse of S α L in (25) follows by a direct computationin Sweedler notation or, equivalently, from the following diagrammatic computation= = = = = = . Explicit expressions for the edge slides for other edge orientations and their inverses are obtainedfrom Definition 5.1 and (7) and given as follows. ab S α L a (2) bS ( a (1) ) β α (27) a b S α − L a (2) ba (1) β α (28)23 b S α R a (1) S − T ( a (2) ) b β α (29) a b S α − R a (1) T ( a (2) ) b β α (30) ab S α R a (1) bT ( a (2) ) β α (31) a b S α − R a (1) bST ( a (2) ) β α (32)In the remainder of this section, we investigate the properties of these edge slides. We start byconsidering their interaction with the H -left module and comodule structures for ciliated verticesand faces of the ribbon graph Γ. Proposition 6.2.
Let v be a ciliated vertex, f a ciliated face of Γ and (cid:66) v : H ⊗ H ⊗ E → H ⊗ E and δ f : H ⊗ E → H ⊗ H ⊗ E the associated H -left module and comodule structures from Definition5.3. Then any edge slide that does not slide edge ends over their cilia is an isomorphism of H -leftmodules and H -left comodules.Proof. That the edge slide from (25) is an automorphism of the object H ⊗ E was already shownabove. The proof for the case where both vertices coincide is analogous, and so are the proofs thatthe slides (27) and (28), (29) and (30) and (31) and (32) are inverse to each other.It remains to check compatibility with the H -module structures for the vertices and with the H -comodule structures for the faces. If the two edges involved in the slide are not incident at v ornot traversed by the face f , this follows directly from the definition of the slide and of the H -moduleand comodule structure. We prove the remaining cases for the slide (28) under the assumptionthat the starting and target vertex of α are different and do not coincide with the target vertex of24 . The proofs for the cases, where some of these vertices coincide are analogous. The proofs forother edge orientations follow by considering the inverses and reversing edge orientations with theinvolution T : H → H from (7).The statement that (28) is an isomorphism of H -left modules and H -left comodules then correspondsto the equality of the following diagrams that arise from the H -left module and -comodule structuresassociated with the three vertices and three faces adjacent to the edges. β α = β α h (2) abS ( h (1) ) ha (2) ba (1) (33) β α = β α aS ( h ) b a (2) S ( h (1) ) ba (1) S ( h (2) ) (34) β α = β α ahb a (2) hba (1) (35) β α = β α a (2) b (2) · · · b (1) a (1) · · · a (3) b (2) a (2) · · · b (1) a (1) · · · (36) β α = β α ab (1) · · · T ( b (2) ) · · · a (2) b (1) a (1) · · · T ( b (2) ) · · · (37) β α = β α a (1) b · · · T ( a (2) ) · · · a (2) ba (1) · · · T ( a (3) ) · · · (38)25dentities (35) and (38) follow directly from the fact that the morphisms (cid:66) β + , (cid:66) β − and themorphisms δ α + , δ α − commute by Lemma 5.2. The proof of the other identities involves thecompatibility conditions for H -left module and H -left comodule structures in a Hopf module, theproperties of the antipode and the properties (8) and (9) of the involution (7). A diagrammatic proofis given in Figure 10. Alternatively, this follows by a direct computation in Sweedler notation.We now show that the edge slides from Definition 6.1 satisfy generalisations of Bene’s relations forslides in (18) to (22). These relations are obtained by replacing each connected component of theouter circle in diagrams in (18) to (22) with a vertex and orienting the edges between these verticesthat replace the chords in (18) to (22). Proposition 6.3.
The edge slides from Definition 6.1 satisfy the involutivity relation, the commu-tativity relation, triangle relation and the left and right pentagon relation from Theorem 4.2.Proof.
The involutivity relations are satisfied trivially, because all edge slides have inverses. Thecommutativity relations follow directly from the fact that each edge slide affects only the copies of H in H ⊗ E for edges involved in the slide.The triangle relation follows by a direct diagrammatic computation in Sweedler notation ab S α L a (2) a (1) b S β − R T ( b (2) ) ab (1) S α − R T ( b ) a (39)The proofs for other edge orientations follow by taking inverses and reversing edge orientation withthe involution T . The proof for the cases where some vertices coincide is analogous.The two pentagon relations follow directly from the compatibility condition between the H -leftmodule and H -left comodule structures for a Hopf module and from the properties of the involution T . A diagrammatic proof for the edge orientations in Figure 9 is given in Figure 11. The proof forother edge orientations follow by taking inverses and reversing edge orientation with the involution T . The proofs for the cases where some vertices coincide are analogous. βγ α S γ L βγ αS β L βγ αS γ L βγ αS β L βγ αS β L β γα S γ R β γαS α R β γαS γ R β γαS α R β γαS α R Figure 9: The left and right pentagon relations for edge slides.26 = = ====== == = = == = =Figure 10: Diagrammatic proof of the identities (33), (34), (36), (37). = = == = == == =
Figure 11: Diagrammatic proof of the left and right pentagon relation in Figure 9.27 orollary 6.4.
The edge slides from Definition 6.1 satisfy generalisations of the opposite endcommutativity relation and the adjacent commutativity relation from Lemma 4.3.
Note that the triangle and the pentagon relations in Proposition 6.3 have a different status. Thediagrammatic proof of the pentagon relations in Figure 11 generalises directly to tensor products ofHopf bimodules. The proof of the triangle relation relies on the fact that edges are decorated withcopies of a Hopf monoid H in C and works only for trivial H -Hopf bimodules.If we restrict attention to ribbon graphs Γ with a single ciliated vertex, the edge slides from (24)coincide with an oriented version of the chord slides in Section 4.2. If additionally, the ribbongraph Γ has only a single face, then the surface Σ obtained by gluing an annulus to the face is anoriented surface with a single boundary component. By Theorem 4.2, the mapping class groupMap(Σ) is then presented by finite sequences of chord slides that preserve Γ up to the cilium, subjectto the relations in Theorem 4.2. As the edge slides from Definition 6.1 satisfy these relations byProposition 6.3, we obtain an action of the mapping class group Map(Σ). As no edge ends slideover cilia, Proposition 6.2 implies that it acts by automorphisms of Yetter-Drinfeld modules withrespect to the Yetter-Drinfeld module structure associated with the cilium. Theorem 6.5.
Let H be a pivotal Hopf monoid in a symmetric monoidal category C , Γ a directedribbon graph with a single ciliated vertex and a single face and Σ the oriented surface obtained bygluing an annulus to the face. Then the edge slides from Definition 6.1 define a group homomorphism ρ : Map (Σ) → Aut
Y D ( H ⊗ E ) . In this section, we consider the simplest examples of mapping class group actions by edge slides,namely the the mapping class groups of the torus T and of the torus T ∗ with a disc removed. Themapping class group of T is the modular group SL(2 , Z ) [FM, Sec. 2.2.4]. A presentation of themapping class group of T ∗ is obtained from Theorem 4.1. It involves only two generators α , δ and relation (iii), which yields the braid group B on three strandsSL(2 , Z ) = h A, B | ( BAB ) = 1 , ABA = BAB i B = h A, B | ABA = BAB i . (40)The element ( BAB ) is central in B . In both cases, the generators A , B correspond to Dehn twistsaround the a - and b -cycle. For more details on these presentations, see for instance [KT, App. A].To describe the tori T and T ∗ we consider a ribbon graph Γ that consists of a single ciliated vertexand two oriented loops representing the generators of the fundamental groups π ( T ) ∼ = Z × Z and π ( T ∗ ) ∼ = F . By cutting the vertex at the cilium, we obtain the associated chord diagram. Theribbon graph Γ and this chord diagram each have a single face, as shown below. b a b a b a (41)Given a pivotal Hopf monoid H in a symmetric monoidal category C , we associate to Γ the object H ⊗ in C , with the first copy of H assigned to a and the second to b . The H -left module structure (cid:66) v : H ⊗ H ⊗ → H ⊗ for the ciliated vertex and the H -left comodule structure δ f : H ⊗ → H ⊗ H ⊗ H ⊗ by Lemma 5.5. They are given by (cid:66) v : b ah h (4) bS ( h (2) ) h (3) aS ( h (1) ) (42) δ f : b a b (2) a (2) T ( a (3) ) b (1) a (1) T ( b (3) )By Theorem 6.5, the edge slides acting on this ribbon graph induce an action of the braid group B on H ⊗ . We will now show that the generators A and B from (40) can be identified with the slidesalong the left side of the two edges to their targets. From expressions (25) and (27) one finds b a D b = S b L b (2) b (1) a (43) b a D a = S a L bS ( a (1) ) a (2) As they do not slide any edge ends over the cilium, Proposition 6.2 implies that the morphisms D a , D b ∈ Aut( H ⊗ ) are automorphisms of Yetter-Drinfeld modules with respect to the Yetter-Drinfeld module structure (42). We will now show they satisfy the braid relation and hence definean action of the group B on H ⊗ by automorphisms of Yetter-Drinfeld modules. We also showthat whenever C is finitely complete and cocomplete, this induces an action of the modular groupSL(2 , Z ) on the object H ⊗ inv from Definition 2.8. Theorem 7.1.
Let H be a pivotal Hopf monoid in a symmetric monoidal category C .1. The slides in (43) define a group homomorphism ρ : B → Aut
Y D ( H ⊗ ) .2. If C is finitely complete and cocomplete, the slides in (43) induce a group homomorphism ρ : SL(2 , Z ) → Aut( H ⊗ inv ) .Proof. We identify A = D a and B = D b and verify the relations in (40). A direct computationusing expressions (43) yields D b ◦ D a ◦ D b = D a ◦ D b ◦ D a : b a b (3) S ( a ) S ( b (1) ) b (2) By applying this morphism four times we obtain( D b ◦ D a ◦ D b ) = (cid:66) v ◦ ( T ⊗ H ⊗ ) ◦ δ f : (44) b a x (4) S ( b (2) ) S ( x (2) ) x (3) S ( a (2) ) S ( x (1) ) x = b (3) S ( a (1) ) S ( b (1) ) S ( a (3) )29y Proposition 6.2, the automorphisms D a and D b are automorphisms of Yetter-Drinfeld moduleswith respect to (42), and by Lemma 2.9 they induce automorphisms D a , D b of H ⊗ inv . As theautomorphism ( D b ◦ D a ◦ D b ) = (cid:66) v ◦ ( T ⊗ H ⊗ ) ◦ δ f induces the identity morphisms on H ⊗ inv , weobtain an SL(2 , Z )-action on H ⊗ inv .Note that the expressions for the Dehn twists in (43) depend only on the Hopf monoid H andnot on the choice of the pivotal structure for H . It will become apparent in Section 9 that this isspecific to the torus and not true for Dehn twists in higher genus mapping class groups. Note alsothat the proof of Theorem 7.1 yields not only an SL(2 , Z )-action on the biinvariants of (43), butalso an SL(2 , Z )-action on the invariants of the H -left module ( H ⊗ , (cid:66) v ) and on the coinvariants ofthe right- H -comodule ( H ⊗ , δ f ) with H -right comodule structure δ f from (12), whenever they aredefined. The restriction to the invariants, coinvariants or biinvariants is sufficient, but not necessaryto obtain actions of the modular group SL(2 , Z ). This depends on the Hopf monoid H . Example 7.2.
Let C = Vect F and H ∼ = F [ G ] for a group G and field F . Then F [ G ] ⊗ F [ G ] ∼ = F [ G × G ] ,and the slides from (43) are given by D b : ( a, b ) ( ba, b ) D a : ( a, b ) ( a, ba − ) , (45) which yields ( D b D a D b ) ( a, b ) = ([ b, a − ] a [ a − , b ] , [ b, a − ] b [ a − , b ]) . This shows that the modular group
SL(2 , Z ) acts on the set M = span F { ( a, b ) ∈ G × G | [ a, b ] ∈ Z ( G ) } If G is nilpotent of nilpotence degree 2, then [ G, G ] is central in G , and Theorem 7.1 defines an SL(2 , Z ) -action on F [ G × G ] . In the previous section we gave concrete expressions for the action of the mapping class group ofthe torus and one-holed torus in terms of generating Dehn twists. In the remainder of this article,we derive analogous descriptions for the mapping class groups of surfaces of genus g ≥ n ≥ a - and b -cycles of the torus from Section 7to twists along more general closed paths in a graph. This requires a number of technical results onedge slides, which we derive in this section. The reader only interested in the results may skip theseat first and proceed to Section 9.To generalise the Dehn twists from Section 7, we proceed in three steps. In Section 8.1 we generalisethe edge slides from Definition 6.1 to slides along face paths (cf. Definition 3.3). In Section 8.2we show how a slide along a face path can be described in terms of a slide along a single edge byadding and removing edges in the graph. In Section 8.3 we then define Dehn twists along closedface paths and investigate their interaction with edge slides. In Section 9 we further generalise theseDehn twists to a number of closed paths that are not face paths. This yields a set of generatingDehn twists that satisfy the relations in Theorem 4.1.30hroughout this section, we assume that H is a pivotal Hopf monoid in a symmetric monoidalcategory C and Γ a directed ribbon graph with edge set E . We start by generalising the edge slides from Section 6 to slides along face paths. For this, notethat the conditions on an edge slide after equation (24) allow one to slide an edge end β along anentire face path γ path via successive edge slides, whenever it can slide along the first edge in γ andis not traversed by γ . This is illustrated in Figure 12. Definition 8.1.
Let γ = γ (cid:15) ◦ . . . ◦ γ (cid:15) n n : v → w be a face path and denote by β the edge end directlyafter the starting end s ( γ (cid:15) n n ) of γ with respect to the cyclic ordering at v . If β is not traversed by γ ,the slide of β along γ is S γ := S ( γ (cid:15) ) L · · · S ( γ (cid:15)nn ) L with S ( γ − i ) L := S γ − Ri . Note that for a face path γ = γ (cid:15) consisting of a single edge γ ∈ E , the slide along γ reduces to theedge slides from Definition 6.1, namely S γ = S γ L if (cid:15) = 1 and S γ = S γ − R if (cid:15) = − H -comodule structure associated to face paths. From Definition 8.1 and the expressions for edgeslides in (25) to (32) one obtains the following alternative definition. Remark 8.2.
Associate to a face path γ in Γ an H -comodule structure δ γ : H ⊗ E → H ⊗ H ⊗ E asin Definition 5.3 and let β be as in Definition 8.1. Then the slide along γ is given by S γ = (cid:66) β ± ◦ δ γ where one takes + if the sliding end of the edge β is incoming and − if it is outgoing. As slides along face paths are defined as composites of edge slides, we obtain a direct generalisationof Proposition 6.2 for slides along face paths.
Corollary 8.3.
Let γ be a face path satisfying the conditions in Definition 8.1. Then S γ is anisomorphism of H -(co)modules for any ciliated vertex v (face f ) whose cilium is not traversed by γ . By considering slides along face paths, one obtains the left- and right pentagon relation and theopposite end and adjacent commutativity relations as special cases of a simpler and more generalrelation. For this, recall from the beginning of Section 6 that edge slides induce isomorphismsbetween the path groupoids of directed ribbon graphs. Because slides along face paths are compositesof edge slides, this statement generalises to face paths.We now consider how a face path ρ in Γ is transformed under the slide of an edge α along a facepath γ . Then there are four possibilities for the relative positions of the paths (cf. Section 3.1):(i) γ and ρ do not overlap,(ii) γ and ρ overlap, but neither is a subpath of the other,(iii) ρ is a subpath of γ ,(iv) γ is a proper subpath of ρ .The transformation of ρ under the slide S γ is depicted in Figure 13. In case (i), one has S γ ( ρ ) = ρ ,unless ρ traverses the edge α . If ρ traverses α , then S γ ( ρ ) = ρ , but S γ ( ρ ) is still a face path, asshown in Figure 13 (i). In case (ii), S γ ( ρ ) is never a face path, as shown in Figure 13 (ii), as facepaths by definition turn maximally left at every vertex. The same holds for case (iv) if γ and ρ have the same starting vertex. In case (iii) S γ ( ρ ) is again a face path, as shown in Figure 13 (iii).The same holds in case (iv) if γ and ρ do not start at the same vertex, as shown in Figure 13 (iv).31 γ γ γ S γ L γ γ γ S γ − R γ γ γ S γ L γ γ γ Figure 12: Sliding an edge end along the face path γ = γ γ − γ .(i) γ ρα S γ S γ ( ρ ) (i) γ ρα S γ S γ ( ρ )(ii) γ ρα S γ S γ ( ρ ) (ii) γρα S γ S γ ( ρ )(iii) γ ρα S γ S γ ( ρ ) (iv) γρα S γ S γ ( ρ )Figure 13: Transformation of a face path ρ under a slide along a face path γ . Lemma 8.4 (Commutativity of slides) . Let γ and ρ be face paths such that S γ ( ρ ) and S ρ ( γ ) alsoare face paths and the slides along them are defined. Then: S S ρ ( γ ) S ρ = S S γ ( ρ ) S γ (46) Proof.
We first prove the claim for paths of the form γ = γ ν , ρ = ρ (cid:15) with γ , ρ ∈ E and ν , (cid:15) ∈ {± } . Then there are three cases: (a) The paths are equal: γ = ρ , (b) Either S γ or S ρ slides the edge traversed by the other path, (c) S γ and S ρ slide different edges α = β or differentends of the same edge α = β , and α and β are distinct from ρ and γ .In case (a) and case (c) we have S γ ( ρ ) = ρ and S ρ ( γ ) = γ . In case (a) the claim is trivial, and in case(c) it follows directly from the adjacent commutativity, opposite end commutativity in Corollary6.4 or from the commutativity relation in Proposition 6.3. In case (b) we can suppose without lossof generality that S γ = S γ L slides the target end of ρ . As we suppose that S γ ( ρ ) is a face path,this can happen only in case (i) in Figure 13, and we have ρ = ρ − , S γ ( ρ ) = ρ − γ , S ρ = S ρ − R and S ρ ( γ ) = γ . (Note that the slide along S γ ( ρ ) is not defined if the starting vertex of γ is bivalent,since in this case S γ ( ρ ) starts at a univalent vertex). We now consider the right pentagon relationfrom Figure 9, with α = γ − and γ = ρ , read against the orientation of its arrows from the top leftpicture to the middle picture on the right. This yields S S γ ( ρ ) S γ = S ρ − R S γ L S γ L = S γ L S ρ − R = S S ρ ( γ ) S ρ . This proves the claim for face paths γ and ρ consisting of one edge. For general paths γ = γ ν · · · γ ν n n and ρ = ρ (cid:15) · · · ρ (cid:15) m m that satisfy the assumptions, it then follows by induction over m and n .Lemma 8.4 allows one to generalise the pentagon equations from an equation involving three edgesto an identity involving three composable face paths, which replace the three edges in Figure 9. Theonly condition is that the last edge of the first face path is not traversed by the second.32 γ γ S γ S γ ( γ )Figure 14: The generalised pentagon relation from Corollary 8.5. η γ γ S γ γγ γ C γ σ ρ C γ ( ρ ) C γ ( σ )Figure 15: Adding an edge γ to a face path γ and the associated transformation of face paths. Corollary 8.5 (Generalised pentagon equation) . Let γ = γ ◦ γ ◦ γ be a face path with face paths γ , γ and a non-trivial face path γ such that the last edge of γ is not traversed by γ . Then S γ ( γ ) = γ ◦ γ and S γ S γ = S S γ ( γ ) S γ . (47) In this section we introduce three additional transformations of directed ribbon graphs that willbe used to simplify slides along face paths. The first is the graph transformation (cid:15) α : Γ → Γ thatdeletes an edge α from Γ. If α is incident at a univalent vertex v , then v is deleted as well.The second is the graph transformation η α : Γ → Γ that adds an oriented edge α at a vertex v of Γ.This edge α may either have a univalent vertex v at its other end or may be a loop based at v . If α has a univalent vertex at its other end, we orient α towards the univalent vertex. If α is a loopbased at v , we require that the target end of α is directly before the starting end of α with respectto the cyclic ordering at v and with respect to the linear ordering at v , if v is ciliated.The third graph transformation is associated with a face path γ in Γ. It first adds a loop γ at thestarting vertex of γ , such that the target end of γ is directly after starting end of γ with respect tothe cyclic ordering. It then slides the target end of γ along γ , as shown in Figure 15. We denote itby C γ = S γ ◦ η γ : Γ → Γ and call it adding an edge to γ .If Γ is obtained from Γ by adding an edge γ to a face path γ , then γ ◦ γ is a ciliated face ofΓ . More generally, every face path ρ of Γ that contains γ as a subpath corresponds to a face path C γ ( ρ ) in Γ that is obtained by replacing γ by γ in the expression for ρ , as in Figure 15. Face pathsof Γ that are proper subpaths of γ also correspond canonically to face paths in Γ , and the sameholds for paths in Γ that do not overlap with γ .33 efinition 8.6. Let γ , ρ face paths of Γ such that either (i) ρ is a proper subpath of γ , (ii) ρ and γ do not overlap or (iii) γ is a subpath of ρ . Let Γ be the directed ribbon graph obtained by adding anedge γ to γ . The path C γ ( ρ ) in Γ is defined as C γ ( ρ ) = ρ in cases (i),(ii) and as C γ ( ρ ) = ρ ◦ γ ◦ ρ if ρ = ρ ◦ γ ◦ ρ with possibly trivial face paths ρ , ρ . We now associate to each of these graph transformations a morphism in C . These morphisms aredenoted by the same letters as the graph transformations and given as follows. Definition 8.7.
Let Γ be a directed ribbon graph and H a pivotal Hopf monoid in C .1. Removing an edge α ∈ E from Γ corresponds to the morphism (cid:15) α : H ⊗ E → H ⊗ E \{ α } , that is the counit on the copy of H for α and the identity morphism on all other components.2. Adding an edge α to Γ corresponds to the morphism η α : H ⊗ E → H ⊗ E ∪{ α } , that is the unit on the copy of H for α and the identity morphism on all other components.3. Adding an edge γ to a face path γ in Γ corresponds to the morphism C γ = S γ ◦ η γ : H ⊗ E → H ⊗ E ∪{ γ } . Note that adding an edge to a face path involves two choices. The first is the orientation of theadded loop γ , which can be reversed by applying the morphism T γ . The second choice is to addthe loop at the starting vertex of γ and slide its target end to the target of γ . With expressions(25) to (32) for the edge slides one finds that adding the loop at the target vertex of γ and slidingits starting end to the starting vertex of γ yields the same morphism C γ . Equivalently, adding aloop at the starting vertex of a face path γ and then sliding it to the target vertex gives the samemorphism as adding the loop at the target vertex of γ .The advantage of adding edges to face paths is that we can reduce slides along face paths to slidesalong edges. This is achieved by adding an edge to the face path, sliding along the added edge andthen deleting this edge. To show that this yields the same result as the slide along the face path, wedetermine the properties of the associated morphism C γ . Lemma 8.8.
Let γ, ρ be face paths and α an edge in a directed ribbon graph Γ . Then the morphism C γ : H ⊗ E → H ⊗ E ∪{ γ } from Definition 8.7 has the following properties:1. Removing the edge γ is left inverse to adding γ : (cid:15) γ ◦ C γ = 1 H ⊗ E . (48)
2. Removing α is left inverse to adding an edge α to α (to α − and reversing orientation): (cid:15) α ◦ C α = 1 H ⊗ E , (cid:15) α ◦ C α − = T α . (49)
3. Adding an edge γ to γ is a morphism of H -modules and H -comodules for all H -(co)modulestructures at ciliated vertices v (faces f ) of Γ whose cilium is not traversed by γ : (cid:66) v ◦ C γ = C γ ◦ (cid:66) v δ C ρ ( f ) ◦ C γ = C γ ◦ δ f . (50)
4. If Γ is obtained from Γ by adding an edge γ to γ , then C γ equalises the trivial comodulestructure and the H -right comodule structure (12) of any ciliated face f of Γ that is a cyclicpermutation of γ ◦ γ : δ f ◦ C γ = η ⊗ C γ . (51)34 . Adding edges commutes with slides along face paths: if ρ, γ satisfy (i),(ii) or (iii) in Definition8.6 and and S γ ( ρ ) is a face path, then S C ρ ( γ ) ◦ C ρ = C S γ ( ρ ) ◦ S γ . (52) In particular, whenever the slide along γ is defined, one has S γ = (cid:15) γ ◦ S γ L ◦ C γ . (53)
6. Adding edges is commutative: if ρ, γ satisfy (i), (ii) or (iii) in Definition 8.6, then C C γ ( ρ ) ◦ C γ = C C ρ ( γ ) ◦ C ρ . (54)
7. If f = α ◦ ρ is a ciliated face and ρ does not traverse α , then T ρ ◦ C ρ ◦ (cid:15) α ◦ ι = ι, S ρ ◦ (cid:15) α ◦ ι = (cid:15) α ◦ S α − R ◦ ι, (55) where ι is the equaliser for the H -right comodule structure δ f from (12) and we identify theedges ρ and α .Proof.
1. and 2. The first and the second claim follow directly from the definition of C γ , theexpressions for the edge slides in (25) to (32) and from the definition of a Hopf monoid in C .3. That C γ is a morphism of H -modules and H -comodules for the cilia that are not traversed by γ follows directly from the fact that this holds for the slide S γ by Corollary 8.3 and it holds for addinga loop. The latter follows directly from Definition 8.7 and from Definition 5.3 of the H -(co)modulestructures, together with the defining properties of a Hopf monoid.4. To prove (51), we first consider the ciliated face f = γ ◦ γ . By Remark 8.2, we can describethe slide along the face path γ in terms of a comodule structure associated with γ . As adding aloop corresponds to the unit of H , the claim then follows from the diagrammatic computation= = = p = p , (56)in which the thick vertical line stands for the comodule structure associated with γ and the firstdiagram on the left is the morphism δ f ◦ C γ . The claim for ciliated face paths that are cyclicpermutations of γ ◦ γ then follows from the diagrammatic computation ι = ι = ι = p ι = p ι = p ι = p ι = ι , γ ◦ γ and ι for the coequaliser of δ f . This showsthat the claim is invariant under cyclic permutations of the edges in f .5. To prove (52), note first that if γ = γ ◦ γ for a face path γ and a non-trivial face path γ andif η α adds a loop directly before the starting end of γ , then one has η α S γ γ = S γ α L γ η α . (57)Otherwise S γ and η α simply commute. By decomposing C ρ = S ρ ◦ η ρ and C S γ ( ρ ) = S S γ ( ρ ) ◦ η S γ ( ρ ) ,using the commutativity relation of slides in (46) and identity (57), one obtains (52).Identity (53) follows by setting γ = ρ in (52), for which C γ ( γ ) = γ by Definition 8.6, composingboth sides of (52) with (cid:15) γ and using (48) .6. To prove Equation (54) we first treat the case where ρ is a proper subpath of γ = γ ◦ ρ ◦ γ .Then equation (57) holds for η α = η ρ and we have C ρ ( γ ) = γ ρ γ and C γ ( ρ ) = ρ . We then obtain C C γ ( ρ ) C γ = C ρ C γ = S ρ η ρ C γ ργ = S ρ η ρ S γ ργ η γ (57) = S ρ S γ ρρ γ η ρ η γ = S ρ S γ ρρ γ η γ η ρ = S ρ C γ ρρ γ η ρ (52) = C S ρ ( γ ρρ γ ) S ρ η ρ = C γ ρ γ S ρ η ρ = C C ρ ( γ ) C ρ . Because of the symmetry of (54) the same holds if γ is a proper subpath of ρ , while for γ = ρ theclaim is tautological. For non-overlapping ρ and γ we have C γ ( ρ ) = ρ , C ρ ( γ ) = γ and η ρ commuteswith C γ . We then obtain C C γ ( ρ ) ◦ C γ = C ρ ◦ C γ = S ρ ◦ η ρ ◦ C γ = S ρ ◦ C γ ◦ η ρ (52) = C S ρ ( γ ) ◦ S ρ ◦ η ρ = C γ ◦ C ρ = C C ρ ( γ ) ◦ C ρ .
7. The first identity in (55) follows from Remark 8.2 and the identity ι = ι = ι = ι = ι , (58)where ι stands for the inclusion morphism of the coinvariants from Definition 2.6, the left line forthe edge α and the other vertical lines for other edges in the face. The second identity in (55) thenfollows from the first, together with (48), (52) and the Definition of the edge slides: S ρ (cid:15) α (48) = (cid:15) ρ C ρ S ρ (cid:15) α (52) = (cid:15) ρ S ρ C ρ (cid:15) α (55) . = (cid:15) α S α T α Def.6 . = (cid:15) α S α − R . Equation (53) allows us to replace slides along face paths or their inverses by slides along edges. Italso extends Definition 8.1 to slides of edge ends that are traversed by the face path γ . Equation(52) states that sliding along a face path γ commutes with the addition of an edge to another facepath ρ , whenever γ remains a face path. Equations (49), (51) and (55) ensure that we can addmultiple edges to face paths and delete any of the added edges without changing the slide. Equation(54) states that the order in which edges are added to different face paths is irrelevant, as long asface paths remain face paths. Together, these identities allow us to replace slides along face pathsby slides along edges that are added to these face paths.Similarly, adding edges to a vertex v allows one to replace slides of multiple edge ends incident at v by a single slide. This is achieved by adding an edge α with a univalent target vertex to the startingvertex of a face path γ , sliding the edge ends in question along α to the target of α , then sliding α along γ and finally sliding the edge ends back to the starting vertex of α , as shown in Figure 16.36m ...... n+m γ S n + mγ η α α S α S γ αS − α(cid:15) α Figure 16: Replacing slides of multiple edge ends along a face path by a single slide.
Lemma 8.9.
Let γ be a face path in Γ and suppose that the first n + m edge ends at s ( γ ) after thestarting end of γ are not traversed by γ . Insert an edge α with a univalent target vertex at s ( γ ) suchthat there are m edge ends between the starting end of γ and of α , as shown in Figure 16. Denoteby S = S nα L S mα R the slide that slides these m + n edge ends to the target of α . Then S m + nγ = (cid:15) α ◦ S − ◦ S γ ◦ S ◦ η α . Proof.
By the generalised pentagon relation (47), sliding the m + n edge ends to the target of α ,sliding α along γ and then sliding the edge ends back to the start of α gives the same result assliding the first m edge ends, then α and then the last n edge ends along γ . Adding the edge α anddeleting the edge α commute with the slides of the m + n edge ends along γ . Expressions (25) to(32) for the slides show that adding an edge α by applying the unit of H , sliding it along γ andthen deleting α gives the identity morphism. Thus we have (cid:15) α ◦ S − ◦ S γ ◦ S ◦ η α (47) = (cid:15) α ◦ S n + m +1 γ ◦ η α = S nγ ◦ (cid:15) α ◦ S γ ◦ η α ◦ S mγ (25) − (32) = S nγ ◦ S mγ = S m + nγ . As already apparent in Section 7, slides along loops or closed face paths γ do not change theunderlying ribbon graph, if they are applied to all edge ends between the starting and target end of γ . In fact, such slides correspond to Dehn twists, and we will will relate them to the Dehn twistsfrom Theorem 4.1 in Section 9.We introduce these twists in three stages. We first define twists along loops in a directed ribbongraph Γ that are based at a ciliated vertex. We then extend this definition to twists along closedface paths, by adding an edge to the face path, twisting around the edge, and then removing it.Finally, in Section 9 we define twists along certain closed paths that are not face paths. Definition 8.10.
Let β be a loop at a ciliated vertex of Γ . The twist along β is the counterclockwiseslide along β , applied to each edge end between the ends of β once. D β = S nβ L s ( β ) < t ( β ) ,S nβ − R s ( β ) > t ( β ) , where n is the number of edge ends between the starting end s ( β ) and the target end t ( β ) of β .
37t is clear from Definition 8.10 and expressions (25) to (32) for the edge slides that the twist D β does not depend on the orientation of β . Reversing the orientation of β and applying the involution T to the associated copy of H yields the same automorphism of H ⊗ E .Note, however, that the twist along a loop β depends on the choice of the cilium at the starting andtarget vertex of β , that is the linear ordering of the edge ends at this vertex. This dependence isinvestigated in more depth at the end of Section 9, where we show that different choices of cilia leadto mapping class group actions related by conjugation.The twist along a loop β acts only on those copies of H that belong to edges with at least one endincident between the starting and the target end of β . With Lemma 8.9, we can restrict attention tothe case where there is only a single such edge end. More precisely, if β is a loop based at a vertex v of Γ, we can add an edge α at v , with the starting end between s ( β ) and t ( β ) and a univalentvertex t ( α ). Then by Lemma 8.9, we have D β = (cid:15) α ◦ S − ◦ D β ◦ S ◦ η α , where S is the compositeslide that slides all other edge ends incident between s ( β ) and t ( β ) along α to its target. This allowsus to replace some or all edge ends between the ends of a loop by a single edge end in diagrammaticcomputations, as shown in Figure 16.We now extend the definition of the twist from loops to closed face paths in a ribbon graph. This isachieved by adding an edge to the face path, twisting along the added edge and then removing it.This reduces to the original definition whenever the closed face path is a single loop. Definition 8.11.
Let φ be a closed face path based at a ciliated vertex in a directed ribbon graph Γ .The twist D φ along φ is the composite D φ = (cid:15) φ ◦ D φ ◦ C φ : H ⊗ E → H ⊗ E obtained by1. Adding an edge φ to φ as in Definition 8.7,2. Performing a twist along the edge φ ,3. Removing the edge φ as in Definition 8.7. Remark 8.12.
Definition 8.11 extends Definition 8.10 of twists along loops. If φ = β ± for a loop β , then D φ = D β . This follows with Lemma 8.9 and the diagrammatic computation ba C φ D β b (2) b (1) D φ b (5) b (4) b (1) S ( b (2) ) b (3) a(cid:15) φ b (2) b (1) a (59)Definition 8.11 extends Definition 8.10 from edges to face paths. In the following, we will alsoneed to consider twists along certain closed paths that are not face paths. These will be defined inSection 9. The basic idea is to slide the edge ends that prevent a closed path from being a face pathout of the way, either along the path or along other edges, twist along the resulting face path, andthen slide the edge ends back in their original position. For this, we require a number of technicalresults on the interaction between twists, slides and adding edges.38 emma 8.13. Let α be a loop based at a ciliated vertex and β an edge whose starting and targetvertex are ciliated, such that the following are defined. Then:1. Sliding both ends of α and all edge ends between them along β commutes with D α .2. Sliding both ends of α in the same direction along different sides of β commutes with D α .3. Sliding an end of an edge γ into or out of a loop α along an edge δ commutes with D α .Proof. Using Lemma 8.9 we replace the edge ends between the ends of α with a single edge labelledwith c . We then verify 1. by a direct diagrammatic computation, b acS − β L D α S − β L b a (1) cST ( a (2) ) S − ( b (3) ) ab (1) b (4) cb (2) D α S − ( b (3) ) a (1) b (1) b (4) cST ( a (2) ) b (2) in which b is drawn as a loop for notational convenience. This proves 1. for the given edge orientations.The claim for other edge orientations follows by applying the involution T . We verify claim 2. bythe following computation, where the loop α carries the label a , the edge β the label b and β isdrawn as a loop for convenience. Note that in this situation, there is only one end β between thestarting and target end of α by assumption. b a D α bS ( a (1) ) a (2) S β L S β R S β L S β R b (1) aT ( b (3) ) b (2) D α b (1) a (1) T ( b (3) ) b (2) S ( a (2) )Claim 3. is the pentagon relation from Figure 9 applied to the twist along α , which moves the endsof γ and δ , and the slide along δ .The third relation in Lemma 8.13 implies a more general relation between slides and twists. Lemma 8.14.
Let γ be a face path and ρ a closed face path based at a ciliated vertex, satisfyingone of the conditions of Definition 8.6 and such that S γ ( ρ ) is a face path. Then one has S γ D ρ = D S γ ( ρ ) S γ . (60) Proof.
1. We first prove (60) under the assumptions that (i) ρ and γ each involve only a single edge,(ii) s ( ρ ) < t ( ρ ), (iii) the slide is along the left of γ to the target, (iv) the sliding edge end is not an39nd of ρ . Note that (ii) and (iii) do not restrict generality as twists and slides are invariant underedge orientation reversal. As a consequence of (i)-(iv) we have S γ L ( ρ ) = ρ and D ρ = S nρ L , where n is the number of edge ends between s ( ρ ) and t ( ρ ).If neither the starting nor the target end of γ is between the ends of ρ , it follows directly that S γ L commutes with D ρ . If only one of the ends of γ is between the ends of ρ , then S γ L moves an edgeend that does not belong to ρ into or out of the loop ρ . The slide S γ L then commutes with D ρ by Lemma 8.13, 3. If both end points of γ are between s ( ρ ) and t ( ρ ), we verify that S γ L and D ρ commute by the following direct computation: b ca D ρ S γ S γ b (4) b (3) cS ( b (1) ) aS ( b (2) ) b c (2) aS ( c (1) ) D S γ ( ρ ) b (4) b (3) c (2) S ( b (1) ) aS ( c (1) ) S ( b (2) )This proves the claim under assumption (i)-(iv) for s ( γ ) < t ( γ ), and an analogous computationproves it for t ( γ ) < s ( γ ).2. If ρ traverses a single edge and γ is a face path such that S γ does not slide an end of ρ , then S γ ( ρ ) = ρ and (60) follows from 1. by factoring S γ into slides along edges as in Definition 8.1.3. Suppose now that γ and ρ are face paths satisfying the conditions of Lemma 8.14. By Definition8.11 the twist D ρ can be decomposed as D ρ = (cid:15) ρ ◦ D ρ ◦ C ρ . We then obtain D S γ ( ρ ) S γ = (cid:15) S γ ( ρ ) D S γ ( ρ ) C S γ ( ρ ) S γ (52) = (cid:15) ρ D ρ S C ρ ( γ ) C ρ = (cid:15) ρ S C ρ ( γ ) D ρ C ρ = S γ (cid:15) ρ D ρ C ρ . = S γ D ρ . Note that we can apply 2. because ρ is an edge and S C ρ ( γ ) does not move an end of ρ . The fourthstep is immediate if γ does not traverse ρ . Otherwise, it follows by decomposing S γ = S γ S ( ρ ) L S γ and applying the second equation in (55) to the face ( ρ ) − ◦ ρ , for which (51) holds. Corollary 8.15.
Let α : v → w, β : w → v be face paths with v ciliated and n edge ends betweenthe target end of β and the starting end of α , such that sliding them along α or β yields a closedface path γ = α ◦ β . Then S − nα ◦ D γ ◦ S nα = S nβ ◦ D γ ◦ S − nβ . (61) Proof.
Identity (61) is equivalent to the identity S γ ◦ D γ = D γ ◦ S γ for the face path γ , whichfollows directly from Lemma 8.14. Corollary 8.16.
Let ρ be a face path and γ a closed face path based at ciliated vertex such that oneof the conditions of Definition 8.6 is satisfied. Then adding an edge to ρ commutes with D γ : C ρ ◦ D γ = D C ρ ( γ ) ◦ C ρ . Proof.
This follows from the identity C ρ = S ρ ◦ η ρ , because adding a loop at the starting end of anface path ρ and sliding both ends of the loop to its target is the same as adding the loop at thetarget of ρ (see the discussion after Definition 8.7) and from identity (60).40 .. ... n − µ n − µ n α g − α g − β µ α β g α g β g − β g − f Figure 17: The generators of the fundamental group π (Σ). In this section we give explicit descriptions of mapping class group actions for surfaces with andwithout boundaries in terms of generating Dehn twists. We prove that these Dehn twists satisfy thedefining relations of Theorem 4.1 for mapping class groups of surfaces with n + 1 ≥ C finitely complete and cocomplete and involutive pivotalstructures they induce actions of mapping class groups of closed surfaces.We consider an oriented surface Σ of genus g ≥ n + 1 ≥ π (Σ) = h µ , ..., µ n , α , β , ..., α g , β g i = F n +2 g , depicted in Figure 17. Viewed as a directed ribbon graph, this set of generators of π (Σ) has asingle vertex and n + 1 faces, of which n correspond to the loops µ i and the last one to the the path f = µ − · · · µ − n α − β α β − · · · α − g β g α g β − g in Figure 17.We add a cilium to the vertex as in Figure 17 and n edges ν , . . . , ν n that connect this vertex tothe boundary components for the loops µ i and carry ciliated univalent vertices at their other ends.This yields a ribbon graph Γ with n + 1 ciliated vertices and 2( n + g ) edges given by the followingchord diagram β g α g β g − α g − . . . β α ν n µ n µ n − ν n − . . . µ ν (62)41 g α g β g − α g − . . . µ i +1 ν i +1 µ i ν i . . .δ i β g α g β g − α g − . . . β j α j . . .δ n +2 j − β g α g β g − α g − . . . β j α j . . .δ n +2 j Figure 18: Paths in the chord diagram (62) representing the paths δ i from (63).Every face and vertex of this graph is assigned to a unique boundary component of Σ and to aunique cilium. The solid boundary component in Figure 17 corresponds to the baseline, while theother boundary components correspond to the cilia at the univalent vertices. By Lemma 5.5, thisdefines n + 1 commuting Yetter-Drinfeld module structures on H ⊗ E = H ⊗ n + g ) .We now consider the curves α i , δ j and γ i,j for the generating Dehn twists from Theorem 4.1 andchoose representatives of these curves in π (Σ). As Σ has n + 1 boundary components, we index thecurves δ j by 0 ≤ j ≤ n + 2 g − γ k,l by 0 ≤ k, l ≤ n + 2 g −
2, with the 0th boundarycomponent associated with the baseline in (62). By comparing Figure 17 with Figures 5 and 6, wefind that the curves from Theorem 4.1 are represented by the following elements of π (Σ) α i i = 1 , ..., g (63) δ = β − g δ i = µ − i · · · µ − n α − β α β − · · · α − g − β g − α g − β − g − α − g β g α g ≤ i ≤ nδ n +2 j − = α − j β j α j β − j · · · α − g − β g − α g − β − g − α − g β g α g ≤ j ≤ g − δ n +2 j = β − j α − j +1 β j +1 α j +1 β − j +1 · · · α − g − β g − α g − β − g − α − g β g α g ≤ j ≤ g − γ i, = δ i β − g ≤ i ≤ n + 2 g − γ i,j = δ j δ − i ≤ j < i ≤ n + 2 g − γ ,j = δ j α − g β − g α g ≤ j ≤ n + 2 g − γ i,j = δ i α g δ − j α − g ≤ i < j ≤ n + 2 g − . Paths in the chord diagram (62) that represent δ i and γ k,l are drawn in Figures 18 and 19. Note thatthe paths α i for 1 ≤ i ≤ g , the paths δ j for 0 ≤ j ≤ n +2 g −
2, the paths γ k,l for 1 ≤ l < k ≤ n +2 g − γ ,j and γ j, for 1 ≤ j ≤ n + 2 g − γ k,l with 1 ≤ k < l ≤ n + 2 g − H in a symmetric monoidal category C , we associate to each of thepaths ρ in (63) an isomorphism D ρ : H ⊗ n + g ) → H ⊗ n + g ) . For the paths ρ in (63) that are facepaths, this isomorphism D ρ is the twist along ρ from Definition 8.11. For the remaining paths γ i,j = δ i ◦ α g ◦ δ − j ◦ α − g with 1 ≤ i < j ≤ n + 2 g − Definition 9.1.
For ≤ i < j ≤ n +2 g − the twist along γ i,j is the morphism D γ i,j : H ⊗ E → H ⊗ E defined as the following composite:1. Add edges δ i and δ j to the face paths δ i and δ j , as shown in Figure 20. g α g β g − α g − . . . µ i +1 ν i +1 µ i ν i . . .γ i, ≤ i ≤ n. . . µ i ν i µ i − ν i − µ i − ν i − . . . µ j +1 ν j +1 µ j ν j µ j − . . .γ i,j ≤ j < i ≤ nβ g α g β g − α g − . . . µ i +1 ν i +1 µ i ν i . . .γ ,j ≤ j ≤ n Figure 19: Paths in the chord diagram (62) representing the paths γ i,j from (63). δ i δ j α g β g ν µ γ Figure 20: The path γ i,j = δ i ◦ α g ◦ δ j ◦ α − g in Definition 9.1
2. Slide the edge ends between the starting end of α g and the target end of δ j along α g and slidethe starting end of β g along δ i . This yields a face path γ ∗ i,j := δ i ◦ α g ◦ δ j ◦ α − g .
3. Perform a twist along the face path γ ∗ i,j .4. Reverse the slides of step 2., moving back all edge ends into their original position.5. Remove the edges δ i and δ j . Note that the choices in Definition 9.1 do not affect the result. Instead of sliding the edge endsbetween the starting end of α g and the target end of δ j along the right of α g , we could also slidethem along the face path α g ◦ δ j and then along δ i together with the starting end of β g . Similarly,instead of sliding the starting end of β g along δ i , we could slide it along the face path α g ◦ δ j andthen along α g together with the edge ends between the starting end of α g and the target end of δ j .Corollary 8.15 ensures that this yields the same morphisms in C .Note also that neither the slides along the face paths in (63) nor the slides along the paths γ i,j in Definition 9.1 slide edge ends over the cilia. Proposition 6.2 then implies that the associatedautomorphisms D ρ of H ⊗ n + g ) are automorphisms of Yetter-Drinfeld modules with respect to theYetter-Drinfeld module structures from Lemma 5.5 for each cilium. Theorem 9.2.
Let H be a pivotal Hopf monoid in a symmetric monoidal category C , Σ a surfaceof genus g ≥ with n + 1 ≥ boundary components and Γ the embedded graph from (62) .Then the twists along the paths α i , δ j , γ k,l from (63) for i ∈ { , . . . , g } , j ∈ { , . . . , n + 2 g − } and k = l ∈ { , . . . , n + 2 g − } satisfy the relations in Theorem 4.1 and define a group homomorphism ρ : Map(Σ) → Aut
Y D ( H ⊗ n + g ) ) . roof. We use the results on twists from Section 8.3 to verify the relations in Theorem 4.1 byexplicit computations and structural arguments.
Relation (i):
For 1 ≤ j ≤ g − γ n +2 j +1 ,n +2 j = δ n +2 j ◦ δ − n +2 j +1 = β − j γ n +2 j,n +2 j − = δ n +2 j − ◦ δ − n +2 j = α − j β j α j , where we set γ n +2 g − ,n +2 g − := γ ,n +2 g − = β − g − , as in Theorem 4.1. We consider the action of theassociated twists for the diagram (62). They affect only the labels of the chords α j and β j , andtheir action on these edge labels is given by D γ n +2 j,n +2 j − : b a b (2) b (1) aD γ n +2 j +1 ,n +2 j : b a pS ( a (3) ) b (1) a (1) b (2) a (2) pS ( a (3) ) b (1) a (1) b (3) b (2) a (2) b (2) b (1) a Relation (ii):
For curves α, β with | α ∩ β | = 0 that are represented by face paths, Definition 8.11and Lemma 8.9 allows us to assume without loss of generality that the chord diagram that definesthe twists D α and D β takes the form β α or β α (64)where the edges α and β are obtained by adding an edge to the face paths α and β as in Definition8.11 and the other edge ends stand for multiple incoming or outgoing edge ends at these positions,as in Lemma 8.9. For the diagram on the left, the twists along α and β commute, since they act ondifferent copies of the Hopf monoid H . For the diagram on the right, the claim follows by applyingLemma 8.13, 1. to D α and the slide S β L .It remains to prove relation (ii) for the cases where α is not a face path, that is α = γ i,j with1 ≤ i < j ≤ n + 2 g − β ∈ G with | α ∩ β | = 0. This involves the following cases:(a) β = α g ,(b) β = α k with n + 2 k = j and n + 2 k = i ,(c) β = γ k,l with l < k ≤ i or j ≤ l < k or i ≤ l < k ≤ j ,(d) β = δ k with i ≤ k ≤ j ,(e) β = γ k,l with i ≤ k < l ≤ j or 1 ≤ k ≤ i < j ≤ l . As this is symmetric in ( i, j ) and ( k, l ) we canrestrict attention to i ≤ k < l ≤ j .(f) β = γ ,k with j ≤ k .To prove relation (ii) for these cases, we use Definition 9.1 and Figure 20 for the twist along α = γ i,j . • Case (a): Sliding the starting end of β g in Figure 20 along δ i commutes with the twist D α g byLemma 8.13, 3. Sliding the edge ends between the starting end of α g and the target of δ j along α g commutes with the twist D α g by adjacent commutativity from Corollary 6.4. The twist D α g alsocommutes with the twist along the face path γ ∗ i,j from Definition 9.1 because adding an edge to γ ∗ i,j yields the second diagram in (64). This shows that D α g ◦ D γ i,j = D γ i,j ◦ D α g for i < j .44 Cases (b), (c) and (f): We use relation (iv), which we prove below without making use of relation(ii), to express the twist D γ i,j for i < j as the product D γ i,j = ( D δ j ◦ D δ i ◦ D α g ) ◦ D − γ j,i . As β is a face path that intersects neither α g nor the face paths δ i , δ j and γ i,j , it follows that D β commuteswith D α g , D δ i , D δ j and D γ i,j . Hence, D β commutes with D γ i,j = ( D δ j ◦ D δ i ◦ D α g ) ◦ D − γ j,i . • Case (d): To verify relation (ii) for β = δ k with i ≤ k ≤ j , we note that adding edges to the facepaths δ i , δ j and δ k yields the chord diagram β g α g δ j δ i δ k ν µσ where µ, ν, σ stand for multiple edge ends incident at these positions, as in Lemma 8.9. The twistalong γ i,j is again given by Definition 9.1 and Figure 20. Sliding the starting end of β g along δ i commutes with the twist D δ k by the commutativity relation. Siding the edge end ν along α g commutes with D δ k by by Lemma 8.13, 3. The twist D γ ∗ i,j commutes with the twist D δ k becausethey are both face paths as in the second diagram in (64). This shows that D γ i,j and D δ k commute. • Case (e): for α = γ i,j and β = γ k,l with i ≤ k < l ≤ j we use relation (iv) to express D γ k,l as D γ k,l = ( D δ k ◦ D δ l ◦ D α g ) ◦ D − γ l,k . The right hand side involves only twists from cases (a) to (d)and hence commutes with D γ i,j . Relation (iii): If α, β ∈ G with | α ∩ β | = 1, then because of (17) we assume w. l. o. g. that α = α k with 1 ≤ k ≤ g and β = δ i or β = γ i,j for suitable i, j ∈ { , ..., n + 2 g − } . If β is a face path, thenadding an edge to β commutes with D α by Corollary 8.16. We can therefore restrict attention to ν σ µ αβ where β is obtained by adding an edge to the face path β and µ, ν, σ stand for multiple edge endsat these positions, as in Lemma 8.9. We can slide them out of the loops α and β by first sliding ν and σ to the starting end of α , then µ and σ to the target end of β . By Lemma 8.13, 3. and by theadjacent commutativity relation, this commutes with the twists D α and D β , and it reduces thechord diagram to the one in (41), for which relation (iii) was shown in Theorem 7.1.It remains to consider the cases, where β is not a face path, namely β = γ i,j with 1 ≤ i < j ≤ n +2 g − | α k ∩ β | = 1 if and only if i = n + 2 k or j = n + 2 k , see Figures 5 and 6 and theintersection numbers in (17). • For α = α k and β = γ i,j with i = n + 2 k < j we consider the chord diagram ν µ β k α k β g α g δ j δ n +2 k δ j and δ n +2 k are obtained by adding edges to the face paths δ j and δ n +2 k and the edge ends µ and ν stand for multiple edge ends incident at these positions, as in Lemma 8.9. The twist along γ n +2 k,j is given by the sequence of slides in Definition 9.1. Sliding the edge end ν along α g commuteswith D α k by the commutativity relation. Sliding the starting end of β g along δ n +2 k commutes with D α k by Lemma 8.13, 3. The twists D γ ∗ n +2 k,j and D α k satisfy relation (iii), since they are face pathsthat intersect in a single point. Hence, D γ n +2 k,j and D α k satisfy relation (iii) as well. • For α = α k and β = γ i,j with i < j = n + 2 k we consider the chord diagram α k β g α g δ n +2 k δ i ν µβ k where δ i and δ n +2 k are obtained by adding edges to the face paths δ i and δ n +2 k , and the edge endslabeled µ and ν stand again for multiple edge ends incident at these positions.The twist along γ i,n +2 k is given by the sequence of edge slides in Definition 9.1. By Corollary 8.15and the discussion after Definition 9.1, sliding the starting end of β k along α g ◦ δ n +2 k instead of α g and then along δ i together with the starting end of β + g yields the same twist. By Lemma 8.13,3. the slide along α g ◦ δ n +2 k commutes with the twist D α k . Sliding the edge end ν and the targetend of β k along α g and sliding the stating end of β g along δ i commute with D α k as well by thecommutativity relation and opposite end commutativity. We can thus omit β g , β k and ν and proverelation (iii) for the diagram µα k α g δ n +2 k δ i γ i,n +2 k (65)The twist along γ i,n +2 k is then given by sliding the target end of α k along α g , twisting along the facepath γ ∗ i,n +2 k = S α Rg ( γ i,n +2 k ) and sliding the target end of α k back along α g in its original position.By Lemma 8.14, this is identical to D γ i,n +2 k = S α − Rg ◦ D γ ◦ S α Rg with γ = S α Rg ( γ i,n +2 k ) . The face path γ is the image of γ i,n +2 k under the slide that slides the starting and target end of α k and the target end of δ n +2 k along α g and is given as follows µα k α g δ n +2 k δ i γ i,n +2 k S α Rg µα k α g δ n +2 k δ i γ (66)46s the slide S α R commutes with the twist D α k by Lemma 8.13,1. it is now sufficient to show thatrelation (iii) holds for the twists along α k and γ in (66). This follows directly because γ is a facepath that intersects α k in a single point. Relation (iv):
Due to the symmetries of relation (iv), it is sufficient to consider the cases j < i < k , j = i < k and j < i = k . We treat the cases j = 0 and j = 0 separately. • For the case case 1 ≤ j < i < k , depicted in Figure 6, we note that all of the paths in relation(iv) except γ j,k are face paths. The twists along these paths are thus obtained by adding edges δ i , δ j , δ k , γ k,i and γ i,j to them, twisting along these added edges and then deleting them. Note thatadding these edges commutes with the twists along the face paths in Relation (iv) by Corollary 8.16.We can apply the same argument to the twist along the non-face path γ j,k by also noting that theslides in step 2. and step 4. of Definition 9.1 commute with adding the edges by (52). The order inwhich these complements are added does not affect the result by (54). From this, we also obtainthat deleting any of the added edges commutes with the twists under consideration.The twist along γ j,k is given by Definition 9.1 and computed from the chord diagram ν σ µα g δ j δ i δ k β g γ k,i γ i,j γ j,k where the edges µ, ν, σ stand again for multiple edge ends incident at this position, as in Lemma 8.9.The twist D γ j,k is obtained by first sliding the starting end of β g along δ j and the edge end ν along α g , which yields the face path γ ∗ j,k = δ j ◦ α g ◦ δ k ◦ α − g , then twisting along γ ∗ j,k and finally slidingthe edge ends back. Sliding the starting end of β g along δ j commutes with the twists D δ i , D δ k , D γ i,j and D γ k,i by the commutativity relation, with D δ j by the adjacent commutativity relation inCorollary 6.4 and with D α g by Lemma 8.13, 3. Sliding the edge end ν along α g commutes with D γ i,j and D γ k,i by the commutativity relation, with D α g by the adjacent commutativity relationand with D δ i , D δ j and D δ k by Lemma 8.13, 3. It is therefore sufficient to consider the followingchord diagram, obtained by omitting β g and ν and adding an edge γ j,k to γ ∗ j,k . α g δ j δ i δ k γ k,i γ i,j γ j,k σ µ (67)We have to show that for this diagram (cid:15) ◦ ( D δ k ◦ D δ i ◦ D δ j ◦ D α g ) = (cid:15) ◦ D γ i,j ◦ D γ j,k ◦ D γ k,i , where (cid:15) deletes the added edges δ i , δ j , γ k,i , γ i,j and γ j,k .The paths γ k,i ◦ δ k ◦ δ i , γ j,k ◦ δ j ◦ α g ◦ δ k ◦ α − g and γ i,j ◦ δ i ◦ δ j are ciliated faces of the chorddiagram, and because they were obtained by adding edges to face paths, the H -right comodulestructures associated to them are trivial by (51). This allows us to use identity (58) to express thelabel of one edge of every face in terms of the other labels of that face.47o simplify the expressions in the following computations, we also reverse the orientation of theedges γ i,j , γ k,i , γ j,k , δ i , δ j , δ k . Assigning the labels a, c, d, e to the edges α g , δ k , γ k,i , γ i,j and labels y, z to the edge ends σ, µ , we then obtain the labels of the edges δ i , δ j and γ j,k from identity (58).We then can delete γ j,k and γ k,i before computing ( D δ k ◦ D δ i ◦ D δ j ◦ D α g ) , and δ i and δ j beforecomputing D γ j,k ◦ D γ k,i ◦ D γ i,j and obtain the following two H -labelled chord diagrams: c (2) a (2) d (1) e (1) y za (3) S ( c (1) ) S ( a (1) ) p − c (3) p − d (2) p − e (2) c (3) p − d (2) p − e c (2) p − d (1) c (1) a y z (68)Using the second diagram in (68), we compute the morphism D δ k ◦ D δ i ◦ D δ j ◦ D α g . D δ k ◦ D δ i ◦ D δ j ◦ D α g : c (3) p − d (2) p − ec (2) p − d (1) c (1) a y zD α g tur (1) a (2) y z r = a (1) cu = r (2) p − d (1) t = r (3) p − d (2) p − eD δ j t (8) t (7) uS ( t (2) ) t (6) r (1) S ( t (4) ) a (2) S ( t (5) ) t (3) y t (1) z r = a (1) cu = r (2) p − d (1) t = r (3) p − d (2) p − e δ i t (8) t (7) u (5) S ( t (2) ) t (6) u (4) r (1) S ( t (4) u (2) ) a (2) S ( t (5) u (3) ) t (3) u (1) y t (1) z r = a (1) cu = r (2) p − d (1) t = r (3) p − d (2) p − eD δ k t (8) t (7) u (5) S ( t (2) ) t (6) u (4) r (2) S ( t (4) u (2) ) a (2) S ( t (5) u (3) r (1) ) t (3) u (1) y t (1) z r = a (1) cu = r (3) p − d (1) t = r (4) p − d (2) p − e Applying this transformation three times, we get( D δ k ◦ D δ i ◦ D δ j ◦ D α g ) : c (3) p − d (2) p − ec (2) p − d (1) c (1) a y z e d c a y z (69)where v = a (5) S ( c (1) ) S ( a (1) ) p − c (5) p − d (3) p − e (2) a = v (9) a (3) S ( v (5) ) y = v (3) d (1) y z = v (1) e (1) zc = v (6) c (3) S ( v (4) ) d = v (7) c (4) p − d (2) S ( v (2) ) e = v (8) a (2) c (2) pS ( a (4) ) . The morphism D γ j,k ◦ D γ k,i ◦ D γ i,j is obtained from the first diagram in (68) and given by49 (3) S ( c (1) ) S ( a (1) ) p − c (3) p − d (2) p − e (2) c (2) a (2) d (1) e (1) y zD γ i,j a (3) S ( c (1) ) S ( a (1) ) p − c (3) p − d (2) p − e (3) c (2) a (2) d (1) e (2) y e (1) zD γ k,i a (3) S ( c (1) ) S ( a (1) ) p − c (3) p − d (3) p − e (3) c (2) a (2) d (2) e (2) d (1) y e (1) zv = a (3) S ( c (1) ) S ( a (1) ) p − c (4) p − d (3) p − e (3) D γ j,k v (9) c (2) S ( v (7) ) v (10) a (2) S ( v (8) ) v (11) v (6) d (2) S ( v (4) ) v (3) e (2) S ( v (1) ) v (5) d (1) y v (2) e (1) z (70)After applying (cid:15) γ i,j , (cid:15) γ k,i and (cid:15) γ j,k to diagram (70) to eliminate the edges γ i,j , γ k,i and γ j,k and apply-ing (cid:15) δ i and (cid:15) δ j to (69) to eliminate the edges δ i and δ j , one finds that the resulting transformationsagree. This proves relation (iv) for 1 ≤ j < i < k .50 The proof for the case 1 ≤ j = i < k is obtained from the same computation by removing theedge labeled z and setting e = 1. Relation (iv) for 1 ≤ j < i = k follows similarly by removing theedge labeled y and setting d = 1. • For the case 0 = j ≤ i ≤ k all of the paths representing the curves in relation (iv) are face paths.The twists along these face paths are given by adding edges to the face paths γ i, , γ k,i , γ ,k , δ i and δ k , which yields the following diagram α g δ i δ k β g = δ γ ,k γ k,i γ i, Sliding the starting and target end of β g along the edge α g commutes with the twist along δ = β g by Lemma 8.13, 2, with the twists along δ i and δ k by Lemma 8.13, 3. and with the twists along γ i, , γ ,k and γ k,i by the commutativity relation. It is therefore sufficient to prove that relation (iv)holds for the diagram obtained by this slide: α g δ i δ k β g = δ γ ,k γ k,i γ i, Up to the labelling of the edges, this coincides with diagram (67), for which we have already proventhe claim. This concludes the proof.Theorem 9.2 extends Theorem 6.5 to surfaces of genus g ≥ of genus g ≥ n ≥ g with n + 1 boundary components in Figure 17 by attaching a disc to thecoloured boundary component. If we work with the graph Γ from (62) and Figure 17, this amountsto attaching a disc to the path γ , = f in (63) and in Figure 17. By Theorem 4.1 the presentationof the mapping class group Map(Σ ) is then obtained from the presentation of Map(Σ) used inTheorem 9.2 by imposing the additional relations D γ , = 1 , D δ = D δ , D γ , = ( D δ D α g ) , D γ ,k = D γ ,k , D γ k, = D γ k, (71)for 1 < k ≤ n + 2 g −
2, see [G01]. In general, they do not hold for the action of Map(Σ) on H ⊗ n + g ) from Theorem 9.2 and neither on the invariants of the H -left module structure nor onthe coinvariants of the H -left comodule structure associated with the baseline of diagram (62)(cf. Example 9.6). Instead, we have to impose both invariance and coinvariance and work with thebiinvariants of the Yetter-Drinfeld module structure associated with the baseline of (62).We first show that the group homomorphism ρ from Theorem 9.2 induces an action of the mappingclass group Map(Σ) on the biinvariants of this Yetter-Drinfeld module. For this, recall that the51enerating Dehn twists from Theorem 9.2 do not slide edge ends over the associated cilium and areautomorphisms of this Yetter-Drinfeld module by Proposition 6.2. From Lemma 2.9 we then obtain Corollary 9.3.
Let C be a finitely complete and cocomplete symmetric monoidal category and H apivotal Hopf monoid in C . Then ρ from Theorem 9.2 induces a group homomorphism ρ inv : Map(Σ) → Aut( H ⊗ n + g ) inv ) (72) Proof.
By Lemma 2.9 every element φ ∈ Map(Σ) induces a unique automorphism ρ inv ( φ ) of H ⊗ n + g ) inv with I ◦ ρ inv ( φ ) ◦ P = π ◦ ρ ( φ ) ◦ ι . This defines a group homomorphism as in (72), because ρ is a group homomorphism, I is a monomorphism and P an epimorphism.We now show that under suitable assumptions the group homomorphism ρ from Theorem 9.2 notonly induces an action of the mapping class group Map(Σ) on the biinvariants of this Yetter-Drinfeldmodule structure but an action of the mapping class group Map(Σ ), where Σ is the surface obtainedby attaching a disc to the face f in Figure 17. For this, we require the following technical lemma. Lemma 9.4.
Let C be a finitely complete and cocomplete symmetric monoidal category and H apivotal Hopf monoid in C with m ◦ ( p ⊗ p ) = η . Let Γ be a directed ribbon graph and γ , γ closedface paths in Γ such that γ = γ γ is a ciliated face. Then we have for the biinvariants of theYetter-Drinfeld module structure at this cilium D invγ = D invγ , D invγ = 1 H ⊗ Einv . Proof.
The second claim follows from the first by setting γ = γ and taking the trivial pathfor γ . To prove the first claim, note that these twists slide no edge ends over the cilium andhence the automorphisms D γ i : H ⊗ E → H ⊗ E are automorphisms of Yetter-Drinfeld modules byProposition 6.2. By Lemma 2.9 they induce unique automorphisms D invγ i : H ⊗ Einv → H ⊗ Einv with I ◦ D invγ i ◦ P = π ◦ D γ i ◦ ι . As I is a monomorphism and P an epimorphism, it is sufficient to showthat π ◦ D γ ◦ ι = π ◦ D γ ◦ ι . As the twist along γ i is obtained by adding an edge γ i to γ i andtwisting along γ i , it is sufficient to verify this for the chord diagram γ γ µ ν , in which we replaced the edge ends between the starting and target ends of γ , γ by single edges µ, ν , as in Lemma 8.9. The Yetter-Drinfeld module structure for the cilium is given by b ac dh (cid:66) h (4) bS ( h (6) ) h (1) aS ( h (3) ) h (5) c h (2) db ac d δ b (2) a (2) c da (1) b (1) δ ◦ ι = ( η ⊗ H ⊗ ) ◦ ι , π ◦ (cid:66) = π ◦ ( (cid:15) ⊗ H ⊗ ) and m ◦ ( p ⊗ p ) = η we then obtain b ac d ι ≡ b (2) (cid:15) ( a ) S ( b (1) ) c dD γ b (3) (cid:15) ( a ) S ( b (2) ) c b (1) pd D − γ b (3) (cid:15) ( a ) S ( b (2) ) b (4) pc b (1) pd = b (5) (cid:15) ( a ) b (2) T ( b (1) ) S ( b (4) p ) b (6) pc b (3) pd = b (5) pb (8) S ( b (7) p ) (cid:15) ( a ) b (2) T ( b (1) ) S ( b (4) p ) b (6) pc b (3) pd π ≡ b (2) (cid:15) ( a ) S ( b (1) ) c d ι ≡ b ac d where we used first the identity δ ◦ ι = ( η ⊗ H ⊗ ) ◦ ι , then the definition of the twists, then theproperties of a Hopf monoid, then the identity π ◦ (cid:66) = π ◦ ( (cid:15) ⊗ H ⊗ ) to pass to the fourth line andthe identity δ ◦ ι = ( η ⊗ H ⊗ ) ◦ ι in the final step. The notations ι ≡ and π ≡ mean that the morphismsassociated to the two labelled diagrams are coequalised by ι and equalised by π , respectively.Note that the condition m ◦ ( p ⊗ p ) = η is required only for the first relation in Lemma 9.4. Thesecond relation holds without any assumptions on the pivotal structure, but we will need bothrelations in the following. Note also that this conditions imposes a restriction on the antipode ofthe Hopf monoid, namely the condition S = 1 H .We now consider the oriented surface Σ of genus g ≥ n + 1 ≥ with n ≥ f in Figure 17. Let Γ be the associated ribbon graph from (62), consider the Yetter-Drinfeldmodule structure for the baseline in (62) and the associated object H ⊗ n + g ) inv from Definition 2.8. Theorem 9.5.
Let C be a finitely complete and cocomplete symmetric monoidal category and H apivotal Hopf monoid in C with m ◦ ( p ⊗ p ) = η . The the group homomorphism ρ from Theorem 9.2induces a group homomorphism ρ inv : Map(Σ ) → Aut( H ⊗ n + g ) inv ) . Proof.
We show that the images of the generating Dehn twists under the group homomorphism ρ inv from Corollary 9.3 satisfy the additional relations in (71). As the face of the ribbon graph in(62) is given in terms of the paths in (63) by f = γ , = δ ◦ δ , we have by Lemma 9.4 ρ inv ( D γ , ) = 1 H ⊗ n + g ) inv , ρ inv ( D δ ) = ρ inv ( D δ ) . (73)53his proves the first two relations in (71). Combining (73) with relation (iv) in Theorem 4.1 forthe mapping class group Map(Σ) for ( j, i, k ) = (0 , , , (0 , , k ) , (1 , , k ) and (0 , , k ) yields for1 < k ≤ n + 2 g − ρ inv ( D γ , ) ( iv ) = ρ inv ( D − γ , ( D δ D δ D α g ) ) , (73) = ρ inv (( D δ D α g ) ) ρ inv (( D δ D δ k D α g ) ) ( iv ) = ρ inv ( D γ ,k D γ k, ) = ρ inv ( D γ ,k D γ k, ) = ρ inv ( D γ ,k D γ k, ) . The first line is the third relation in (71). The second line implies the last two relations in (71).In particular, Theorem 9.5 defines actions of mapping class groups of closed surfaces of genus g ≥ H ⊗ ginv . That the condition on the pivotal element is necessary is already apparent in the simplestexample for a surface of genus g = 2. Example 9.6.
Let Σ be a surface of genus g = 2 with one boundary component, Σ the closedsurface obtained by attaching a disc to the boundary of Σ and Γ the ribbon graph from Figure 17and (62) . Let G be a finite group, H = C [ G ] and p ∈ Z ( G ) a central element.The left-left Yetter-Drinfeld module structure on C [ G × ] associated with the cilium in Figure 17 andthe baseline of the chord diagram (62) is given by (cid:66) : C [ G × ] → C [ G × ] , ( h, a , b , a , b ) ( ha h − , hb h − , ha h − , hb h − ) , (74) δ : C [ G × ] → C [ G × ] , ( a , b , a , b ) ([ a − , b ][ a − , b ] , a , b , a , b ) , where the arguments a i and b i are associated with the loops α i and β i , respectively. The linearsubspace of coinvariants of this Yetter-Drinfeld module structure is M coH = span C { ( a , b , a , b ) | [ a − , b ][ a − , b ] = 1 } ⊂ C [ G × ] , and the subspace M inv ⊂ C [ G × ] of elements that are both invariants and coinvariants with respectto (74) is the image of M coH under the projector π : M → M , x | G | Σ h ∈ G h (cid:66) x. By Theorem 4.1, the action of
Map(Σ) on C [ G × ] via the group homomorphism ρ in Theorem 9.2is generated by the Dehn twists along α , α and along the following paths from (63) δ = β − , δ = [ α − , β ] α − β α , δ = β − α − β α , γ , = [ α − , β ] , γ , = β − ,γ , = [ α − , β ][ α − , β ] , γ , = β − [ α − , β ] , γ , = α − β α , γ , = [ α − , β ][ α − , β ] α β α − . From Definitions 8.10 and 9.1 and formulas (25) to (32) for the slides, one computes the action ofthe associated twists on C [ G × ] . Listing only those arguments that transform nontrivially, we obtain D α : b b a − , (75) D α : b b a − ,D δ : a b a ,D δ : x a − b − a [ b , a − ] x [ a − , b ] a − b a , x ∈ { a , b } ,a p a [ a − , b ] a − b a ,D δ : a p − a − b − a b a ,b a − b − a b a − b a ,a pa b − a − b a , γ , : x [ b , a − ] x [ a − , b ] , x ∈ { a , b } ,D γ , : a b a ,D γ , : x [ b , a − ][ b , a − ] x [ a − , b ][ a − , b ] , x ∈ { a , b , a , b } ,D γ , : a p − [ b , a − ] b a ,x [ b , a − ] b xb − [ a − , b ] , x ∈ { b , a , b } ,D γ , : a b a ,D γ , : a p [ w, a − ] a w − , w = a b − a − [ b , a − ][ b , a − ] ,b [ w, a − ] b [ a − , w ] , u = a − b − a [ b , a − ] ,a wa w − ,b b [ u, w ] . By a direct but lengthy computation, one can verify that these generating Dehn twists indeed satisfythe relations from Theorem 4.1 for the mapping class group
Map(Σ) . The mapping class group
Map(Σ ) is obtained by imposing the additional relations from (71) D γ , = id C [ G × ] , D δ = D δ , D γ , = ( D δ D α ) , D γ , = D γ , , D γ , = D γ , , (76) which follow from the first two relations in (76) . From the expressions for the Dehn twists in (75) it is apparent that the first relation holds both on the invariants and on the coinvariants ofthe Yetter-Drinfeld module structure (74) and for all elements p ∈ Z ( G ) . In general, the relation D δ = D δ holds only if one imposes both, invariance and coinvariance and p = 1 . Example 9.7.
Let Σ , Σ and Γ be as in Example 9.6. Let G be a group, viewed as a Hopf monoidin the cartesian monoidal category C = Set , and p ∈ Z ( G ) .Then the Yetter-Drinfeld module structure on G × for the cilium in Figure 17 and the baseline of (62) is again given by (74) . Its biinvariants are the moduli space of flat G -connections on Σ G × inv = { ( a , b , a , b ) ∈ G × | [ a − , b ][ a − , b ] = 1 } /G = Hom( π (Σ ) , G ) /G. The generating Dehn twists of the mapping class group
Map(Σ) are again given by (75) , and onefinds that they induce an action of
Map(Σ ) on M inv if p = 1 . For p = 1 , this gives the action ofthe mapping class group Map(Σ ) on Hom( π (Σ ) , G ) /G . Theorems 9.2 and 9.5 associate mapping class group actions to specific ribbon graphs equippedwith additional structures, namely choices of cilia at each vertex and face. In the remainder of thissection we show that these mapping class group actions are not limited to the specific graphs inFigure 17 and in (62) and only depend mildly on these choices. Similar actions of the mapping classgroups Map(Σ) and Map(Σ ) are obtained for any ribbon graph Γ, as long as each vertex and eachface of Γ carries exactly one cilium. This follows because any such graph can be transformed intothe graph from (62) by a sequence of edge slides. Lemma 9.8.
Let Γ be a ribbon graph with n + 1 vertices and faces, such that every vertex and facecarries exactly one cilium. Then there is a sequence of slides that do not slide edges over cilia andof edge reversals that transforms Γ into the graph from (62) with g = | E | − n .Proof. Note that the edge slides preserve the bijection between the faces and vertices defined by thecilia, as long as no edges slide over cilia. As a first step we transform Γ into a chord diagram. Firstwe choose a vertex x . For every neighbouring vertex w , we choose an edge ν w connecting x and w ,orient ν w towards x , and slide every other edge end incident at w along ν w to x , such that no edge55a) µ ν γβ α µνγβ α (b) µν µν Figure 21: Transforming a ribbon graph into the graph from Figure 17.end slides over the cilium at w . We repeat this until every vertex except x is univalent. We thenorient every loop γ at x such that t ( γ ) > s ( γ ) for the ordering at x and obtain a chord diagram Γ .As a second step we form the pairs of loops labeled α i and β i in (62). For any pair of edges α and β in Γ with t ( β ) > t ( α ) > s ( β ) > s ( α ), we slide all other edge ends between their starting and targetend along α and β to the right, as shown in Figure 21 (a). This yields a chord diagram Γ .As a third step we form the loops labeled µ i in (62). Suppose w is a univalent vertex of Γ and ν the unique edge that connects w and x . Because w carries a cilium that also belongs to a uniqueface of Γ , there is a loop µ with t ( µ ) < t ( ν ) < s ( µ ), as shown in Figure 21 (b). We then slide bothends ends of µ next to the target vertex of ν , as shown in Figure 21 (b). We repeat this step untilall univalent vertices are associated with an edge ν i to the baseline and a loop µ i as in (62). Finallywe slide all these pairs ( µ i , ν i ) to the right of all pairs ( α j , β j ) and obtain the graph from (62). Theidentity for g follows from the assumptions and the formula for the Euler characteristic.Suppose now that Γ is a directed ribbon graph as in Lemma 9.8. Then attaching annuli to the facesyields a surface Σ with n + 1 boundary components. By Lemma 9.8 for any vertex x of Γ there is asequence of edge slides and edge reversals that transforms Γ into the graph (62) and the cilium at x to its baseline. As no edges slide over cilia and orientation reversal is compatible with the Yetter-Drinfeld module structure by definition, this defines an automorphism φ : H ⊗ n + g ) → H ⊗ n + g ) ofYetter-Drinfeld modules. By conjugating the generating Dehn twists from Theorem 9.2 with φ , wethen obtain another set of automorphisms for each curve σ in (63) that satisfy the relations fromTheorem 4.1 and define an action of Map(Σ) by automorphisms of Yetter-Drinfeld modules.If C is finitely complete and cocomplete and m ◦ ( p ⊗ p ) = η , by Lemma 2.9 the automorphism φ induces an isomorphism φ inv between the biinvariants associated with the Yetter-Drinfeld modulestructure at x and with the baseline of (62). Conjugating the action of the mapping class groupMap(Σ ) from Theorem 9.5 on the biinvariants of (62) with φ inv then defines an action of Map(Σ )on the biinvariants at x . In particular, this implies that the choice of the cilium at the multivalentvertex in Figure 17 affects the associated mapping class group actions from Theorem 9.2 andTheorem 9.5 only by conjugation with an automorphism. Concluding Remarks
The mapping class group actions constructed in this article are distinct from the ones obtained byLyubashenko [Ly95a, Ly95b, Ly96], from the mapping class group actions [AS, Fa18a, Fa18b] inthe context of Chern-Simons theory, as they are based on different data.As our mapping class group actions are obtained from a generalisation of Kitaev lattice models56Ki, BMCA] which were in turn related to Turaev-Viro TQFTs in [BK, KKR], it is plausible thatthey should be related to mapping class group actions in Turaev-Viro TQFTs when our pivotalHopf monoid H is a finite-dimensional semisimple Hopf algebra over C . The relations betweenTuraev-Viro TQFTs and Reshetikhin-Turaev TQFTs [Ba, KB, TVi] and between Kitaev latticemodels and the quantum moduli algebra [Me] then suggest a relation to the mapping class groupactions from [AS, Fa18a, Fa18b, Ly95a, Ly95b, Ly96] for the Drinfeld double D ( H ). However, thisquestion is beyond the scope of this article.If H is a finite-dimensional semisimple Hopf algebra over C , the results in Section 6 are also relatedto an earlier observation by Kashaev [Ka95]. It is shown in [Ka95] that the action of the universal R -matrix of the Drinfeld double D ( H ) on three modules over its Heisenberg double H ( H ) satisfiesthe pentagon relation, which is related to the mapping class group action on a triangulation viaflip or Whitehead moves [P]. Under these assumptions, modules over the Heisenberg double H ( H )correspond bijectively to H -Hopf modules. A direct computation shows that the action of theuniversal R -matrix of D ( H ) on H ( H ) ⊗ H ( H ) then gives the expressions for the edge slides in (25)to (32) for the pivotal element η . The pentagon relations from Figure 9 for the edge slides and thediagrammatic proof in Figure 11 can thus be viewed as a generalisation of [Ka95]. Acknowledgements
This research was supported in part by Perimeter Institute for Theoretical Physics. Researchat Perimeter Institute is supported by the Government of Canada through the Department ofInnovation, Science and Economic Development and by the Province of Ontario through the Ministryof Research and Innovation.