Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups
Simon Lentner, Svea Nora Mierach, Christoph Schweigert, Yorck Sommerhaeuser
aa r X i v : . [ m a t h . QA ] M a r ZMP-HH/20-5Hamburger Beitr¨age zur Mathematik Nr. 825
Hochschild Cohomology, Modular TensorCategories, and Mapping Class Groups
I. General Theory
Simon Lentner, Svea Nora Mierach,Christoph Schweigert, Yorck Sommerh¨auser
Abstract
Given a finite modular tensor category, we associate with each compact surfacewith boundary a cochain complex in such a way that the mapping class groupof the surface acts projectively on its cohomology groups. In degree zero, thisaction coincides with the known projective action of the mapping class groupon the space of chiral conformal blocks. In the case that the surface is a torusand the category is the representation category of a factorizable ribbon Hopfalgebra, we recover our previous result on the projective action of the modulargroup on the Hochschild cohomology groups of the Hopf algebra. ontents Introduction 41 Mapping class groups 6 ntroduction
It is by now well-understood that a semisimple modular tensor category gives riseto a topological field theory. In particular, such a topological field theory assignsfinite-dimensional vector spaces to surfaces, possibly with boundary. These vectorspaces are constructed from the homomorphism spaces of the category and are calledthe spaces of chiral conformal blocks, or briefly the block spaces. They dependfunctorially on the objects of the category assigned to the boundary components, sothat it is more appropriate to say that a topological field theory assigns to a surfacenot only a space, but rather a functor. To diffeomorphisms of surfaces, the topologicalfield theory assigns natural transformations between these functors, which gives riseto projective representations of mapping class groups. These data obey factorizationconstraints related to the cutting and gluing of surfaces, properties formalized in thenotion of a modular functor.Already early in the development of the theory, it was noted that such projectiverepresentations can be obtained even if the modular category is not semisimple. Inthis case, the arising functors are no longer exact, but they are still left exact, sothat it is natural to study their derived functors. The main result of the presentwork is that the cohomology groups arising from these derived functors still carry aprojective action of the mapping class group in such a way that the original actionis recovered in degree zero. It is therefore appropriate to call these spaces derivedblock spaces.This result can be applied to a special case: For the category, one can use therepresentation category of a factorizable ribbon Hopf algebra, and for the surface, onecan use the torus. In this case, the mapping class group is isomorphic to the modulargroup, and the cohomology groups become the Hochschild cohomology groups of theHopf algebra. In this way, we recover the main result of our previous article [LMSS1].The approach used to reach our present result is inspired by the principle of ‘propaga-tion of vacua’ (cf. [TUY, Par. 2.2, p. 476]). This technique introduces an additionalboundary component on the surface that, if labeled with the monoidal unit of thecategory, leads to block spaces that are canonically isomorphic to the block spaceswithout the additional boundary component. In our construction, the new boundarycomponent serves as the position where we insert a projective resolution of the unitobject. By the functoriality of the block spaces, we then obtain a cochain complexon which the mapping class group of the surface with one additional boundary com-ponent acts. The main point of the argument will be that this additional boundary4omponent can be closed again when passing to cohomology. To establish this, weproceed in two steps, using standard techniques from the theory of mapping classgroups, namely the capping sequence in the first step and the Birman sequence inthe second step.The present work sets out the general theory needed to establish the result just de-scribed, but also prepares the ground for the computation of explicit examples ofthese mapping class group representations, which will be addressed in our forthcom-ing article [LMSS2]. A new aspect of our approach is that we consider an actionof the mapping class group on the fundamental group by requiring that the basepoint of the fundamental group be kept fixed. In this way, it is possible to avoid thenecessity to identify homomorphisms that are related by simultaneous conjugationthat can be found in many of the other articles on topological or conformal fieldtheory.The material is organized as follows: Section 1 reviews surfaces, fundamental groups,and mapping class groups as well as the capping sequence and the Birman sequence.It also introduces the action of the mapping class group on the fundamental groupjust mentioned. Section 2 explains how representations of mapping class groups areassigned to certain tensor categories in topological field theory. For this, we use theframework created by V. Lyubashenko in his articles [L1] and [L2]; in particular, weuse the approach to surfaces via nets and ribbon graphs described in his articles.In Section 3, we first state and prove our main result and then explain why thisresult generalizes our previous one from [LMSS1]. In fact, our present result wasalready mentioned in [LMSS1], and it was also described in [FS2]. Here, we are nowsupplying proofs for our claims.Throughout the text, the word ‘projective’ is used frequently. It has two very differentmeanings: When speaking about projective modules and projective resolutions, theterm is used in the sense of ring theory and homological algebra. When speakingabout projective space and projective representations or actions, the term is used inthe sense of projective geometry. In particular, for a vector space V , we denote theassociated projective space, i.e., the set of its one-dimensional subspaces, by P ( V ),and the projectivity or homography induced by a bijective linear map f by P ( f ). Theset of all projectivities from P ( V ) to itself forms the projective linear group PGL( V ),which is isomorphic to the general linear group GL( V ) modulo the scalar multiplesof the identity transformation. By a projective representation or projective action,we mean a group homomorphism from a given group to the projective linear group.5e will assume throughout that our base field K is perfect, i.e., that all of itsalgebraic extensions are separable (cf. [J, Chap. IV, §
1, p. 146]). In particular,algebraically closed fields are perfect, as are fields of characteristic zero.In general, we compose mappings and morphisms in a category from right to left, i.e.,we have ( g ◦ f )( x ) = g ( f ( x )), so that f is applied first. In contrast, we concatenatepaths in the fundamental group from left to right, i.e., in the concatenation κγ , thepath κ is traced out first, while the path γ is traced out afterwards.We use the symbol V for the category of finite-dimensional vector spaces and thesymbol S n for the symmetric group in n letters. Additional notation will be explainedin the text.While carrying out this research, the first and the third author were partially sup-ported by the RTG 1670 ‘Mathematics inspired by String theory and Quantum FieldTheory’ and by the ‘Deutsche Forschungsgemeinschaft’ under Germany’s ExcellenceStrategy EXC 2121 ‘Quantum Universe’ 390833306, while the second and the fourthauthor were partially supported by NSERC grant RGPIN-2017-06543. The principal result of the classification of surfaces asserts that every compact, con-nected, orientable, smooth 2-dimensional manifold is diffeomorphic to the connectedsum Σ g of a sphere with g ≥ g of the attached tori, which are often called handles, is uniquely deter-mined by the surface and is called its genus. If we include manifolds with boundary,then the classification theorem asserts that a compact, connected, orientable, smooth2-dimensional manifold with boundary is diffeomorphic to Σ g with n ≥ g,n , so that Σ g = Σ g, . If the boundaryis not empty, it is the finite disjoint union of n connected components, called theboundary components. We assume that on each of the boundary components, a pointhas been distinguished, or marked, as we also say. The Euler characteristic of Σ g,n is χ (Σ g,n ) = 2 − g − n.
6y assumption, the surfaces above are orientable, but when we refer to Σ g,n , wewill always assume that an orientation has been chosen. In the pictures of Σ g,n thatwe will encounter below, Σ g,n will be drawn embedded into 3-dimensional space, inwhich case we will assume that the orientation is represented by the outward-pointingnormal vector field.In the standard proof of the classification theorem, the surface is realized not as aconnected sum, but rather as a quotient space of a polygon whose edges are labeledin a certain normal form. In the normal form for the closed surface Σ g , the edges arelabeled consecutively as α , β , α − , β − , α , β , α − , β − , . . . , α g , β g , α − g , β − g , while for the surface Σ g,n with boundary, the edges are labeled consecutively as α , β , α − , β − , . . . , α g , β g , α − g , β − g , ξ , ρ , ξ − , . . . , ξ n , ρ n , ξ − n . In the case g = n = 3, this polygon has the form α α α α α α β β β β β β ξ ξ ξ ξ ξ ξ ρ ρ ρ where the inverse signs have been depicted by reversing the orientation of the corre-sponding arrow. 7n the case of a closed surface, the quotient space is then realized in such a waythat all the vertices of the polygon become a single point x , while the edges labeled α i and β i are, respectively, identified with their counterparts labeled α − i and β − i ,which are traced out in the opposite direction, as we use the general conventionthat γ − ( t ) := γ (1 − t ) for a path defined on the unit interval. In the case of thesurface Σ g,n with boundary, in addition an edge labeled ξ j is identified in the sameway with its counterpart labeled ξ − j . The edges labeled ρ j then become closed curvesin the quotient that represent the n boundary components (cf. [ST, Kap. 6, § x , but only those that are not the start point or the endpoint of one of the edges ρ , . . . , ρ n . The start point of ρ j is instead identified withthe end point of ρ j and in this way yields the marked point on the j -th boundarycomponent.If we carry out only the second identification in the case depicted above, i.e., theidentification of ξ j with ξ − j , we obtain the following picture of the intermediate stage: α α α α α α β β β β β β ξ ξ ξ ρ ρ ρ When we carry out all the identifications, the edges labeled α i and β i become in thequotient the closed curves that appear in the following picture:8 α α α β β β ρ ρ ρ ξ ξ ξ This realization of the surface is called the polygon model of the surface, as opposedto the connected sum model that we briefly described at the beginning of the para-graph. Note that, in order to obtain the outward-pointing orientation of the surfacementioned above, the surface of the polygon needs to be oriented by a normal vectorfield that is orthogonal to the page and is pointing away from the reader, which isnot the standard orientation of the plane used elsewhere in the text, for example inParagraph 1.6. Furthermore, in this model the boundary curves ρ j do not carry theorientation that is induced by the surface on its boundary, but rather the oppositeorientation. In the polygon model of a closed surface, we use the point x in the quotient that allvertices of the polygon map to as the base point of the fundamental group π (Σ g , x ).The elements of the fundamental group are the homotopy classes [ γ ] of paths γ defined on the unit interval, where homotopy is relative to this base point x . It9urns out that the relative homotopy classes of the 2 g elements α , β , . . . , α g , β g justdefined generate the fundamental group. In the connected sum model, each pair α i , β i lies on one of the g tori that we attached to a sphere at the beginning ofParagraph 1.1. Two neighboring tori are connected by the path µ i := α − i +1 β i α i β − i ,where it is understood that µ g := α − β g α g β − g if i = g . To see this, we first observethat the path β i α i β − i is homotopic to a kind of mirror image of α i : ... ... x i − ii + 1 g α i β i β − i ... ... x i − ii + 1 g ... ... x i − ii + 1 g ... ... x i − ii + 1 g
10f we now also concatenate with the curve α − i +1 , we see that the curve µ i is homotopicto a curve that connects the i -th and ( i + 1)-st handle: ... ... x i − ii + 1 g α − i +1 ... ... x i − ii + 1 g µ i These pictures in fact illustrate that the concatenated path µ g µ g − · · · µ is homotopicto the constant path based at x , which means that( α − β g α g β − g )( α − g β g − α g − β − g − ) · · · ( α − β α β − )represents the unit element in the fundamental group. Alternatively, by conjugatingwith α and inverting, we see that α β α − β − α β α − β − · · · α g β g α − g β − g represents the unit element in the fundamental group. This fact is even easier tosee in the polygon model, where this path exactly traces out the boundary of thepolygon and can therefore be contracted to a point in the interior of the polygon.Using the Seifert-van Kampen theorem, one can show that this relation is a definingrelation for the fundamental group (cf. [F, Prop. 17.6, p. 242]).For the surfaces Σ g,n with boundary, additional generators are needed for the funda-mental group, which we denote by δ , . . . , δ n . Starting at the base point, δ j circlesaround the j -th boundary component. Together δ , . . . , δ n form a so-called bouquet11f circles. In the polygon model, δ j arises by mapping the concatenation of ξ j , ρ j ,and ξ − j to the quotient. We therefore see that in this case, α β α − β − α β α − β − · · · α g β g α − g β − g δ · · · δ n is homotopic to the constant path based at x , and again this yields a defining relationfor the fundamental group (cf. [AS, Chap. I, No. 43B, p. 100]). Because it is possibleto solve in this relation for the last of the new generators, this implies that, in thepresence of boundary components, the fundamental group of Σ g,n is a free group on2 g + n − δ j . An important object associated with the surface Σ g,n is its mapping class group Γ g,n .There are several variants for the definition of this group in the literature; we willnow explain which one is used in this article and how it compares to other variants.By the boundary invariance theorem (cf. [D, Chap. IV, Prop. 3.9, p. 61]), a diffeo-morphism of Σ g,n will necessarily map a boundary component to a boundary com-ponent, but we require that the diffeomorphisms we consider also map the markedpoint on each boundary component to the corresponding marked point on the otherboundary component. The group Diffeo + (Σ g,n ) of orientation-preserving diffeomor-phisms of Σ g,n that permute the marked points becomes a topological group whenendowed with the compact-open topology (cf. [A, Thm. 4, p. 598]). By the fun-damental property of the compact-open topology (cf. [Q, Satz 14.17, p. 167]), thepath-component Diffeo +0 (Σ g,n ) of the identity mapping consists precisely of thoseorientation-preserving diffeomorphisms that are isotopic to the identity. It is a nor-mal subgroup. Continuity implies that the elements of Diffeo +0 (Σ g,n ) cannot permutethe marked points, but rather need to preserve them individually. We define themapping class group as the corresponding quotient group: Definition 1.1.
The mapping class group Γ g,n of Σ g,n is the quotient groupΓ g,n := Diffeo + (Σ g,n ) / Diffeo +0 (Σ g,n )In accordance with our notation for surfaces in Paragraph 1.1, we will also write Γ g if n = 0. The mapping class of a diffeomorphism ψ ∈ Diffeo + (Σ g,n ) will be denotedby [ ψ ]. 12n view of this definition, two diffeomorphisms represent the same element in themapping class group if and only if they are contained in the same path-componentof Diffeo + (Σ g,n ); i.e., if and only if they are isotopic. Because the path connectingthem is contained in Diffeo + (Σ g,n ), the corresponding isotopy must permute themarked points at each time of the deformation process. By continuity, this is onlypossible if all diffeomorphisms occurring in this process, and in particular those at thebeginning and the end, permute the marked points in the same way. By enumeratingthe marked points, we can therefore obtain a group homomorphism p : Γ g,n → S n to the symmetric group S n . We call the kernel of p the pure mapping class group anddenote it by PΓ g,n . An element in the pure mapping class group can be representedby a diffeomorphism that restricts to the identity on the entire boundary, not onlyon the marked points. As explained in [FM, Sec. 1.4, p. 42], the mapping class groupcan alternatively be defined by using homeomorphisms instead of diffeomorphisms.For a subset U of Σ g,n , we also consider the subgroup of Diffeo + (Σ g,n ) consisting ofthose diffeomorphisms that restrict to the identity on U . If we divide it by its pathcomponent of the identity mapping, the arising quotient group Γ g,n ( U ) is called therelative mapping class group modulo U . It comes with a canonical group homomor-phism F U : Γ g,n ( U ) → Γ g,n called the forgetful map, because it arises by forgetting the information about therestriction to U . This homomorphism is not necessarily injective, because it mighthappen that a diffeomorphism is isotopic to the identity, although the diffeomor-phisms that occur during this deformation process cannot be chosen so that theyrestrict to the identity on U . If U = { u } consists of a single point u , we will alsowrite Γ g,n ( u ) and F u instead of Γ g,n ( { u } ) and F { u } .The definition of mapping class groups used here is the one from [L1, Sec. 4, p. 485].Often, other definitions are used, for example in [FM, Sec. 2.1, p. 44] or [Ko, Sec. 1,p. 101]. In these definitions, there are additional marked points in the interior, noton the boundary like in our definition. These points are frequently called punctures.In contrast to our definition, both the definition in [FM] and the definition in [Ko]require that the diffeomorphisms restrict to the identity on the boundary. In [FM],the punctures in the interior may be permuted by a diffeomorphism, while they arerequired to be preserved individually in [Ko].13 .4 Dehn twists An annulus can be defined as the Cartesian product A := S × [0 , π ] of the unitcircle S := { z ∈ C | | z | = 1 } and the interval [0 , π ] ⊂ R . A is an orientable2-manifold with boundary, which is diffeomorphic to Σ , and therefore sometimescalled a binion. We single out one of the two possible orientations by requiring thatin the tangent space of A at the point (1 , π ) the tangent vector v to the curve t (1 , π + t ) and the tangent vector v to the curve t ( e it , π ) form a positivelyoriented basis v , v . Note that both curves start at the specified point for theparameter value t = 0. On A , we define the twist map d : A → A, ( z, t ) ( ze it , t )Now suppose that γ : S → Σ g,n is a simple closed curve that does not intersect theboundary. It follows from the existence of tubular neighborhoods (cf. [H, Chap. 4,Thm. 5.2, p. 110]) that there is an orientation-preserving embedding φ : A → Σ g,n with the property that φ ( z,
0) = γ ( z ). Definition 1.2.
We define d γ : Σ g,n → Σ g,n as the map p ( ( φ ◦ d ◦ φ − )( p ) : p ∈ φ ( A ) p : p / ∈ φ ( A )Note that this map is a homeomorphism, not a diffeomorphism. But as we mentionedabove, the mapping class group can also be defined using homeomorphisms, so that d γ determines an element [ d γ ] in the mapping class group, called the Dehn twist along γ .Since d ( z,
0) = ( z, d γ ◦ γ = γ .It follows from the isotopy of tubular neighborhoods (cf. [H, Chap. 4, Thm. 5.3,p. 112]) that the mapping class of d γ does not depend on φ . Furthermore, any curveisotopic to γ yields the same Dehn twist in the mapping class group. Further detailson Dehn twists and their geometric meaning can be found in [FM, Sec. 3.1, p. 64ff]. Itis important to note that in some references the other orientation for the annulus A isused in the definition of a Dehn twist, for example in [L2, Par. 2.2, p. 316f] and [Ko,Sec. 2, p. 102]. Dehn twists defined using the other orientation are the inverses ofthe Dehn twists as they are defined here.It is almost immediate from this definition that non-intersecting curves γ and γ give rise to commuting Dehn-twists d γ and d γ . This and other basic properties ofDehn-twists are discussed in [FM, Sec. 3.2, p. 73ff].14e will introduce some special notation for a number of Dehn twists along certaincurves. The first of these are the following: In the construction of Σ g,n from Σ g ,the removal of the j -th open disk from Σ g leaves a connected component of theboundary that is diffeomorphic to the unit circle S . In the polygon model discussedin Paragraph 1.1, this curve is parametrized by ρ j . By moving this curve slightlyto the interior, for example by using a collar (cf. [H, Chap. 4, Thm. 6.1, p. 113]),we obtain a simple closed curve ∂ j that is freely homotopic to ρ j , and therefore alsoto δ j , but does not intersect the boundary. We will use the notation d j for the Dehntwist along ∂ j . As explained in [FM, Par. 4.2.5, p. 102], d j is contained in the centerof the mapping class group. Besides Dehn twists, the second type of elements of the mapping class group that wewill need are the braidings of the boundary components. To define them, we considerthe surface B in R that consists of the closed unit disk from which two open diskswith radius 1 / / ,
0) and ( − / ,
0) on the x -axis, respectively, havebeen removed. B is an orientable 2-manifold with boundary, which is sometimescalled a trinion, as it is diffeomorphic to Σ , . We orient B by requiring that thecanonical basis at the origin is positively oriented. On the boundary components,i.e., the unit circle and the two smaller circles, we choose the base points (0 , / , / − / , / b : B → B , which we call thebraiding map, that is the identity on the unit circle, interchanges the two smallercircles as well as their base points while preserving their orientations, and transformsthe line segments connecting the base point of the unit circle with the base points ofthe smaller circles as indicated in the picture b (cf. also [T, Fig. 2.9, p. 254]). As also stated in [T, loc. cit.], these conditionsdetermine b up to isotopy. It should be noted that the braiding map we define here15s the inverse of the braiding map in [T, Sec. V.2.5, p. 251], but coincides with themap used in [FM, Par. 5.1.1, p. 119].We will now use the braiding map to define the braiding of two boundary compo-nents of Σ g,n . It should be noted that it is not possible to define such a braiding ifthe boundary components are distributed over the surface in a somewhat arbitraryfashion, as in our description of the surface in terms of the connected sum model,as this definition depends not only on these two boundary components alone, butalso on their position relative to the other boundary components. Rather, it is nec-essary that the boundary components are not only numbered, but in fact arrangedin a certain order, like in the polygon model described in Paragraph 1.1. We usethe standard arrangement of the boundary components in this model as follows: Fortwo indices satisfying 1 ≤ i < j ≤ n , we choose an orientation-preserving embedding ϕ i,j : B → Σ g,n of surfaces that maps the circle with center ( − / ,
0) to the i -thboundary component of Σ g,n and the circle with center (1 / ,
0) to the j -th boundarycomponent of Σ g,n , mapping base points to base points. The unit circle is mappedto a smooth curve γ i,j on Σ g,n that is defined as follows: Starting at x , we followa path slightly right from ξ i until we reach the curve ∂ i , which we follow until weneed to return along ξ i , which we now do on its left side, which is the right side forthe reversed orientation of ξ i . Shortly before reaching x , however, we turn right andfollow a path slightly right from ξ j until we again reach ∂ j , which we follow in thesame way as before until we need to return along ξ j , which we now do on its left sideuntil we reach the base point x . The curve γ i,j is illustrated in the following picture: x i l j n . . . . . . . . . . . . γ i,j We now define the braiding of the i -th and the j -th boundary component in the sameway as we defined Dehn twists in Paragraph 1.4: Definition 1.3.
We define b i,j : Σ g,n → Σ g,n as the map p ( ( ϕ i,j ◦ b ◦ ϕ − i,j )( p ) : p ∈ ϕ i,j ( B ) p : p / ∈ ϕ i,j ( B )16gain, this map is a homeomorphism, not a diffeomorphism, but as for Dehn twists,it nonetheless determines an element [ b i,j ] in the mapping class group, called thebraiding of the i -th and the j -th boundary component. It is obvious that under theprojection p : Γ g,n → S n to the symmetric group introduced in Paragraph 1.3, [ b i,j ]maps to the transposition of i and j .We note that a mapping class very similar to [ b i,j ], using punctures instead of bound-ary components, is called a half-twist in [FM, Sec. 9.4, p. 255]. There are severalrelations between half-twists and Dehn twists; one is explained right there, anotherone will be discussed in Paragraph 1.8. During certain steps of our argument, it will be necessary to cut out small disksfrom our surface or at least to avoid certain points. For a Dehn twist, it makes adifference how we avoid this point. Suppose that a homotopy class is represented bya smooth simple closed curve γ that starts and ends at a point x on our surface. Wechoose a neighborhood U of x that is diffeomorphic to the open unit disk in R viaan orientation-preserving diffeomorphism φ that sends x to the origin. As before,we assume that the unit disk has the orientation in which the canonical basis is apositively oriented basis for the tangent space at the origin. By perhaps passingto a smaller neighborhood, we can assume that γ passes through the unit disk asindicated on the left of the picture below. Because φ is orientation-preserving, it ismeaningful to talk about pushing off γ to the left or to the right, as indicated in themiddle and right picture below: x x xγ γ ′ γ ′′ In this way, we obtain two curves γ ′ and γ ′′ whose free homotopy classes in Σ g,n \ { x } do not depend on our choices. Both curves avoid a small neighborhood of the point x d γ ′ ] = [ d γ ′′ ] ∈ Γ g,n ( x ), we clearly have F x ([ d γ ′ ]) = F x ([ d γ ′′ ]) = [ d γ ]for their images under the forgetful map F x : Γ g,n ( x ) → Γ g,n .As explained in [FM, Par. 8.2.7, p. 235], we can choose for x the base point of thefundamental group to get an action of Γ g,n ( x ) on π (Σ g,n , x ). In particular, the Dehntwists d γ ′ and d γ ′′ act on π (Σ g,n , x ). We now give explicit formulas for this actionin the case where the homotopy class acted upon can be represented by a curve κ that does not intersect γ , except at the base point x . It should be noted that, inthe present situation, the curves κ and γ , and therefore also γ ′ and γ ′′ , are oriented,although, according to our construction, Dehn twists like d γ ′ do not depend on theorientation of γ ′ . In each homotopy class, we can choose smooth representatives(cf. [FM, Par. 1.2.2, p. 26]), which by definition of smoothness have nowhere avanishing tangent vector. If we in addition can choose these representatives so thattheir tangent vectors ˙ κ x and ˙ γ x at x are linearly independent, we say that κ and γ intersect transversely at x . If ˙ κ x and ˙ γ x form a positively oriented basis of the tangentspace at x , we define the algebraic intersection number i A ( κ, γ ) as +1, and if theyform a negatively oriented basis of the tangent space at x , we define the algebraicintersection number i A ( κ, γ ) as −
1. This definition can be extended to curves withfinitely many intersection points by defining i A ( κ, γ ) as the sum of the so-determinedsigns at all intersection points. This definition is illustrated in the picture i A ( κ, γ ) = +1 i A ( κ, γ ) = − γ κγ κ κ γγ κ in which the standard orientation of the plane has been used, i.e., the orientationin which the canonical basis is positive and the normal vector points towards thereader.The action of the two Dehn twists on the homotopy class [ κ ] of κ is summarized inthe following table: 18 A ( κ, γ ) = 1 i A ( κ, γ ) = − d γ ′ ([ κ ]) [ κγ ] [ γ − κ ] d γ ′′ ([ κ ]) [ γκ ] [ κγ − ]Here, expressions like [ κγ ] = [ κ ][ γ ] mean the product in the fundamental group: Theconcatenation of first κ , followed by γ . To understand the table, let us consider howthe entry for d γ ′ ([ κ ]) in the case i A ( κ, γ ) = 1 comes about: The curve starts at x with κ . In view of how γ ′ has been pushed off γ and how the curves intersect, κ is moving away from γ ′ and traces out κ almost completely before approaching thebase point x upon returning. But before reaching x , it encounters γ ′ and at thispoint is forced to turn left in view of our definition of a Dehn twist. But because i A ( κ, γ ) = 1, turning left is in this case the same as following γ in the direction ofits orientation. The other entries arise from analogous considerations.It is of course possible that we cannot choose representatives whose tangent vectors ˙ κ x and ˙ γ x at x are linearly independent. In this case, the two curves do not intersecttransversely, and by pushing off γ either to the left or to the right, we can avoidintersection completely. Depending on the orientation of γ , the situation is describedby one of the following two pictures: κ γγ ′ γ ′′ κ γγ ′′ γ ′ In the first case, the actions of the Dehn twists are given by d γ ′ ([ κ ]) = [ γ − κγ ] and d γ ′′ ([ κ ]) = [ κ ] , while in the second case they are given by d γ ′ ([ κ ]) = [ κ ] and d γ ′′ ([ κ ]) = [ γκγ − ] . In both cases, these formulas are valid regardless of the orientation of κ .19 .7 Dehn twists for special curves In Paragraph 1.4, we have already introduced the notation d j for the Dehn twistdetermined by the curve ∂ j . We now also assign names to the Dehn twists derivedfrom the other curves stemming from the polygon model discussed in Paragraph 1.1and define t i := d α i , r i := d β i , n i := d µ i . Because the curves α i and β i intersect exactly once, the corresponding Dehn twistssatisfy the braid relation [ t i r i t i ] = [ r i t i r i ] (cf. [FM, Par. 3.5.1, Prop. 3.11, p. 77]). Forthe elements s i := t − i r − i t − i , the braid relation implies that [ s i t i s − i ] = [ r i ].These special Dehn twists are important in view of the following result, which isknown as the Dehn-Lickorish theorem: Theorem 1.4. If g ≥
1, the Dehn twists [ t i ], [ r i ], and [ n l ], for i = 1 , . . . , g and l = 1 , . . . , g −
1, which are called the Lickorish generators, generate the mappingclass groups Γ g = Γ g, and Γ g, .A proof of this result can be found in [FM, Par. 4.4.4, p. 113ff]. We note that inthe case g = 0, the groups Γ g and Γ g, are trivial anyway as a consequence of theAlexander lemma (cf. [FM, Par. 2.2.1, p. 47ff]). The case g = 1 will be discussed inmore detail in Paragraph 1.12.In the case where there is more than one boundary component, we need additionalDehn twists around the curves ζ l , which we denote by z l for l = 1 , . . . , n −
1. Thecurve ζ l separates the l -th and l + 1-st boundary component and connects to the g -thhandle as shown in the picture for the case g = 3 and n = 2: X X ζ Theorem 1.5. If g ≥ n ≥
2, the mapping classes [ t i ], [ r i ], [ n j ], [ d k ], [ z l ],and [ b l,l +1 ], for i = 1 , . . . , g , j = 1 , . . . , g − k = 1 , . . . , n , and l = 1 , . . . , n − g,n .We can assume without loss of generality that the curve ∂ j does not only avoidthe n boundary components, but also the base point x . Then the corresponding Dehntwist d j is contained in Γ g,n ( x ) and therefore can act on π (Σ g,n , x ), as explained inParagraph 1.6. Since any homotopy class in π (Σ g,n , x ) can be represented by a curvethat does not intersect ∂ j , this action is trivial. However, the curves α i , β i , and µ i start and end at the base point x , and as explained in Paragraph 1.6, it makes adifference for the action on π (Σ g,n , x ) whether the curves meet before or after thebase point x . We will now make specific choices for these curves. For t i = d α i , wechoose t ′ i := d α ′ i , for r i = d β i , we choose r ′′ i := d β ′′ i , and for n i = d µ i , we choose n ′′ i := d µ ′′ i . For s i , we mix the choices and define s ′ i := t ′− i r ′′− i t ′− i . These maps acton the generators of the fundamental group as follows: Proposition 1.6.
Suppose that i ≤ g .1. For j ≤ n , the Dehn twists t ′ i ([ δ j ]) = r ′′ i ([ δ j ]) = [ δ j ]. If i = g , we also have n ′′ i ([ δ j ]) = [ δ j ].2. We have t ′ i ([ β i ]) = [ β i α i ] and t ′ i ([ α i ]) = [ α i ]. For j = i , we have t ′ i ([ β j ]) = [ β j ]and t ′ i ([ α j ]) = [ α j ].3. We have r ′′ i ([ α i ]) = [ α i β − i ] and r ′′ i ([ β i ]) = [ β i ]. For j = i , we have r ′′ i ([ α j ]) = [ α j ]and r ′′ i ([ β j ]) = [ β j ].4. For i = 1 , . . . , g −
1, we have n ′′ i ([ α i ]) = [ α i ] as well as n ′′ i ([ β i ]) = [ µ i β i ] , n ′′ i ([ α i +1 ]) = [ µ i α i +1 µ − i ] , n ′′ i ([ β i +1 ]) = [ β i +1 µ − i ] . For j = i and j = i + 1, we have n ′′ i ([ α j ]) = [ α j ] and n ′′ i ([ β j ]) = [ β j ].5. We have s ′ i ([ α i ]) = [ α i β i α − i ] and s ′ i ([ β i ]) = [ α − i ]. For j = i , we have that s ′ i ([ α j ]) = [ α j ] and s ′ i ([ β j ]) = [ β j ]. 21 roof. (1) We first note that α i and β i are oriented so that α ′ i and β ′′ i arise bypushing α i and β i off the base point x in the direction of the i -th attached torus,so that we could cut the i -th attached torus off again in such a way that the basepoint x on the one hand and α ′ i and β ′′ i on the other hand would lie on differentconnected components. If j = i , this implies that the curves α ′ i and β ′′ i do notintersect the curves α j and β j . This implies that t ′ i ([ β j ]) = [ β j ] and t ′ i ([ α j ]) = [ α j ]as well as r ′′ i ([ α j ]) = [ α j ] and r ′′ i ([ β j ]) = [ β j ]. For the same reason, t ′ i and r ′′ i preservethe homotopy class of δ j .(2) We have i A ( β i , α i ) = − i A ( α i , β i ) = 1, so that it follows from the table in Para-graph 1.6 that t ′ i ([ β i ]) = [ β i α i ] and r ′′ i ([ α i ]) = [ α i β − i ]. Clearly, we have t ′ i ([ α i ]) = [ α i ]and r ′′ i ([ β i ]) = [ β i ]. Therefore, the second and the third assertions are now completelyproved.(3) For the fourth assertion, we see from the discussion in Paragraph 1.2 that µ i is oriented in such a way that µ ′′ i arises by pushing µ i off the base point x in thedirection of the i -th and the i + 1-st attached tori, so that we could cut these two torioff again in such a way that the base point x and µ ′′ i would lie on different connectedcomponents. For j = i and j = i + 1, this implies that µ ′′ i does not intersect α j or β j , so that n ′′ i ([ α j ]) = [ α j ] and n ′′ i ([ β j ]) = [ β j ]. This discussion also shows that µ ′′ i does not intersect α i , so that n ′′ i ([ α i ]) = [ α i ]. On the other hand, µ i intersects β i and β i +1 exactly once, the intersection is transversal, and the intersection numbersare i A ( β i , µ i ) = 1 and i A ( β i +1 , µ i ) = −
1, respectively. It therefore follows from thetable in Paragraph 1.6 that n ′′ i ([ β i ]) = [ µ i β i ] and n ′′ i ([ β i +1 ]) = [ β i +1 µ − i ] . In contrast, the curves µ i and α i +1 do not intersect transversally, but rather as inthe second picture in Paragraph 1.6, so that n ′′ i ([ α i +1 ]) = [ µ i α i +1 µ − i ]. Since i = g ,the curve µ ′′ i does not intersect δ j , so that n ′′ i ([ δ j ]) = [ δ j ].(4) It remains to show the fifth assertion. For j = i , the claim follows easily fromthe second and the third assertion. For the case j = i , we argue as follows: Byinverting part of the third assertion, we have r ′′− i ([ α i β − i ]) = [ α i ], which impliesthat r ′′− i ([ α i ])[ β i ] − = r ′′− i ([ α i ]) r ′′− i ([ β i ] − ) = [ α i ]. This in turn yields s ′ i ([ α i ]) = ( t ′− i r ′′− i t ′− i )([ α i ]) = ( t ′− i r ′′− i )([ α i ]) = t ′− i ([ α i β i ]) = t ′− i ([ α i ]) t ′− i ([ β i ]) . A second inversion yields t ′− i ([ β i ]) = [ β i α − i ], so that s ′ i ([ α i ]) = [ α i ] t ′− i ([ β i ]) = [ α i β i α − i ]as asserted. The formula for s ′ i ([ β i ]) follows from a similar computation.22 .8 Dehn twists related to two boundary components According to our conventions in Paragraph 1.4, the curve δ i δ j , for i < j , cannotbe used to define a Dehn twist for two reasons: On the one hand, it intersects theboundary, and on the other hand, it is not a simple closed curve. The first problemcan be addressed by moving it slightly to the interior with the help of a collar, in thesame way as in our treatment of ∂ j at the end of Paragraph 1.4. As a consequence ofthe orientation of ρ i and ρ j , a movement to the interior is a movement to the rightof the given curve.The second problem arises not only from the fact that the curve returns to the basepoint x at the time of concatenation, when δ i ends and δ j begins, but also fromthe fact that the paths ξ i and ξ j are both traced out twice, in opposite directions.To address this problem, we consider the curve γ i,j defined in Paragraph 1.5, whichdoes not intersect the boundary, starts and ends in x , and represents the samerelative homotopy class as δ i δ j in the fundamental group π (Σ g,n , x ). Passing to γ ′ i,j as indicated in Paragraph 1.6, we in addition avoid the base point x . In contrastto δ i δ j , the modification γ ′ i,j is a simple closed curve that can be used to define aDehn twist. This curve is shown in the picture that appears in the proof below. Wedenote the Dehn twist along γ ′ i,j by d i,j .As explained in [FM, Par. 5.1.1, p. 118f] or [PS, §
7, p. 63], the Dehn twist d i,j isrelated to the braiding introduced in Paragraph 1.5: If we define the double braidingas q i,j := b j,i b i,j , it is related to our Dehn twist via the formula[ q i,j ] = [ d i,j d − i d − j ] . This formula holds because the Dehn twist d i,j not only interchanges the i -th andthe j -th boundary components twice, but also twists these boundary componentsthemselves, while the double braiding moves the boundary components in a parallelfashion, without introducing a twist.The action of q i,j on the generators of the fundamental group is given as follows: Proposition 1.7.
Suppose that 1 ≤ i < j ≤ n .1. We have q i,j ([ δ i ]) = [( δ i δ j ) − δ i ( δ i δ j )] = [ δ − j δ i δ j ] and q i,j ([ δ j ]) = [( δ i δ j ) − δ j ( δ i δ j )] .
2. If l = i and l = j , we have q i,j ([ δ l ]) = [ δ l ].3. The double braiding q i,j acts trivially on the homotopy classes of α , . . . , α g and β , . . . , β g in π (Σ g,n , x ). 23 roof. We have already seen in Paragraph 1.7 that the Dehn twists d i and d j acttrivially, so that the action of q i,j coincides with the action of d i,j . If l = i or l = j ,then γ i,j and δ l intersect only in x , and the intersection is described by the firstpicture at the end of Paragraph 1.6. Therefore, the discussion of the first case thereyields that d i,j ([ δ l ]) = [ γ − i,j δ l γ i,j ]. Because γ i,j is homotopic to δ i δ j relative to thebase point x , this proves the first assertion.If l < i or l > j , the generator δ l also intersects γ i,j only in x , but now the intersectionis described by the second picture at the end of Paragraph 1.6, so that γ ′ i,j does notintersect δ l , which implies the second assertion in these cases. The case i < l < j ismore complicated and is illustrated by the picture x i l j n . . . . . . . . . . . . γ ′ i,j δ i δ l δ j in which we have replaced the curves δ i , δ l , and δ j by slight modifications within theirrelative homotopy class. It shows that the curve d i,j ( δ l ) comes about as follows: Itstarts at x with δ l , but then soon meets γ ′ i,j . At that point, it turns left and follows γ ′ i,j against its orientation. Upon returning, it follows δ l again, but soon encounters γ ′ i,j a second time. Again, it turns left, but now follows γ ′ i,j in the direction of itsorientation. Up to relative homotopy, these two encounters with γ ′ i,j cancel eachother. While continuing along δ l , the curve has two similar encounters with γ ′ i,j ,which again cancel each other. In summary, the curve is homotopic to δ l , whichfinishes the proof of the second assertion.The curves α i and β i intersect γ i,j again only in x , and the intersection is againdescribed by the second picture at the end of Paragraph 1.6, so that γ ′ i,j does notintersect them at all. This yields the third assertion. In the process of defining Σ g,n , we have removed n open disks from Σ g . If we assumethat our surface is realized via the polygon model, the boundary components come24ith a natural enumeration, and the j -th boundary component is parametrized by ρ j .In the place of the j -th removed disk, we now glue a punctured disk back, i.e., a diskwhose interior contains a marked point denoted by y . If a homeomorphism of Σ g,n restricts to the identity on the image of ρ j , we can extend it to Σ g,n − by requiringthat the extension restricts to the identity on the newly inserted punctured disk. Inthis way, we obtain a group homomorphism C j : Γ g,n ( ρ j ) → Γ g,n − ( y )which we call the j -th capping homomorphism. Here, we have not only writ-ten Γ g,n − ( y ) for Γ g,n − ( { y } ), as indicated in Paragraph 1.3, but also briefly Γ g,n ( ρ j )for Γ g,n (Im( ρ j )).It is a consequence of the Alexander lemma (cf. [FM, Sec. 2.2, Lem. 2.1, p. 47f])that d j is contained in the kernel of C j . The following result, which can be foundin [FM, Sec. 3.6, Prop. 3.19, p. 85] or [Ko, Sec. 3, p. 104f], states that d j in factgenerates the kernel: Proposition 1.8.
The sequence1 −→ h d j i −→ Γ g,n ( ρ j ) C j −→ Γ g,n − ( y ) −→ d j ∈ Γ g,n ( ρ j ) is a nontrivial mapping class, in which case h d j i isisomorphic to Z . The only exception is the case g = 0 and n = 1: Here, it is easyto see that Γ , ( y ) is trivial (cf. [FM, p. 49]), and Γ , is trivial by the Alexanderlemma.Clearly, we can compose C j with the forgetful map F y : Γ g,n − ( y ) → Γ g,n − to obtaina homomorphism D j := F y ◦ C j from Γ g,n ( ρ j ) to Γ g,n − that caps off the boundarycomponent not with a punctured, but rather with a full disk. The effect of theadditional mapping F y will be studied in the next paragraph. Because we can always modify a diffeomorphism by isotopy so that it fixes a point,the forgetful map F x : Γ g,n ( x ) → Γ g,n
25s surjective. If φ ∈ Diffeo + (Σ g,n ) not only fixes x , but in addition represents amapping class in the kernel of F x , it can be connected to the identity in Diffeo + (Σ g,n ),i.e., there is a path [0 , → Diffeo + (Σ g,n ) , t φ t with φ = id and φ = φ . Since φ t is not required to fix the point x , we get a closedpath γ : [0 , → Σ g,n , t φ t ( x )based at x , which represents a homotopy class in the fundamental group π (Σ g,n , x ).Using the long exact sequence of a fibration (cf. [Ro1, Thm. 11.48, p. 358]), onecan show that the homotopy class of γ determines the mapping class of φ . Moreprecisely, there is group antihomomorphism P x : π (Σ g,n , x ) → Γ g,n ( x )that maps the class of γ to the class of φ . This map is called the point-pushing map,or briefly the pushing map. The arising exact sequence π (Σ g,n , x ) P x −→ Γ g,n ( x ) F x −→ Γ g,n −→ , is called the Birman sequence. If the Euler characteristic of Σ g,n is strictly negative,the Birman sequence is in fact short exact, i.e., P x is injective. The Birman sequenceis discussed in greater detail in [FM, Sec. 4.2, p. 96ff].As we have explained in Paragraph 1.6, the mapping class group Γ g,n ( x ) acts on thefundamental group π (Σ g,n , x ), and so in particular P x ([ γ ]) acts on the fundamentalgroup. If P x ([ γ ]) = [ φ ] and [ β ] ∈ π (Σ g,n , x ), it can be shown that φ ( β ), i.e., theclosed path t φ ( β ( t )), is homotopic to γ − βγ relative to the basepoint x , so that P x ([ γ ])([ β ]) = [ γ − βγ ] . In other words, P x ([ γ ]) acts on the fundamental group by an inner automorphism.The Birman sequence therefore implies that we have a homomorphism from Γ g,n toOut( π (Σ g,n , x )) that makes the following diagram commutative:Γ g,n ( x ) Aut( π (Σ g,n , x ))Γ g,n Out( π (Σ g,n , x )) F x Further aspects of this diagram are discussed in [FM, Par. 8.2.7, p. 235].26he action of the mapping class group Γ g,n ( x ) on the fundamental group π (Σ g,n , x )is compatible with the pushing map in another way: Clearly, Γ g,n ( x ) acts on thekernel of the group homomorphism F x by conjugation. This action is compatiblewith the pushing map in the sense that P x ([ ψ ( γ )]) = [ ψ ] P x ([ γ ])[ ψ ] − for [ ψ ] ∈ Γ g,n ( x ) (cf. [FM, Par. 4.2.2, Fact 4.8, p. 99]). This relation is a directconsequence of our description above.If a homotopy class in the fundamental group can be represented by a simple closedcurve γ , there is a formula for its image under the pushing map in terms of Dehntwists: Using the notation from Paragraph 1.6, we have P x ([ γ ]) = [ d γ ′ d − γ ′′ ](cf. [FM, Par. 4.2.2, Fact 4.7, p. 99]). If we think of a closed curve as being defined on the interval [0 , π (Σ g,n , x ) → H (Σ g,n , Z )from the fundamental group to the first singular homology group that is called the(first) Hurewicz homomorphism. Hurewicz’ theorem asserts in this situation thatthe Hurewicz homomorphism is surjective and induces an isomorphism from thecommutator factor group of the fundamental group to the first singular homologygroup (cf. [Ro1, Thm. 4.29, p. 83]).It follows from its universal property that the commutator factor group of a finitelypresented group is presented by the same relations, but now understood as a pre-sentation of an abelian group. Therefore, Hurewicz’ theorem and the presenta-tions of the fundamental group discussed in Paragraph 1.2 together imply that H (Σ g , Z ) is a free abelian group of rank 2 g with the homology classes of thecurves α , β , α , β , . . . , α g , β g as a basis. In the presence of boundary components,Hurewicz’ theorem yields that H (Σ g,n , Z ) is generated by the homology classes of thecurves α , β , α , β , . . . , α g , β g , δ , . . . , δ n subject to the relation that the homology27lasses of the curves δ , . . . , δ n sum up to zero. By solving for one of these generators,we see that the first homology group is free abelian of rank 2 g + n −
1. We note thatthe homology class of δ j is equal to the homology class of ∂ j , because these curvesare freely homotopic.The algebraic intersection number introduced in Paragraph 1.6 depends only on thehomology classes of the curves involved. It therefore defines a symplectic form onthe abelian group H (Σ g,n , Z ). For the generators, we have on the one hand i A ( β i , α j ) = − i A ( α j , β i ) = δ i,j and i A ( α i , α j ) = 0 = i A ( β i , β j )(cf. [FM, Par. 6.1.2, p. 165]), while on the other hand we have i A ( δ j , γ ) = 0 for everyclosed curve γ , because its free homotopy class has a representative that does notintersect δ j .Because the outer automorphism group of any group evidently acts on the corre-sponding commutator factor group, Hurewicz’ theorem implies that the commutativesquare obtained in Paragraph 1.10 can be enlarged to the diagramΓ g,n ( x ) Aut( π (Σ g,n , x ))Γ g,n Out( π (Σ g,n , x ))Γ g,n Aut( H (Σ g,n , Z )) F x It is immediate from the construction that the action of Γ g,n on H (Σ g,n ) preservesthe intersection form and therefore takes values in the symplectic group. The discus-sion in Paragraph 1.7 and Paragraph 1.8 also implies how the special mapping classesintroduced there act on the first homology group. The mapping classes [ d j ] and themapping classes [ q i,j ] act as the identity on the entire first homology group. FromProposition 1.6, we see that the mapping classes [ t i ], [ r i ], and [ s i ] preserve the twosubgroups generated by the homology classes of the curves α , β , α , β , . . . , α g , β g on the one hand and the curves δ , . . . , δ n on the other hand, and act as the identityon the second subgroup. As an abelian group, the first homology group is the directsum of these two subgroups, and the first subgroup is free of rank 2 g on the givengenerating set. We can therefore represent the action on the first subgroup by ma-trices in Sp(2 g, Z ). It follows from Proposition 1.6 that these matrix representations28re block-diagonal with g blocks of 2 × t i , r i , and s i , the blockscorresponding to the generators α i and β i are t := (cid:18) (cid:19) r := (cid:18) − (cid:19) s := (cid:18) −
11 0 (cid:19) respectively, while the other blocks are 2 × i = 1 , . . . , g − n i is also block-diagonal, but now contains a 4 × n := −
10 1 0 00 − corresponding to the homology classes of the curves α i , β i , α i +1 , and β i +1 . The surface Σ is a torus. Because the defining relation of the fundamental groupdiscussed in Paragraph 1.2 in this case states that the relative homotopy class of α β α − β − is trivial, the fundamental group is abelian here, and therefore Hurewicz’theorem mentioned in Paragraph 1.11 yields that it is isomorphic to the first sin-gular homology group. According to the Dehn-Lickorish theorem 1.4, the Dehntwists [ t ] and [ r ], or alternatively [ t ] and [ s ] = [ t − r − t − ], generate the mappingclass group Γ . As we have just seen, the actions of these generators on the firsthomology group are, respectively, represented by the matrices t = (cid:18) (cid:19) r = (cid:18) − (cid:19) s = (cid:18) −
11 0 (cid:19) with respect to the basis consisting of the singular homology classes of α and β .Because Sp(2 g, Z ) = SL(2 , Z ) if g = 1, the group homomorphism Γ g,n → Sp(2 g, Z )described in Paragraph 1.11 becomes in this situation a group homomorphismΓ → SL(2 , Z )that maps [ t ], [ r ], and [ s ] to t , r , and s , respectively. It is a classical fact that themodular group SL(2 , Z ) is generated by the matrices s and t and that the relations s = 1 and sts = t − st − that they satisfy are defining (cf. [KT, Thm. A.2, p. 312]).29hese relations imply that s is central, which can also easily be seen directly. Alter-natively, the modular group is generated by the elements r and t , and the definingrelations for s and t just stated translate into the defining relations trt = rtr and( rt ) = 1 for the generators r and t (cf. [SZ, Prop. 1.1, p. 7]). This implies immedi-ately that our group homomorphism Γ → SL(2 , Z ) is surjective, a fact that is alsoa special case of a more general result for mapping class groups (cf. [FM, Par. 6.3.2,Thm. 6.4, p. 170]). In the case of the torus, the homomorphism is even bijective(cf. [FM, Par. 2.2.4, Thm. 2.5, p. 53]), so that Γ ∼ = SL(2 , Z ).To discuss the mapping class group Γ , , we need to introduce the braid group B on three strands, which we define as the group generated by two generators r and t subject to the one defining relation rtr = trt . We will refer to this relation as thebraid relation. Geometrically, r can be interpreted as the interchange of the firsttwo strands, while t can be interpreted as the interchange of the last two strands(cf. [FM, Sec. 9.2, p. 246f]). In view of the second presentation of the modular groupgiven above, there is a surjective group homomorphism B → SL(2 , Z )that maps r to r and t to t . If we define s := ( trt ) − , then on the one hand s ismapped to s under our homomorphism, and on the other hand the braid relationis equivalent to the relation sts = t − st − for the generators s and t of B . Thebraid relation implies relatively easily that the element ( rt ) = ( rtr )( trt ) is centralin B . This yields, again in view of the second presentation of the modular group,not only that ( rt ) is contained in the kernel of this homomorphism, but that it evengenerates the kernel. We therefore have the short exact sequence1 −→ h ( rt ) i −→ B −→ SL(2 , Z ) −→ . We want to compare this homomorphism from B to SL(2 , Z ) with the homomor-phism D : Γ , → Γ that caps off the one boundary component with a full disk. Asdiscussed in Paragraph 1.9, D is the composition of C : Γ , → Γ , ( y ), which capsoff the boundary component with a punctured disk that contains a marked point y in its interior, and the forgetful map F y : Γ , ( y ) → Γ , = Γ . At this point, there is a small conceptual difficulty with the polygon model. In thepolygon model, the capped-off boundary component is parametrized by ρ , and the30oint y in its interior is different from the base point x that is the common imageof all the vertices of the polygon that are different from the start point and theend point of the edge labeled by ρ . To account for this difference, we choose adiffeomorphism φ : Σ , → Σ , that is isotopic to the identity and satisfies φ ( x ) = y .We then have the isomorphismΓ , ( x ) → Γ , ( y ) , [ ψ ] [ φ ◦ ψ ◦ φ − ]that satisfies F y ([ φ ◦ ψ ◦ φ − ]) = F x ([ ψ ]). Moreover, the map π (Σ , , x ) → π (Σ , , y ) , [ γ ] [ φ ◦ γ ]is an isomorphism of fundamental groups. Because the curves γ and φ ◦ γ arefreely homotopic, they map to the same singular homology class under the respectiveHurewicz maps. If γ is a smooth simple closed curve, we have[ d φ ◦ γ ] = [ φ ][ d γ ][ φ − ] = [ d γ ]in Γ . These considerations show that the diagramΓ , ( y ) Aut( π (Σ , , y )) Aut( H (Σ , , Z )) ∼ = SL(2 , Z )Γ , Aut( H (Σ , , Z )) ∼ = SL(2 , Z ) F y commutes. Although it does not prove this fact, this diagram suggests that themap Γ , ( y ) → Aut( H (Σ , , Z )) in the top row might be an isomorphism, and thisindeed turns out to be the case (cf. [FM, Par. 2.2.4, p. 54f]). Therefore the forgetfulmap F y must be bijective in our present case. For the kernel of the homomorphism D = F y ◦ C , we therefore have Ker( D ) = Ker( C ) = h d i by Proposition 1.8. Forthe Birman sequence π (Σ , y ) P y −→ Γ ( y ) F y −→ Γ −→ P y vanishes identi-cally, and is in particular not injective. This does not contradict the claims madethere, because the Euler characteristic χ (Σ ) = 0 is not strictly negative.If we compose the isomorphism Γ , ( y ) → Aut( H (Σ , , Z )) just discussed withthe capping map C : Γ , → Γ , ( y ), we obtain a homomorphism from Γ , to31ut( H (Σ , , Z )). This map can also be constructed in another way. As discussedin Paragraph 1.2, the fundamental group π (Σ , , x ) of a torus with one boundarycomponent is a free (nonabelian) group whose generators are the relative homotopyclasses [ α ] and [ β ]. But by Hurewicz’ theorem, we then have that the first homol-ogy group H (Σ , , Z ) is again free abelian with the homology classes of α and β asgenerators. This shows that the inclusion map Σ , → Σ induces an isomorphismbetween the first homology groups, so that also H (Σ , , Z ) ∼ = SL(2 , Z ) with respectto these generators. In view of our considerations above, the Dehn twists t and r in Γ , are then represented by the same matrices as the corresponding Dehn twistsin Γ .As we said at the beginning of Paragraph 1.7, the Dehn twists r and t satisfy thebraid relation, so that we obtain a group homomorphism B → Γ , that maps r to r and t to t . By the Dehn-Lickorish theorem 1.4, this homomorphismis surjective. Now the 2-chain relation (cf. [FM, Par. 4.4.1, p. 107f]) states that( r t ) = d (cf. [S, Par. 3.4.1, Fig. 141, p. 124] for an illustration). We thereforehave the commutative diagram1 h ( rt ) i B SL(2 , Z ) 11 h d i Γ , Γ ∼ = D in which the rightmost vertical map is the isomorphism described above. The five-lemma now implies that Γ , ∼ = B (cf. [FM, Par. 3.6.4, p. 87f]). In the preceding paragraph, we have introduced the braid group on three strands.More generally, there is, for n ≥
2, a braid group B n on n strands, which can bedefined as the group generated by elements σ , . . . , σ n − subject to the relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 i = 1 , . . . , n − σ i σ j = σ j σ i if j > i + 1. Geometrically, σ i can be interpretedas the interchange of the i -th and the i + 1-st strand (cf. [FM, Sec. 9.2, p. 246f]). Inthe case n = 3 considered in the preceding paragraph, we have r = σ and t = σ .From this definition, we see that there is a group homomorphism from the braidgroup B n to the symmetric group S n that maps σ i to the transposition of i and i + 1.Pulling the natural action of S n back along this homomorphism, we get an actionof B n on the set { , , . . . , n } that we denote by i σ.i . If we view Z n as the setof functions from { , , . . . , n } to Z , we can form the corresponding wreath product,which we denote by Z n ⋊ B n . If E i = ( e i , B n ) denotes the i -th canonical basisvector, considered as an element of the wreath product, and if we identify σ ∈ B n with (0 , σ ) ∈ Z n ⋊ B n , we have the commutation relation σE i = E σ.i σ . The wreathproduct is therefore generated by the elements σ , . . . , σ n − together with E , . . . , E n .Besides the defining relations of the braid group stated above, these generators satisfythe relations E i E j = E j E i as well as σ i E j = E i +1 σ i : j = iE i σ i : j = i + 1 E j σ i : j = i and j = i + 1It is not too complicated to show that these relations are defining.It turns out that the mapping class group Γ ,n is a quotient group of this wreathproduct. If we map σ i to b i,i +1 and E i to d i , we obtain a surjective group homo-morphism from Z n ⋊ B n to Γ ,n . However, the map is not bijective, because thegenerators b i,i +1 and d i satisfy additional relations: Besides the relations b i,i +1 b i +1 ,i +2 b i,i +1 = b i +1 ,i +2 b i,i +1 b i +1 ,i +2 for i = 1 , . . . , n − b i,i +1 b j,j +1 = b j,j +1 b i,i +1 if j > i + 1 for the braidings as wellas d i d j = d j d i and b i,i +1 d j = d i +1 b i,i +1 : j = i d i b i,i +1 : j = i + 1 d j b i,i +1 : j = i and j = i + 1for the Dehn twists and their interaction with the braidings, we have the additionalrelations b , b , · · · b n − ,n · · · b , b , = d − b , b , · · · b n − ,n ) n = d − d − · · · d − n . These relations all together are againdefining (cf. [L1, Par. 4.2, p. 487], see also [L2, Par. 3.1, p. 322f]).Clearly, we can embed the wreath product Z n − ⋊ B n − into the wreath product Z n ⋊ B n by sending σ i to σ i and E i to E i . By composing this injection with thesurjection just described, we get a group homomorphism from Z n − ⋊ B n − to Γ ,n . Itclearly takes values in the subgroup Γ ,n ( ρ n ) that fixes the last boundary componentpointwise. The fundamental fact about this map is the following: Proposition 1.9.
The group homomorphism Z n − ⋊ B n − → Γ ,n ( ρ n )that maps σ i to b i,i +1 and E i to d i is an isomorphism.This proposition follows from the fact that we can view a sphere with one boundarycomponent as a closed disk, say of radius 1. The mapping class group Γ ,n ( ρ n ) istherefore isomorphic to the mapping class group of a disk with n − Z n − ⋊ B n − in the indicated way (cf. [PS, §
7, p. 64], see also [FM, Par. 9.1.3, p. 243f]).
Let K be a perfect field and C be an essentially small abelian K -linear category.Here, C is called essentially small if there is a set, not a class, of objects with theproperty that every object in C is isomorphic to an object in this set. We assumethat C is finite in the sense of [EGNO, Def. 1.8.6, p. 9], i.e., that it has finite-dimensional spaces of morphisms, that every object has finite length, that it hasenough projectives, and that there are only finitely many isomorphism classes ofsimple objects. That it has enough projectives means that for every object X ,there is an epimorphism f : P → X from a projective object P (cf. [Mi, Sec. II.14,p. 70]). A standard, but not completely trivial argument shows that, under theassumption on finite length, the epimorphism f can be chosen to be essential, where34n epimorphism f : P → X is called essential if, given a morphism g : Y → P , wecan conclude that g is an epimorphism if f ◦ g is an epimorphism. An essentialepimorphism f : P → X from a projective object P is usually called a projectivecover of X ; it follows from the assumption on finite length that this definition isequivalent to the one given in [EGNO, Def. 1.6.6, p. 6].As explained in [DSS, Prop. 1.4, p. 3], an essentially small abelian K -linear categoryis finite if and only if it is equivalent to the category of finite-dimensional modulesover a finite-dimensional K -algebra (cf. also [EGNO, p. 9f]). We note that finitecategories are called bounded categories in [KL].We will assume that C is a strict tensor category in the sense of [Ka, Def. XI.2.1,p. 282], so that we have in particular a tensor product functor ⊗ from C × C to C and a unit object . We note that tensor categories are called monoidal categoriesin [EGNO, Def. 2.2.8, p. 25]; the term ‘tensor category’ is used there for a categorythat satisfies additional restrictions (cf. [EGNO, Def. 4.1.1, p. 65]), all of which willbe satisfied in our situation, as we start to discuss now: First, we require the tensorproduct to be K -bilinear. Second, we require that the unit object is simple andthat End( ) is one-dimensional over K .In addition, we require that every object X has a left dual X ∗ (cf. [EGNO, Def. 2.10.1,p. 40]; [Ka, Def. XIV.2.1, p. 342]). The corresponding evaluation and coevaluationmorphisms will be denoted byev X : X ∗ ⊗ X → and coev X : → X ⊗ X ∗ . We also assume that C is braided with braiding c X,Y : X ⊗ Y → Y ⊗ X (cf. [EGNO, Def. 8.1.1, p. 195]; [Ka, Def. XIII.1.1, p. 315]) and that it carries aribbon structure θ X : X → X (cf. [EGNO, Def. 8.10.1, p. 216]; [Ka, Def. XIV.3.2, p. 349]), which by definitionsatisfies the equations θ X ⊗ Y = ( θ X ⊗ θ Y ) ◦ c Y,X ◦ c X,Y and θ X ∗ = θ ∗ X . Under these assumptions, every object X does not only have aleft dual X ∗ , but also a right dual ∗ X , which is defined by using the same ob-ject ∗ X := X ∗ , but introducing the evaluation morphismev ′ X := ev X ◦ c X,X ∗ ◦ ( θ X ⊗ id X ∗ ) : X ⊗ ∗ X → ′ X := (id X ∗ ⊗ θ X ) ◦ c X,X ∗ ◦ coev X : → ∗ X ⊗ X (cf. [Ka, Prop. XIV.3.5, p. 352]). The axioms of left and right duality imply thatfor every object X , the functor X ⊗ – has the left adjoint X ∗ ⊗ – and the rightadjoint ∗ X ⊗ – , so thatHom( X ∗ ⊗ Y, Z ) ∼ = Hom( Y, X ⊗ Z ) and Hom( X ⊗ Y, Z ) ∼ = Hom( Y, ∗ X ⊗ Z ) . Similarly, the functor – ⊗ X has the right adjoint – ⊗ X ∗ and the left adjoint – ⊗ ∗ X ,so thatHom( Y ⊗ X, Z ) ∼ = Hom( Y, Z ⊗ X ∗ ) and Hom( Y ⊗ ∗ X, Z ) ∼ = Hom( Y, Z ⊗ X )(cf. [EGNO, Prop. 2.10.8, p. 42]; [Ka, Prop. XIV.2.2, p. 343f and p. 346f]). Thisimplies that the functors X ⊗ – and – ⊗ X are exact (cf. [HS, Chap. II, Thm. 7.7,p. 68]; [P, Par. 2.7, Satz 3, p. 72]). This in turn implies that X ⊗ P and P ⊗ X areprojective if P is projective (cf. [HS, Chap. II, Prop. 10.2, p. 82]). We will also needthe following related result on adjunctions: Proposition 2.1.
Suppose that F : C → D is a K -linear functor between finite K -linear categories. Then F has a right adjoint if and only if it is right exact. Proof. If F has a right adjoint, it is right exact (cf. [HS, Chap. II, Thm. 7.7,p. 68]). As pointed out above, we can assume for the converse that C and D arethe categories of finite-dimensional left modules over finite-dimensional algebras A and B , respectively. Then the Eilenberg-Watts theorem (cf. [Ro2, Thm. 5.45, p. 261];[I, Thm. 2.4, p. 677]) states that F is naturally equivalent to the functor M ⊗ A − fora B - A -bimodule M . But by the standard adjunction (cf. [Ro2, Thm. 2.76, p. 93]),this functor has the adjoint Hom B ( M, − ).A different proof of this proposition that connects it to the adjoint functor theoremis given in [DSS, Cor. 1.9, p. 6].An object X is called transparent if c Y,X ◦ c X,Y = id X ⊗ Y for all objects Y ∈ C (cf. [B,p. 224]). Transparent objects are called central in [Mue, Rem. 2.10, p. 296]. The fullsubcategory consisting of all transparent objects is called the M¨uger center of C .The unit object is transparent (cf. [EGNO, Exerc. 8.1.6, p. 196]; [Ka, Prop. XIII.1.2,p. 316]), and the direct sum of two transparent objects is transparent. Therefore, an36bject that is isomorphic to a finite direct sum of copies of the unit object is containedin the M¨uger center. We say that C is modular if the M¨uger center does not containany other objects. In this definition, we do not require that C be semisimple. Inthe semisimple case, this condition is equivalent to the invertibility of the modularS-matrix (cf. [EGNO, Prop. 8.20.12, p. 243]; [Mue, Cor. 2.16, p. 297]), which is in thiscase usually taken as the definition of modularity (cf. [EGNO, Def. 8.13.4, p. 224]). A fundamental example for categories with the properties that we have just listed arethe representation categories of factorizable ribbon Hopf algebras. Suppose that A isa Hopf algebra with coproduct ∆, counit ε , and antipode S . For the coproduct, wewill use the sigma notation of R. Heyneman and M. Sweedler in the modified form∆( a ) = a (1) ⊗ a (2) . If we take for C the category of finite-dimensional left A -modules,it is not only a finite K -linear abelian category, but also a tensor category, where thetensor product of two A -modules X and Y becomes an A -module via a. ( x ⊗ y ) := a (1) .x ⊗ a (2) .y for x ∈ X , y ∈ Y , and a ∈ A . The unit object is the base field K , endowed withthe trivial A -module structure coming from the counit. The left dual X ∗ of X is thedual vector space X ∗ := Hom K ( X, K ), endowed with structures described in [Ka,Examp. XIV.2.1, p. 347]. Although this category is not strict, our considerationsbelow apply to this category, because we have the notion of the tensor product X ⊗ X ⊗ · · · ⊗ X n of n vector spaces or A -modules, which is not build as aniteration of the tensor product of two vector spaces and therefore does not dependon the insertion of parentheses. The tensor products that appear below have to beunderstood in this way.We assume that A is quasitriangular with R-matrix R = R ⊗ R ∈ A ⊗ A , where thenotation follows the spirit of the notation for the coproduct above and writes thistensor, which is in general not decomposable, as if it were decomposable (cf. [Ka,Def. VIII.2.1, p. 173]). From the R-matrix, we obtain a braiding on C via the formula c X,Y ( x ⊗ y ) := R .y ⊗ R .x for x ∈ X and y ∈ Y (cf. [Ka, Prop. XIII.1.2, p. 318]). From the R-matrix, we alsoobtain the monodromy matrix Q := R ′ R , where R ′ := R ⊗ R . As in the case of37he R-matrix, we write Q = Q ⊗ Q . A is called factorizable if the mapΦ : A ∗ → A, ϕ (id A ⊗ ϕ )( Q )is bijective (cf. [SZ, Par. 3.2, p. 26] and the references given there). Note that thiscondition implies that A is finite-dimensional. We will require that A is factorizable,which is equivalent to the requirement that the category C is modular, as we willexplain in Paragraph 2.3.Furthermore, we assume that A contains a ribbon element, i.e., a nonzero centralelement v that satisfies∆( v ) = Q ( v ⊗ v ) and S ( v ) = v (cf. [SZ, Par. 4.3, p. 37]; note the difference to [Ka, Def. XIV.6.1, p. 361]). Ourribbon element gives rise to a ribbon structure of C by defining θ X : X → X, x v.x (cf. [Ka, Prop. XIV.6.2, p. 361]). If C and D are categories and F : C op × C → D is a bifunctor, a coend of F is anobject L of D together with a morphism ι X : F ( X, X ) → L for every object X ∈ C which is dinatural in the sense that, for every morphism f : X → Y , the diagram F ( Y, X ) F ( Y, Y ) F ( X, X ) L F (id Y ,f ) F ( f, id X ) ι Y ι X commutes (cf. [ML2, Chap. IX, Sec. 6, p. 226f]). Moreover, this dinatural transfor-mation is required to be universal, which means that for another dinatural transfor-mation κ X : F ( X, X ) → Z , there is a unique morphism g : L → Z that makes for all X ∈ C the diagram F ( X, X ) LZ ι X κ X g commutative. 38rom this universality, we see that coends behave well with respect to natural trans-formations: Lemma 2.2.
Suppose that F ′ : C op × C → D is a second bifunctor with coend ι ′ X : F ′ ( X, X ) → L ′ , and that η X,Y : F ( X, Y ) → F ′ ( X, Y ) is a natural transformation.Then there is a unique morphism g : L → L ′ such that the diagram F ( X, X ) LF ′ ( X, X ) L ′ ι X η X,X gι ′ X commutes for all objects X ∈ C . Proof.
This lemma is the version of [ML2, Chap. IX, Sec. 7, Prop. 1, p. 228] forcoends instead of ends.When we speak of a coend without further specifications, we think of the casewhere C = D is our base category satisfying the requirements stated in Paragraph 2.1,and the bifunctor F is given by F ( X, Y ) = X ∗ ⊗ Y . It is shown in [KL, Par. 5.1.3,p. 266ff] that, under our assumptions on C , a coend for this bifunctor F exists.It is not too difficult to see that a functor that possesses a right adjoint preservescoends. This together with the Fubini theorem for coends (cf. [ML2, Chap. IX,Sec. 8, p. 230f]) implies that the dinatural transformation ι X ⊗ ι Y : X ∗ ⊗ X ⊗ Y ∗ ⊗ Y → L ⊗ L is a coend. Comparing this transformation with the dinatural transformation(ev X ⊗ ev Y ) ◦ (id X ∗ ⊗ ( c Y ∗ ,X ◦ c X,Y ∗ ) ⊗ id Y ) : X ∗ ⊗ X ⊗ Y ∗ ⊗ Y → , the universal property yields a morphism ω L : L ⊗ L → that, via the adjunctionsstated in Paragraph 2.1, determines homomorphisms ω ′ L : L → L ∗ and ω ′′ L : L → ∗ L ,which are duals of each other. It is shown in [Sh1, Thm. 1.1, p. 3] that the modularityof C is equivalent to the property that ω ′ L , or alternatively ω ′′ L , is an isomorphism.This property is used in [KL, Def. 5.2.7, p. 276] as the definition of modularity.The same method can be used to introduce other morphisms that will be needed inthe sequel. The easiest of these morphisms arises from the dinatural transformation ι X ◦ (id X ∗ ⊗ θ X ) : X ∗ ⊗ X → L . Applying the universal property of the coend yields39 morphism T : L → L that makes all diagrams of the form X ∗ ⊗ X LL ι X ι X ◦ (id X ∗ ⊗ θ X ) T commutative. Similarly, the universal property of L ⊗ L explained above can be usedto obtain a morphism N ′ : L ⊗ L → L ⊗ L that makes all diagrams of the form X ∗ ⊗ X ⊗ Y ∗ ⊗ Y L ⊗ LL ⊗ L ι X ⊗ ι Y ( ι X ⊗ ι Y ) ◦ (id X ∗ ⊗ ( c Y ∗ ,X ◦ c X,Y ∗ ) ⊗ id Y ) N ′ commutative. Both of these morphisms are used when defining N := N ′ ◦ ( T ⊗ T ).An in a sense hybrid form of the dinatural transformations used in the definitionof ω L and of N ′ is used in the definition of S ′ from L ⊗ L to L : It arises by applyingthe universal property of L ⊗ L to a dinatural transformation in such a way that thediagram X ∗ ⊗ X ⊗ Y ∗ ⊗ Y L ⊗ LL ι X ⊗ ι Y (ev X ⊗ ι Y ) ◦ (id X ∗ ⊗ ( c Y ∗ ,X ◦ c X,Y ∗ ) ⊗ id Y ) S ′ becomes commutative.Given an object Z ∈ C , we define another morphism that is similar to N ′ : Again,because a functor that possesses a right adjoint preserves coends, the dinatural trans-formation id Z ⊗ ι X : Z ⊗ X ∗ ⊗ X → Z ⊗ L is a coend. Its universal property yieldsa morphism N lZ,L : Z ⊗ L → Z ⊗ L that makes all diagrams of the form Z ⊗ X ∗ ⊗ X Z ⊗ LZ ⊗ L id Z ⊗ ι X (id Z ⊗ ι X ) ◦ (( c X ∗ ,Z ◦ c Z,X ∗ ) ⊗ id X ) N lZ,L commutative. 40he coend L is a Hopf algebra inside the category C . We will not need the arisingproduct, coproduct, unit, counit, and antipode, which are constructed in a similarway as the morphisms ω L , T , N ′ , and S ′ by using the universal property of a coendand are described in [V, Par. 1.6, p. 478f] (cf. also [KL, Par. 5.2.2, p. 271ff]; notethat the conventions are slightly different there). We will, however, need that, as aconsequence of the Hopf algebra structure, there are two-sided integrals Λ L : → L and λ L : L → . Here, the assumption on modularity is used for two points, namelyon the one hand for the fact that the unit object can indeed be used as the domainof Λ L and the codomain of λ L , and on the other hand for the fact that λ L is two-sided(cf. [KL, Sec. 5.2, p. 270ff]). The integral Λ L is used to define S ∈ End( L ) as thecomposition L ∼ = −−→ L ⊗ id L ⊗ Λ L −−−−−→ L ⊗ L S ′ −−→ L (cf. [L1, Par. 1.3, p. 473]; the setting there is slightly more general as the modularityhypothesis is weakened). In the case where C is the category of finite-dimensional left modules over a factoriz-able ribbon Hopf algebra A , the coend can be described explicitly: The dual vectorspace L := A ∗ , viewed as a left A -module via the left coadjoint action( a.ϕ )( a ′ ) := ϕ ( S ( a (1) ) a ′ a (2) )for a, a ′ ∈ A and ϕ ∈ A ∗ , becomes a coend when endowed with the dinatural trans-formation ι X ( ξ ⊗ x )( a ) = ξ ( a.x )for a left A -module X and elements ξ ∈ X ∗ , x ∈ X , and a ∈ A (cf. [V, Lem. 4.3,p. 498], see also [KL, Thm. 7.4.13, p. 331]). It is not difficult to find the explicit formof the morphisms introduced in Paragraph 2.3 in this model of the coend: Proposition 2.3.
Suppose that ϕ, ψ ∈ A ∗ and that a, a ′ ∈ A . Then we have1. T ( ϕ )( a ) = ϕ ( av )2. N ′ ( ϕ ⊗ ψ )( a ⊗ a ′ ) = ϕ ( aQ ) ψ ( S ( Q ) a ′ )3. S ′ ( ϕ ⊗ ψ )( a ) = ϕ ( Q ) ψ ( S ( Q ) a )4. ω L ( ϕ ⊗ ψ ) = ϕ ( Q ) ψ ( S ( Q )) 41 roof. We prove these assertions by showing that the respective right-hand sideindeed has the required universal property. For the first assertion, this holds sincewe have for an A -module X and ξ ∈ X ∗ , x ∈ X that(( ι X ◦ (id X ∗ ⊗ θ X ))( ξ ⊗ x ))( a ) = ι X ( ξ ⊗ v.x )( a ) = ξ ( av.x ) = T ( ι X ( ξ ⊗ x ))( a ) . The proof of the second assertion is similar: If Y is another A -module and ζ ∈ Y ∗ , y ∈ X are elements, we have([( ι X ⊗ ι Y ) ◦ (id X ∗ ⊗ ( c Y ∗ ,X ◦ c X,Y ∗ ) ⊗ id Y )]( ξ ⊗ x ⊗ ζ ⊗ y ))( a ⊗ a ′ )= (( ι X ⊗ ι Y )( ξ ⊗ Q .x ⊗ Q .ζ ⊗ y ))( a ⊗ a ′ ) = ξ ( hQ .x )( Q .ζ )( a ′ .y )= ξ ( hQ .x ) ζ ( S ( Q ) a ′ .y ) = N ′ ( ι X ( ξ ⊗ x ) ⊗ ι Y ( ζ ⊗ y ))( a ⊗ a ′ ) . The proofs of the third and the fourth assertion are again very similar.Because the antipode of a finite-dimensional Hopf algebra is bijective (cf. [Mo,Thm. 2.1.3, p. 18]), the fourth assertion shows that ω L is nondegenerate if and only ifthe map Φ introduced in Paragraph 2.2 is bijective. This shows that the modularityof C is equivalent to the factorizability of A (cf. [KL, Par. 7.4.6, p. 332]).We note that, because these maps are defined via the universal property of the coend,they must automatically be morphisms in C , i.e., they must be A -linear. We alsonote that in the second assertion, we have chosen a special identification of A ∗ ⊗ A ∗ with ( A ⊗ A ) ∗ , which is determined by the equation ( ϕ ⊗ ψ )( a ⊗ a ′ ) = ϕ ( a ) ψ ( a ′ ).It is important to stress that the Hopf algebra structure on L is not the usual Hopfalgebra structure on A ∗ . In fact, A ∗ is a Hopf algebra in the category of vector spaces,whereas L is a Hopf algebra in the category C , which has a different braiding. TheHopf algebra structure on L is rather dual to the so-called transmuted Hopf algebrastructure of A (cf. [Maj, Examp. 9.4.9, p. 504]; [V, Lem. 4.4, p. 499]). However, thereis a very direct relation between the integrals of A ∗ and L : Because a factorizableHopf algebra is unimodular (cf. [L1, Prop. 3.7.4, p. 482] and [Ra, Prop. 12.4.2,p. 405f]), there is a nonzero two-sided integral Λ A ∈ A . This leads to the two-sidedintegral λ L : L → = K, ϕ ϕ (Λ A ) . On the other hand, a right integral ρ : A → K is by definition contained in L = A ∗ ,and it can be shown that the unique morphism Λ L : = K → L that maps 1 K to ρ is even a two-sided integral (cf. [BKLT, Prop. 6.6, p. 153]). As a consequence, themorphism S is given by the formula S ( ϕ )( a ) = ϕ ( Q ) ρ ( S ( Q ) a ) . .5 The block spaces In topological field theory, one associates to a modular category projective repre-sentations of mapping class groups of surfaces. There is a vast literature on thistopic; our approach here is based on a construction given by V. Lyubashenko in hisarticles [L1] and [L2]. We will now briefly review some key aspects of his construc-tion to the extent that we need them. Suppose that C is a category that satisfiesthe requirements stated in Paragraph 2.1. We assume that each of the n boundarycomponents of the surface Σ g,n described in Paragraph 1.1 is labeled by an object X i from C . Associated with such a labeled surface is the space Z (Σ X ,...,X n g,n ) := Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g ) , where L denotes the coend defined in Paragraph 2.3. This space, which we callthe space of chiral conformal blocks or briefly the block space, obviously dependsfunctorially on the labels and can therefore be viewed as the value of a left exactcontravariant functor Z (Σ g,n ) on the object ( X , . . . , X n ) ∈ C n , by which we meanthat the functor is left exact in each argument (cf. [P, Par. 4.6, p. 130f]).The construction of a projective action of the mapping class group then starts froman oriented net (cf. [L2, Sec. 6, p. 355] and [L1, Par. 4.1, p. 486f]). We consider thefollowing oriented net that encodes the structure of our surface: X n X n − X X Y · · · Y g − Y g Z n − · · · Z N N N g − N g − A · · · A g − A A g The block space arises from the oriented net by taking suitable coends over internaledges (cf. [L2, Par. 8.2, p. 374]). For our chosen net, the construction gives the space T := Z A i ,N j ,Z l ,Y m Hom( A ⊗ Y , N ) ⊗ Hom( Y , Y ⊗ N ) ⊗ · · · ⊗ Hom( Y g , Y g − ⊗ N g − ) ⊗ Hom( X n ⊗ Z n − , Y g ) ⊗ Hom( X n − ⊗ Z n − , Z n − ) ⊗ · · · ⊗ Hom( X ⊗ Z , Z ) ⊗ Hom( X , Z ⊗ A g ) ⊗ Hom( N g − ⊗ A g , A g − ) ⊗ · · · ⊗ Hom( N ⊗ A , A )43here in the upper limit of the integral sign the abbreviation A i has been usedfor A , A , . . . , A g . Similarly, the abbreviations N j , Z l , and Y m have been used forthe objects with indices j = 1 , . . . , g − l = 1 , . . . , n −
1, and m = 2 , . . . , g . Thecoends are taken in the category of left exact functors: Obviously, the argument ofthe coend depends functorially on the objects, and the corresponding functor is leftexact. A more detailed discussion of this aspect can be found in [L2, App. B, p. 398]and [FS, Sec. 3, p. 72ff].In view of [L2, Lem. B.1, p. 398] and the functor adjunctions discussed in Para-graph 2.1, the space T is naturally isomorphic to T := Z A i ,N j ,Y ,Y g ,Z Hom( Y , ∗ A ⊗ N ) ⊗ Hom( Y g , Y ⊗ N ⊗ · · · ⊗ N g − ) ⊗ Hom( X n ⊗ · · · ⊗ X ⊗ Z , Y g ) ⊗ Hom( X ⊗ ∗ A g , Z ) ⊗ Hom( N g − , A g − ⊗ A ∗ g ) ⊗ · · · ⊗ Hom( N , A ⊗ A ∗ )Taking also the coend over Y g , we obtain a natural isomorphism with the space T := Z A i ,N j ,Z ,Y Hom( Y , ∗ A ⊗ N ) ⊗ Hom( X n ⊗ · · · ⊗ X ⊗ Z , Y ⊗ N ⊗ · · · ⊗ N g − ) ⊗ Hom( X ⊗ ∗ A g , Z ) ⊗ Hom( N g − , A g − ⊗ A ∗ g ) ⊗ · · · ⊗ Hom( N , A ⊗ A ∗ )Treating N , . . . , N g − in the same way and moreover taking the coend over Z , weobtain the space T := Z A i ,Y Hom( Y , ∗ A ⊗ A ⊗ A ∗ ) ⊗ Hom( X n ⊗ · · · ⊗ X ⊗ ∗ A g , Y ⊗ A ⊗ A ∗ ⊗ · · · ⊗ A g − ⊗ A ∗ g )Finally, if we also take the coend over Y and apply the functor adjunctions to A g ,we arrive at the space T := Z A i Hom( X n ⊗ · · · ⊗ X , ∗ A ⊗ A ⊗ A ∗ ⊗ A ⊗ · · · ⊗ A g − ⊗ A ∗ g ⊗ A g )It is now important to realize that we have introduced right duals in Paragraph 2.1in such a way that left and right duals have the same underlying object. They44iffer only in their evaluation and coevaluation morphisms, which affect the functoradjunctions, but not the objects themselves. The space T is therefore also equal to T = Z A i Hom( X n ⊗ · · · ⊗ X , A ∗ ⊗ A ⊗ · · · ⊗ A ∗ g ⊗ A g )Because the coends are taken in the category of left exact functors, the space T is naturally isomorphic to our block space Hom( X n ⊗ · · · ⊗ X , L ⊗ g ) defined above(cf. [FS, Prop. 3.4, p. 74]).We have claimed that the specific oriented net given above encodes the structure ofthe surface introduced in Paragraph 1.1. We now explain this relation in the specialcase g = 3 and n = 2; the general case is not essentially different. The first stepconsists in the application of the fattening functor (cf. [L2, p. 341]), which turns thenet into a so-called ribbon graph by replacing each trivalent vertex by a hexagon inwhich every second side is represented by a double line. These hexagons are arrangedso that their double lines come to lie on the edges of the net, and if an edge connectstwo vertices, the corresponding hexagons are glued along their now common doubleline. If we apply this procedure to our net above, we obtain the ribbon graph X X Y Y Z N N A A A in which the colored lines indicate the glued double lines.The second step consists in the application of the duplication functor (cf. [L2, Par. 2.2and Par. 2.3, p. 315ff]). The duplication functor turns a ribbon graph into a surfaceby taking two copies of the graph and gluing them along the boundary componentsthat do not arise from the double lines mentioned above. Applied to our ribbongraph, it yields a surface of genus 3 that has two boundary components, which arisefrom the two double lines still present in our ribbon graph and are labeled by X X . If we deform the arising surface slightly and bring the first and the thirdhandle closer together by pulling the middle handle to the bottom, we obtain thestandard surface X X described in Paragraph 1.1, in which the curves arise from the duplication of the linesin the ribbon graph that have the same color. It should be noted that the red curvesare freely homotopic to the curves denoted by α , α , and α in Paragraph 1.1, andthe orange curves are freely homotopic to the curves µ and µ from Paragraph 1.2,respectively.It is possible to compare this discussion with the corresponding one in [L1, Par. 4.5,p. 494f]. To do that, it is necessary to deform the surface given in [L1, Fig. 6, p. 495]by pulling down the middle handles to the bottom in order to bring the first and the g -th handle together at the top, exactly as in our discussion of the relation betweenthe net and the surface above. This leads to a surface in which the handles are labeledcounterclockwise, but the boundary components are labeled clockwise, in contrastto the surface used in Paragraph 1.1, where the boundary components arose fromthe polygon and therefore were also labeled counterclockwise. As a consequence, theorder of the labels X , . . . , X n is reversed. The following table explains the relationbetween our surface and the surface in [L1, Fig. 2, p. 490 and Fig. 6, p. 495]:46urves α i β i µ j ζ l ∂ k Color red blue orange turquoise purpleLabel in the net A i N j Z l X k Dehn twists t i r i n j z l d k [L1] e i b i a j +1 t n − l,g R g − k +1 Here, the index ranges are i = 1 , . . . , g , j = 1 , . . . , g − k = 1 , . . . , n , and l = 1 , . . . , n −
1. The light green curves in our surface correspond to the curveslabeled d j in [L1, loc. cit.]. As we just explained, the theory associates with a surface a certain left exact func-tor. But in addition, the theory associates with a mapping class [ ψ ] ∈ Γ g,n a projec-tive class [ Z ( ψ )] = P ( Z ( ψ )) of natural equivalences between two of these functors,namely the functor ( X , . . . , X n )
7→ Z (Σ X ,...,X n g,n )on the one hand and the functor( X , . . . , X n )
7→ Z (Σ X τ (1) ,...,X τ ( n ) g,n )on the other hand, where τ := p ([ ψ ]) − ∈ S n is the inverse permutation of the markedpoints introduced in Paragraph 1.3. A representative Z ( ψ ) of this projective class isa natural equivalence between these functors, i.e., a family of linear isomorphisms Z ( ψ ) : Z (Σ X ,...,X n g,n ) → Z (Σ X τ (1) ,...,X τ ( n ) g,n )that are natural in the following sense: If, for i = 1 , . . . , n , we are given morphisms f i : X i → Y i , then the diagram Z (Σ Y ,...,Y n g,n ) Z (Σ Y τ (1) ,...,Y τ ( n ) g,n ) Z (Σ X ,...,X n g,n ) Z (Σ X τ (1) ,...,X τ ( n ) g,n ) Z ( ψ ) ◦ ( f n ⊗···⊗ f ) ◦ ( f τ ( n ) ⊗···⊗ f τ (1) ) Z ( ψ ) Z ( ψ )] = P ( Z ( ψ )).This assignment is also compatible with composition in the sense that, for a secondmapping class [ φ ], we have [ Z ( φ ◦ ψ )] = [ Z ( φ ) ◦ Z ( ψ )]. However, this does not implythat the mapping class group Γ g,n acts projectively on each block space Z (Σ X ,...,X n g,n ),because the block spaces are not preserved by Z ( ψ ) if [ ψ ] permutes two boundarycomponents with different labels. In general, only the pure mapping class group PΓ g,n acts projectively on each block space. However, the entire mapping class group Γ g,n acts projectively on the direct sum M τ ∈ S n Z (Σ X τ (1) ,...,X τ ( n ) g,n )of block spaces. In the case where all boundary components are labeled with the sameobject X , we also have an action of Γ g,n on the space Hom C ( X ⊗ n , L ⊗ g ). We can in factrealize the direct sum above as a subspace of this space by setting X := X ⊕· · ·⊕ X n .Via a sophisticated system of rules laid out in [L1] and [L2], the description of asurface in terms of a net makes it possible to describe the actions of the elements ofthe mapping class group on the block spaces explicitly. According to this formalism,the elements of the mapping class group introduced in Paragraph 1.4, Paragraph 1.5,Paragraph 1.7, and Paragraph 1.8 act on the block spaces as follows:(1) For j = 1 , . . . , n , the Dehn twist d j acts by precomposition with the morphismid X n ⊗···⊗ X j +1 ⊗ θ X j ⊗ id X j − ⊗···⊗ X , i.e., via the map Z ( d j ) : Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g ) → Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g ) f f ◦ (id X n ⊗···⊗ X j +1 ⊗ θ X j ⊗ id X j − ⊗···⊗ X ) . (2) For j = 1 , . . . , n −
1, the braiding b j,j +1 acts by precomposition with the mor-phism id X n ⊗···⊗ X j +2 ⊗ c X j ,X j +1 ⊗ id X j − ⊗···⊗ X , i.e., it induces a map Z ( b j,j +1 ) : Hom C ( X n ⊗ · · · ⊗ X j +1 ⊗ X j ⊗ · · · ⊗ X , L ⊗ g ) → Hom C ( X n ⊗ · · · ⊗ X j ⊗ X j +1 ⊗ · · · ⊗ X , L ⊗ g ) f f ◦ (id X n ⊗···⊗ X j +2 ⊗ c X j ,X j +1 ⊗ id X j − ⊗···⊗ X ) . (3) For j = 1 , . . . , n −
1, the element d j,j +1 acts by precomposition with the morphismid X n ⊗···⊗ X j +2 ⊗ θ X j +1 ⊗ X j ⊗ id X j − ⊗···⊗ X , a map that we denote by Z ( d j,j +1 ).484) For j = 1 , . . . , n −
1, the element q j,j +1 acts by precomposition with the morphismid X n ⊗···⊗ X j +2 ⊗ ( c X j ,X j +1 ◦ c X j +1 ,X j ) ⊗ id X j − ⊗···⊗ X , a map that we denote by Z ( q j,j +1 ).(5) For i = 1 , . . . , g , the Dehn twist t i acts by postcomposition with the morphismid L ⊗ ( i − ⊗ T ⊗ id L ⊗ ( g − i ) , i.e., via the map Z ( t i ) : Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g ) → Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g ) f (id L ⊗ ( i − ⊗ T ⊗ id L ⊗ ( g − i ) ) ◦ f. (6) For i = 1 , . . . , g , the element s i acts by postcomposition with the morphismid L ⊗ ( i − ⊗ S ⊗ id L ⊗ ( g − i ) , a map that we denote by Z ( s i ).(7) For i = 1 , . . . , g −
1, the Dehn twist n i acts by postcomposition with the morphismid L ⊗ ( i − ⊗ N ⊗ id L ⊗ ( g − i − , a map that we denote by Z ( n i ).(8) For j = 1 , . . . , n −
1, the Dehn twist z j acts via the map Z ( z j ) : Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g ) → Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g )which is the unique linear map that makes the diagramHom C ( X j ⊗ · · · ⊗ X , ∗ ( X n ⊗ · · · ⊗ X j +1 ) ⊗ L ⊗ g ) Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g )Hom C ( X j ⊗ · · · ⊗ X , ∗ ( X n ⊗ · · · ⊗ X j +1 ) ⊗ L ⊗ g ) Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g ) Z ( z j ) commutative. In this diagram, the horizontal arrows are the functor adjunctionsfrom Paragraph 2.1, and the left vertical map is given by h ( θ ∗ ( X n ⊗···⊗ X j +1 ) ⊗ L ⊗ ( g − ⊗ T ) ◦ N l ∗ ( X n ⊗···⊗ X j +1 ) ⊗ L ⊗ ( g − ,L ◦ h. We now explain how some of the formulas come about. For any edge of a trivalentnet, there is an automorphism called the twist (cf. [L2, Par. 4.3, p. 350]) that cor-responds to a Dehn twist around the corresponding curve on the surface (cf. [L2,Prop. 2.2, p. 319]). Recall that, as already mentioned in Paragraph 1.4, Dehn twistsas defined here are inverse Dehn twists as defined in [L1] and [L2]. By [L2, Par. 8.1,No. (x), p. 374], the twist acts by applying the ribbon twist to the correspondingvariable. Because the isomorphisms between the spaces T to T introduced in Para-graph 2.5 and our block space are natural, this implies Claim (1). Similarly, the49enerators t i , n j , and z l act by applying a ribbon twist to the internal variables A i , N j , and Z l (cf. [L1, p. 493]). We calculate the action of these generators explicitlyin the case g = 3 and n = 2 to explain Claim (5), Claim (7), and Claim (8).We first calculate the action of t : On the space T , it acts by postcomposition withthe twist θ A on the last tensor factor, which corresponds to postcomposition with θ A ⊗ id A ∗ on the last tensor factor on the spaces T and T . Under the isomorphismto the space T , this corresponds to postcomposition with id ∗ A ⊗ θ A ⊗ id A ∗ on thefirst tensor factor. This in turn corresponds on the space T to postcomposition withid ∗ A ⊗ θ A ⊗ id A ∗ ⊗ A ⊗ A ∗ ⊗ A . In view of the definition of T in Paragraph 2.3, thisbecomes postcomposition with T ⊗ id L ⊗ L on our block space Hom( X ⊗ X , L ⊗ L ⊗ L ).Very similar calculations yield the claim for t and t .Next, we calculate the action of n : On the space T , it acts by postcomposition withthe twist θ N on the first tensor factor. Under the isomorphism to the space T , thiscorresponds to postcomposition with id ∗ A ⊗ θ N on the first tensor factor, which doesnot change under the isomorphism to the space T . On the space T , it becomes post-composition with id ∗ A ⊗ θ A ⊗ A ∗ on the first tensor factor. Under the isomorphism tothe space T , this corresponds to postcomposition with id ∗ A ⊗ θ A ⊗ A ∗ ⊗ id A ⊗ A ∗ . Thisis equal to postcomposition with id ∗ A ⊗ (( θ A ⊗ θ A ∗ ) ◦ c A ∗ ,A ◦ c A ,A ∗ ) ⊗ id A ⊗ A ∗ ⊗ A ,which corresponds on our block space Hom( X ⊗ X , L ⊗ L ⊗ L ) to postcompositionwith N ⊗ id L . The calculations for n are similar.For the action of z , we proceed differently: On the space T , it acts by postcompo-sition with θ Z ⊗ id A on the tensor factor Hom( X , Z ⊗ A ), which on the spaces T and T becomes postcomposition with θ Z on the fourth and the third tensor factor,respectively. Under the isomorphism to the space T , this corresponds to precompo-sition with id X ⊗ θ X ⊗ ∗ A . This space is isomorphic to Z A i Hom( X ⊗ ∗ A , ∗ X ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ⊗ A ∗ ) , where the action becomes precomposition with θ X ⊗ ∗ A , which equals postcomposi-tion with θ ∗ X ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ⊗ A ∗ . Under the isomorphism to the space Z A i Hom( X , ∗ X ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ) , it becomes postcomposition with θ ∗ X ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ id A , which equals postcom-position with(( θ ∗ X ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ⊗ θ A ∗ ) ◦ c A ∗ , ∗ X ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ◦ c ∗ X ⊗ A ∗ ⊗ A ⊗ A ∗ ⊗ A ,A ∗ ) ⊗ id A . X , ∗ X ⊗ L ⊗ ), this equals postcompositionwith ( θ ∗ X ⊗ L ⊗ L ⊗ T ) ◦ N l ∗ X ⊗ L ⊗ L,L . But this means precisely that Z ( z ) acts as assertedon our block space Hom( X ⊗ X , L ⊗ ).Using a fusing morphism (cf. [L2, Eq. (6.8), p. 359]), the braiding of two neigh-boring boundary components corresponds to a braiding morphism of a trivalent net(cf. [L2, Par. 4.3, p. 350f]), which in turn corresponds to the braiding of the corre-sponding labels of the boundary components on our block space (cf. [L2, Par. 8.1,No. (viii) and (ix), p. 373f]). In this way, we arrive at Claim (2) and Claim (4).Claim (3) follows from Claim (1), because we have seen in Paragraph 1.8 that[ q j,j +1 ] = [ d j,j +1 d − j d − j +1 ], and in a ribbon category we have c X j ,X j +1 ◦ c X j +1 ,X j = ( θ − X j +1 ⊗ θ − X j ) ◦ θ X j +1 ⊗ X j = θ X j +1 ⊗ X j ◦ ( θ − X j +1 ⊗ θ − X j ) . The element s i corresponds to a switch (cf. [L2, Prop. 2.2, p. 319]) that acts by S onthe corresponding tensor factor of L ⊗ g (cf. [L2, Prop. 8.8, p. 391]). This establishesClaim (6) and concludes our discussion about how the different elements of themapping class group act.It is important to note that, as we use a different net for the description of thesurface, our formulas for the actions of the generators of the mapping class groupon the block space are different from those in [L1, Par. 4.5, p. 494f]. Starting fromthe net in [L1, Fig. 5, p. 491] and using the internal variables called there E i insteadof those called there D i to form the coends in the last step would give formulassimilar to ours. To facilitate the comparison, we include a small table that relatesour notation to the one used in [L1]:Present notation Γ g,n L N ′ N lX,L S T b l,l +1 q l,l +1 s i [L1] M ′ g,n f Ω Ω lX, f S T ω l ω l S i It must be emphasized that the morphisms in the first row are only analogous, butnot strictly equal, to the ones in the second row.In the case where n = 1, i.e., when there is only one boundary component, theprojective action on the block space can be obtained by postcomposition from a pro-jective action of Γ g, on L ⊗ g : For a mapping class [ ψ ] ∈ Γ g, , a representative Z ( ψ ) ofthe associated projective class [ Z ( ψ )] is a natural equivalence from the contravariantHom-functor X
7→ Z (Σ Xg, ) = Hom C ( X, L ⊗ g )51o itself, which according to the Yoneda lemma (cf. [ML2, Chap. III, Sec. 2, p. 61];[P, Par. 1.15, p. 37]) is given by postcomposition with an automorphism of L ⊗ g . Forthe generators of the mapping class group whose projective action is described in thelist above, this can be seen explicitly: The only generator that appears in this caseand is not already given by postcomposition is d , which according to Claim (1) isgiven by precomposition with θ X . However, by the naturality of the twist, this isequal to postcomposition with θ L ⊗ g . The various mapping class group representations just described are not unrelated,but rather together form a modular functor (cf. [L2, Par. 8.1, p. 372ff]). In particular,they are compatible with gluing of surfaces (cf. [L2, Par. 8.1, No. (xi), p. 374]). Wewill need only one very special instance of this general property: Consider the lasttwo boundary components of the surface Σ g,n +2 . They carry an orientation that isinduced from the orientation of the surface, which is opposite to the orientation of thecurves ρ n +1 and ρ n +2 in Paragraph 1.1. Up to isotopy, there is a unique orientation-reversing diffeomorphism between these two boundary components that maps thedistinguished point on one boundary component to the distinguished point on theother. If we identify the two boundary components along this diffeomorphism, thearising quotient space is diffeomorphic to Σ g +1 ,n , so that we have effectively attacheda handle, which we consider as the first one. A diffeomorphism ϕ : Σ g,n +2 → Σ g,n +2 that restricts to the identity on the last two boundary components then induces adiffeomorphism ψ : Σ g +1 ,n → Σ g +1 ,n . For τ := p ([ ϕ ]) − ∈ S n +2 , our hypothesis meansthat τ fixes n + 1 and n + 2.Suppose we are given n objects X , . . . , X n and another object X . From the adjunc-tions recalled in Paragraph 2.1, we get an isomorphismHom C ( X ∗ ⊗ X ∗∗ ⊗ X n ⊗ · · · ⊗ X , L ⊗ g ) → Hom C ( X n ⊗ · · · ⊗ X , X ∗ ⊗ X ⊗ L ⊗ g ) . If we postcompose with ι X ⊗ id L ⊗ g , we obtain a morphismHom C ( X ∗ ⊗ X ∗∗ ⊗ X n ⊗ · · · ⊗ X , L ⊗ g ) → Hom C ( X n ⊗ · · · ⊗ X , L ⊗ ( g +1) )that we call the handle gluing homomorphism. By construction, this homomorphismis natural with respect to the objects X , . . . , X n of C in the sense that for givenmorphisms f i : X i → Y i , the diagram 52 (Σ Y ,...,Y n ,X ∗∗ ,X ∗ g,n +2 ) Z (Σ Y ,...,Y n g +1 ,n ) Z (Σ X ,...,X n ,X ∗∗ ,X ∗ g,n +2 ) Z (Σ X ,...,X n g +1 ,n ) ◦ (id X ∗⊗ X ∗∗ ⊗ f n ⊗···⊗ f ) ◦ ( f n ⊗···⊗ f ) commutes.The special case of the gluing property under consideration here then states that thediagram Z (Σ X ,...,X n ,X ∗∗ ,X ∗ g,n +2 ) Z (Σ X ,...,X n g +1 ,n ) Z (Σ X τ (1) ,...,X τ ( n ) ,X ∗∗ ,X ∗ g,n +2 ) Z (Σ X τ (1) ,...,X τ ( n ) g +1 ,n ) Z ( ϕ ) Z ( ψ ) commutes for suitably chosen representatives Z ( ϕ ) and Z ( ψ ) within their respectiveprojective classes. As above, Z ( ϕ ) and Z ( ψ ) are viewed here as natural transforma-tions of the corresponding functors, so that the nonzero scalar used in passing to adifferent representative does not depend on the spaces X , . . . , X n .To illustrate this property, we consider the example ϕ = d n +1 , which also plays animportant role later. Then the corresponding map on the glued surface is ψ = t .The actions of ϕ and ψ are described in the list in Paragraph 2.6. By its definitionin Paragraph 2.3, we have T ◦ ι X = ι X ◦ (id X ∗ ⊗ θ X ) = ι X ◦ ( θ ∗ X ⊗ id X ), where thesecond equality follows from the dinaturality of ι . Therefore, the diagramHom C ( X n ⊗ · · · ⊗ X , X ∗ ⊗ X ⊗ L ⊗ g ) Hom C ( X n ⊗ · · · ⊗ X , L ⊗ ( g +1) )Hom C ( X n ⊗ · · · ⊗ X , X ∗ ⊗ X ⊗ L ⊗ g ) Hom C ( X n ⊗ · · · ⊗ X , L ⊗ ( g +1) ) ( θ ∗ X ⊗ id X ⊗ id L ⊗ g ) ◦ ( ι X ⊗ id L ⊗ g ) ◦ ( T ⊗ id L ⊗ g ) ◦ ( ι X ⊗ id L ⊗ g ) ◦ commutes.Combining this with the naturality of the adjunction, we see that the gluing propertyholds in this case.We note that the choice ϕ = d n +2 leads to the same map on the glued surface,namely ψ = t . A very similar reasoning shows that the gluing property also holdsin this case. 53 .8 The case of the torus In order to understand how the results of this article generalize the results of ourprevious one (cf. [LMSS1]), it will be important to consider the projective represen-tation reviewed in Paragraph 2.6 in the case of the torus, where g = 1, and the casewhere C is the category of representations of a Hopf algebra A with the propertiesdescribed in Paragraph 2.2. Important ingredients of this projective representationare the endomorphisms S and T of L = A ∗ , whose explicit form we have deter-mined in Paragraph 2.4. In our previous article, we have used the same symbols forendomorphisms of A , which we will now denote by ˆ S and ˆ T and which are given byˆ S ( a ) = ρ ( aQ ) S ( Q ) and ˆ T ( a ) = va (cf. [LMSS1, Sec. 3, p. 410]). In order to understand how these endomorphisms arerelated, we use a variant of the Radford map ι : A → A ∗ , a ι ( a )defined by ι ( a )( a ′ ) = ρ ( aa ′ ), which was also introduced on the page just cited. Here ρ is the right integral already considered in Paragraph 2.4. As recalled in [LMSS1],loc. cit., ρ is contained in the space¯ C ( A ) := { ϕ ∈ A ∗ | ϕ ( aa ′ ) = ϕ ( S ( a ′ ) a ) for all a, a ′ ∈ A } , which can be viewed as the set of invariants ( A ∗ ) A for the left coadjoint actiondiscussed in Paragraph 2.4. The variant of the Radford map that we will need is themap ¯ ι := ι ◦ S , which then satisfies¯ ι ( a )( a ′ ) = ι ( S ( a ))( a ′ ) = ρ ( S ( a ) a ′ ) = ρ ( a ′ a ) . With the help of ¯ ι , we see that ˆ S and ˆ T are conjugate to S and T : Lemma 2.4.
The diagrams A ∗ A ∗ A A S ¯ ι ˆ S ¯ ι and A ∗ A ∗ A A T ¯ ι ˆ T ¯ ι commute. Moreover, ¯ ι is an isomorphism of A -modules, where A ∗ carries the leftcoadjoint action and A carries the left adjoint action of the coopposite Hopf algebra,given by a.a ′ := a (2) a ′ S − ( a (1) ). 54 roof. To show the commutativity of the first diagram, we use that ρ ∈ ¯ C ( A ) tocompute ¯ ι ( ˆ S ( a ))( a ′ ) = ρ ( aQ )¯ ι ( S ( Q ))( a ′ ) = ρ ( aQ ) ρ ( a ′ S ( Q ))= ρ ( S ( Q ) a ) ρ ( S ( Q ) a ′ ) = ρ ( Q a ) ρ ( S ( Q ) a ′ )= ¯ ι ( a )( Q ) ρ ( S ( Q ) a ′ ) = S (¯ ι ( a ))( a ′ ) . The proof of the commutativity of the second diagram is even simpler: We have¯ ι ( ˆ T ( a ))( a ′ ) = ¯ ι ( va )( a ′ ) = ρ ( a ′ va ) = ¯ ι ( a )( a ′ v ) = T (¯ ι ( a ))( a ′ )by Proposition 2.3. Because ρ is a Frobenius homomorphism, ι and ¯ ι are bijective.To see that ¯ ι is linear with respect to the specified A -actions, we compute¯ ι ( a.a ′ )( a ′′ ) = ρ ( a ′′ ( a.a ′ )) = ρ ( a ′′ a (2) a ′ S − ( a (1) )) = ρ ( S ( a (1) ) a ′′ a (2) a ′ )= ¯ ι ( a ′ )( S ( a (1) ) a ′′ a (2) ) = ( a. (¯ ι ( a ′ )))( a ′′ )and get the assertion.By [SZ, Prop. 4.3, p. 37], we have ˆ S ◦ ˆ T ◦ ˆ S = ρ ( v ) ˆ T − ◦ ˆ S ◦ ˆ T − . The precedinglemma therefore implies that we also have S ◦ T ◦ S = ρ ( v ) T − ◦ S ◦ T − . Itthen follows from our discussion of the braid group B in Paragraph 1.12 that theassignment s S and t T yields a projective representation of B in Aut A ( L ). If X is an A -module, postcom-position with S and T therefore leads to a projective representation of B on thespace Hom A ( X, L ). On the other hand, we have seen in Paragraph 1.12 that theassignment r r and t t yields an isomorphism between the braid group B and the mapping class group Γ , . By construction, this isomorphism maps s to s .Therefore, if we transport the projective action of B to Γ , along this isomorphism,we obtain exactly the projective representation considered in Paragraph 2.6 in thecase g = 1 and n = 1.It is instructive to see why the 2-chain relation ( r t ) = d in the mapping classgroup Γ , , which we discussed in Paragraph 1.12, is satisfied in our situation.We have already used in Paragraph 1.12 that ( r t ) = s − . Now suppose that f ∈ Hom A ( X, L ) is an A -linear map. For x ∈ X , we choose a ∈ A so that f ( x ) = ¯ ι ( a ).For example from [LMSS1, Prop. 3.2, p. 411] combined with [LMSS1, Lem. 3.3,p. 412], we know that ˆ S ( a ) = (( ρ ⊗ ρ )( Q )) S ( v − ) av − . ρ ∈ ¯ C ( A ), we then get(( s .f )( x ))( a ′ ) = S ( f ( x ))( a ′ ) = S (¯ ι ( a ))( a ′ ) = ¯ ι ( ˆ S ( a ))( a ′ ) = ρ ( a ′ ˆ S ( a ))= (( ρ ⊗ ρ )( Q )) ρ ( a ′ S ( v − ) av − )= (( ρ ⊗ ρ )( Q )) ρ ( S ( v − ) a ′ S ( v − ) a )= (( ρ ⊗ ρ )( Q )) f ( x )( S ( v − ) a ′ S ( v − ))= (( ρ ⊗ ρ )( Q )) ( S ( v − ) .f ( x ))( a ′ ) = (( ρ ⊗ ρ )( Q )) ( v − .f ( x ))( a ′ )= (( ρ ⊗ ρ )( Q )) f ( v − .x )( a ′ ) . On the other hand, we have(( d − .f )( x ))( a ′ ) = (( f ◦ θ − X )( x ))( a ′ ) = f ( v − .x )( a ′ ) , so that the actions of s and d − agree up to a scalar, as required.In the case g = 1, but n = 0, the definition in Paragraph 2.5 is to be understoodin such a way that the action is on the block space Hom A ( K, L ), because the basefield is the unit object of the category, as explained in Paragraph 2.2. For any left A -module Y , the map Hom A ( K, Y ) → Y A , f f (1 K ) yields a bijection with thespace of invariants Y A = { y ∈ Y | a.y = ε ( a ) y for all a ∈ A } . As we said above, in the case Y = L we have L A = ¯ C ( A ), while in the case Y = A ,endowed with the action described in Lemma 2.4, we have Y A = Z ( A ), the centerof A . In view of its A -linearity, ¯ ι restricts to an isomorphism between Z ( A ) and ¯ C ( A ).In the case X = K , we have that θ X is the identity map, because ε ( v ) = 1. Thismeans that d acts trivially on Hom A ( K, L ). By the 2-chain relation that we havejust checked explicitly, this implies that s acts trivially. This in turn means, inview of the discussion in Paragraph 1.12, that the projective action of the braidgroup B restricts to a projective action of the modular group SL(2 , Z ) on ¯ C ( A ), asit is required for the construction in Paragraph 2.6, because Γ ∼ = SL(2 , Z ). Via ¯ ι , thisprojective action on ¯ C ( A ) is isomorphic to an action of SL(2 , Z ) on the center Z ( A ).This is the action that was considered in [LMSS1, Cor. 3.4, p. 412].Let us mention that using ¯ ι is not the only way to relate the projective representationson ¯ C ( A ) and Z ( A ). Other ways are discussed in [SZ, Par. 9.1, p. 87ff].56 Derived functors
There are various approaches to the definition of the Ext-functors in abelian cate-gories. We assume here that our category satisfies the assumptions listed in Para-graph 2.1; one of these assumptions was that it has enough projectives. In thiscase, we can use the approach via projective resolutions described in [Mi, Chap. VII, §
7, p. 182ff] and denoted there by Ext, but here just denoted by Ext. In this ap-proach, the group Ext m ( X, Y ) for two objects X and Y of C is defined by choosinga projective resolution X ξ ←−− P d ←−− P d ←−− P d ←−− · · · of X , which exists by the above hypothesis. The group Ext m ( X, Y ) is then definedas the m -cohomology group of the cochain complexHom C ( P , Y ) −−→ Hom C ( P , Y ) −−→ Hom C ( P , Y ) −−→ · · · of abelian groups. Although the notation does not reflect this, the definition dependson the chosen resolution; different resolutions lead to Ext-groups that are canonicallyisomorphic, but not equal (cf. [ML1, Chap. XII, §
9, p. 390]; [LMSS1, Sec. 1, p. 402]).Note that Ext ( X, Y ) ∼ = Hom C ( X, Y ), because the Hom-functor is left exact.The following lemma about the ribbon structure θ will be important in the sequel: Lemma 3.1.
Suppose that ξ ←−− P d ←−− P d ←−− P d ←−− · · · is a projective resolution of the unit object. Then there exists, for every object X ∈ C and every m ≥
0, a morphism h m ( X ) : P m ⊗ X → P m +1 ⊗ X such that θ P m ⊗ X − id P m ⊗ θ X = ( d m +1 ⊗ id X ) ◦ h m ( X ) + h m − ( X ) ◦ ( d m ⊗ id X )for all m ≥
1, and θ P ⊗ X − id P ⊗ θ X = ( d ⊗ id X ) ◦ h ( X ). These morphisms arenatural in X in the sense that, for a morphism f : X → Y , the diagram P m ⊗ X P m ⊗ YP m +1 ⊗ X P m +1 ⊗ Y h m ( X ) id Pm ⊗ f h m ( Y )id Pm +1 ⊗ f commutes. 57 roof. We denote by Rex( C , C ) the category of K -linear right exact functors from C to C , whose morphisms are natural transformations. By [Sh2, Cor. 2.6, p. 466],a result that relies on our assumption that the base field is perfect, the functor X P m ⊗ X is a projective object in the K -linear category Rex( C , C ), and thereforethese functors form a projective resolution of the functor X ⊗ X ∼ = X in thecategory Rex( C , C ). The result now follows from the comparison theorem (cf. [Ro2,Thm. 6.16, p. 340]) applied to the category Rex( C , C ).If C is the category of finite-dimensional modules over a factorizable Hopf algebradiscussed in Paragraph 2.2, the preceding proof can be given a more explicit form.By the Eilenberg-Watts theorem already mentioned in the proof of Proposition 2.1,right exact functors can in this case be represented by tensoring with bimodules. Inparticular, the functor X P m ⊗ X is represented as X ( P m ⊗ A ) ⊗ A X , wherethe space P m ⊗ A carries the A -bimodule structure a. ( p ⊗ a ′ ) .a ′′ = a (1) .p ⊗ a (2) a ′ a ′′ for p ∈ P m and a, a ′ , a ′′ ∈ A . We then have two liftings of θ A , which appear in thediagram A P ⊗ A P ⊗ A · · · A P ⊗ A P ⊗ A · · · θ A ξ ⊗ id A id P ⊗ θ A θ P ⊗ A d ⊗ id A id P ⊗ θ A θ P ⊗ A d ⊗ id A ξ ⊗ id A d ⊗ id A d ⊗ id A and have the explicit form θ P m ⊗ A ( p ⊗ a ) = v (1) .p ⊗ v (2) a and (id P m ⊗ θ A )( p ⊗ a ) = p ⊗ va in terms of the ribbon element v .Now we can apply the more standard comparison theorem for bimodules, consideredas A ⊗ A op -modules, to see that these two liftings are chain-homotopic via a chainhomotopy h ′ m : P m ⊗ A → P m +1 ⊗ A . The mappings h m ( X ) are then induced fromthe mappings h ′ m ⊗ A id X via the isomorphism ( P m ⊗ A ) ⊗ A X ∼ = P m ⊗ X .58 .2 Derived block spaces Our goal now is to extend the projective action of the pure mapping class group PΓ g,n on the block spaces Z (Σ X ,...,X n g,n ) := Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g )to the derived block spaces Z m (Σ X ,...,X n g,n ) := Ext m ( X n ⊗ · · · ⊗ X , L ⊗ g ) . To do this, we consider the surface Σ g,n +1 with one additional boundary component,and choose a projective resolution ξ ←−− P d ←−− P d ←−− P d ←−− · · · of the unit object. As we mentioned in Paragraph 2.1, tensoring is exact and preservesprojectives, so that ⊗ X n ⊗ · · · ⊗ X ←−− P ⊗ X n ⊗ · · · ⊗ X ←−− P ⊗ X n ⊗ · · · ⊗ X ←−− · · · is a projective resolution of X n ⊗ · · · ⊗ X ∼ = ⊗ X n ⊗ · · · ⊗ X . If we use theabbreviation Γ g,n +1 ( n + 1) for the mapping class group Γ g,n +1 (Im( ρ n +1 )), an ele-ment [ ψ ] of this group fixes the n + 1-st boundary component. Then the permutation τ := p ([ ψ ]) − ∈ S n +1 introduced in Paragraph 1.3 fixes n + 1, and therefore eachrepresentative Z ( ψ ) of the associated projective class [ Z ( ψ )] yields by naturality acochain homomorphism between the cochain complexes Z (Σ X ,...,X n ,P m g,n +1 ) = Hom C ( P m ⊗ X n ⊗ · · · ⊗ X , L ⊗ g )and Z (Σ X τ (1) ,...,X τ ( n ) ,P m g,n +1 ) = Hom C ( P m ⊗ X τ ( n ) ⊗ · · · ⊗ X τ (1) , L ⊗ g )and so induces a homomorphism Z m ( ψ ) : Ext m ( X n ⊗ · · · ⊗ X , L ⊗ g ) → Ext m ( X τ ( n ) ⊗ · · · ⊗ X τ (1) , L ⊗ g )between the derived block spaces. Choosing a different representative of the pro-jective class clearly rescales Z m ( ψ ) by a nonzero scalar, so that the projectiveclass [ Z m ( ψ )] is well-defined. 59owever, we have associated this homomorphism between derived block spaces with amapping class [ ψ ] ∈ Γ g,n +1 ( n + 1), whereas the homomorphisms between the originalblock spaces were associated with a mapping class [ ψ ] ∈ Γ g,n . As we will shownow, we can also associate homomorphisms between the derived block spaces with amapping class [ ψ ] ∈ Γ g,n , namely by choosing a preimage in Γ g,n +1 ( n + 1) under thehomomorphism D n +1 defined at the end of Paragraph 1.9. For this, we obviously needto show that the arising homomorphisms are independent of the chosen preimage.We begin with a few auxiliary results: Lemma 3.2.
For the Dehn twist d n +1 ∈ Γ g,n +1 ( n + 1), we have [ Z m ( d n +1 )] = [id]. Proof.
According to Paragraph 2.6, the Dehn twist d n +1 acts by precompositionwith the morphism θ X n +1 ⊗ id X n ⊗···⊗ X . By [Ka, Lem. XIV.3.3, p. 350], we have θ = id . As we have discussed already at the end of Paragraph 3.1, the diagram P P · · · P P · · · θ ξ θ P d θ P d ξ d d commutes by the naturality of the twist. On the other hand, it is obvious that thediagram P P · · · P P · · · id ξ id P d id P d ξ d d commutes. Therefore, both the family ( θ P m ) and the family (id P m ) lift the morphism θ = id to the projective resolution. By the comparison theorem, the two lifts arechain-homotopic. This chain homotopy induces a cochain homotopy on the cochaincomplex Hom C ( P m ⊗ X n ⊗ · · · ⊗ X , L ⊗ g ), which yields that the two maps induce thesame map in cohomology, namely the identity.We will need another lemma of a similar nature: Lemma 3.3.
For the two Dehn twists d n and d n,n +1 in Γ g,n +1 ( n + 1), we have[ Z m ( d n )] = [ Z m ( d n,n +1 )]. 60 roof. From Lemma 3.1, we know that the chain maps θ P m ⊗ X n and id P m ⊗ θ X n arechain-homotopic. This implies that the chain maps θ P m ⊗ X n ⊗ id X n − ⊗···⊗ X and id P m ⊗ θ X n ⊗ id X n − ⊗···⊗ X are chain-homotopic. Because d n,n +1 acts by precomposition with the first one and d n acts by precomposition with the second one, this implies the assertion.We now consider the curve α i introduced in Paragraph 1.1, which begins and ends inthe base point x for the surface Γ g,n . If we push it off the base point to the left andto the right as described in Paragraph 1.6, we can cut out a small disk centered at x without intersecting these two curves, which we consider as the n + 1-st boundarycomponent. In this way, we obtain two curves α ′ i and α ′′ i on the surface Γ g,n +1 : ... ... x i − ii + 1 g ... α ′ i α ′′ i σ We now label the new boundary component with the elements of our projectiveresolution; i.e., for a given m , we set X n +1 := P m . We want to show that the twoDehn twists t ′ i := d α ′ i and t ′′ i := d α ′′ i induce the same map in cohomology. As we willsee below, it is sufficient to consider the case i = 1: Lemma 3.4.
For the two Dehn twists t ′ := d α ′ and t ′′ := d α ′′ in Γ g,n +1 ( n + 1), wehave [ Z m ( t ′ )] = [ Z m ( t ′′ )]. 61 roof. (1) If we cut the surface along the curve that is denoted by σ in the pictureabove, we obtain a surface that is diffeomorphic to Σ g − ,n +3 : ... ... x ... γ ′ γ ′′ Here, we consider the lower boundary component arising from the cut in the pictureabove as the n + 2-nd one and the upper boundary component as the n + 3-rd one.The surface Σ g,n +1 can be reconstructed from the surface Σ g − ,n +3 by gluing in ahandle, as described in Paragraph 2.7. Upon gluing, the Dehn twists along thecurves denoted by γ ′ and γ ′′ in the second picture become the Dehn twists alongthe curves denoted by α ′ and α ′′ in the first picture, respectively. As we saw inParagraph 2.7, the diagram Z (Σ X ,...,X n ,P m ,X ∗∗ ,X ∗ g − ,n +3 ) Z (Σ X ,...,X n ,P m g,n +1 ) Z (Σ X ,...,X n ,P m ,X ∗∗ ,X ∗ g − ,n +3 ) Z (Σ X ,...,X n ,P m g,n +1 ) Z ( d γ ′ ) Z ( t ′ ) commutes for our choices of the representatives Z ( d γ ′ ) and Z ( t ′ ) of the projectiveclasses, because upon appropriate labeling d γ ′ is d n +2 and t ′ is t in the notationused there. From the discussion in Paragraph 2.7, we know that a similar diagramcommutes for d γ ′′ and t ′′ . Our goal is to show that Z ( t ′ ) and Z ( t ′′ ) induce the samemap in cohomology.(2) To see this, we apply Lemma 3.1 to the so-called reversed category, in whichtensor products are taken in the opposite order. In this way, we obtain for each62bject X ∈ C a chain homotopy h m ( X ) : X ⊗ P m → X ⊗ P m +1 between the chainmaps ( θ X ⊗ P m ) and ( θ X ⊗ id P m ) that is natural in X . For two objects X and Y of C , theDehn twists d γ ′ and d γ ′′ act on the space Hom C ( Y ∗ ⊗ X ∗∗ ⊗ P m ⊗ X n ⊗· · ·⊗ X , L ⊗ ( g − )by precomposition with id Y ∗ ⊗ θ X ∗∗ ⊗ id P m ⊗ Z and id Y ∗ ⊗ θ X ∗∗ ⊗ P m ⊗ id Z , respectively,where, for brevity, we have used the notation Z := X n ⊗ · · · ⊗ X . By naturality, theadjunction isomorphismHom C ( Y ∗ ⊗ X ∗∗ ⊗ P m ⊗ Z, L ⊗ ( g − ) → Hom C ( P m ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − )is an isomorphism of cochain complexes, so that we obtain a cochain map ( f ′ mX,Y )of the cochain complex on the right-hand side whose defining property is that thediagramHom C ( Y ∗ ⊗ X ∗∗ ⊗ P m ⊗ Z, L ⊗ ( g − ) Hom C ( P m ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − )Hom C ( Y ∗ ⊗ X ∗∗ ⊗ P m ⊗ Z, L ⊗ ( g − ) Hom C ( P m ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − ) = ◦ (id Y ∗ ⊗ θ X ∗∗ ⊗ id Pm ⊗ Z ) Z ( d γ ′ ) f ′ mX,Y commutes. For γ ′′ , there is a second cochain map ( f ′′ mX,Y ) that makes a very similardiagram commutative. For γ ′ , the naturality of the adjunction isomorphism im-plies that f ′ mX,Y is given by postcomposition with θ X ∗ ⊗ id Y ⊗ id L ⊗ ( g − . The chainhomotopy (id Y ∗ ⊗ h m ( X ∗∗ ) ⊗ id Z ) induces a cochain homotopy h ′ m ( X, Y ) : Hom C ( P m ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − ) → Hom C ( P m − ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − )that is natural in X and Y and makes the diagramHom C ( Y ∗ ⊗ X ∗∗ ⊗ P m ⊗ Z, L ⊗ ( g − ) Hom C ( P m ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − )Hom C ( Y ∗ ⊗ X ∗∗ ⊗ P m − ⊗ Z, L ⊗ ( g − ) Hom C ( P m − ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − ) ◦ (id Y ∗ ⊗ h m − ( X ∗∗ ) ⊗ id Z ) h ′ m ( X,Y ) commutative.(3) Now the functor Hom C ( P m ⊗ Z, − ⊗ L ⊗ ( g − ) is an exact functor from the cat-egory C to the category V of finite-dimensional vector spaces. By Proposition 2.1,it has a right adjoint and therefore preserves coends, as already mentioned in Para-graph 2.3. Therefore, the family of morphisms( ι X ⊗ id L ⊗ ( g − ) ◦ : Hom C ( P m ⊗ Z, X ∗ ⊗ X ⊗ L ⊗ ( g − ) → Hom C ( P m ⊗ Z, L ⊗ g )63s a coend for the bifunctor C op × C → V , ( X, Y ) Hom C ( P m ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − ) . So we can apply Lemma 2.2 to this bifunctor and the corresponding bifunctorwith m − m to obtain a K -linear map h ′ m : Hom C ( P m ⊗ Z, L ⊗ g ) → Hom C ( P m − ⊗ Z, L ⊗ g )that makes the diagramHom C ( P m ⊗ Z, X ∗ ⊗ X ⊗ L ⊗ ( g − ) Hom C ( P m ⊗ Z, L ⊗ g )Hom C ( P m − ⊗ Z, X ∗ ⊗ X ⊗ L ⊗ ( g − ) Hom C ( P m − ⊗ Z, L ⊗ g ) h ′ m ( X,X ) h ′ m commutative, i.e., satisfies h ′ m (( ι X ⊗ id L ⊗ ( g − ) ◦ k ) = ( ι X ⊗ id L ⊗ ( g − ) ◦ h ′ m ( X, X )( k )for all morphisms k : P m ⊗ Z → X ∗ ⊗ X ⊗ L ⊗ ( g − .(4) By the definition of h m ( X ∗∗ ), we have θ X ∗∗ ⊗ P m − θ X ∗∗ ⊗ id P m = (id X ∗∗ ⊗ d m +1 ) ◦ h m ( X ∗∗ ) + h m − ( X ∗∗ ) ◦ (id X ∗∗ ⊗ d m )For a morphism k ′ : Y ∗ ⊗ X ∗∗ ⊗ P m ⊗ Z → L ⊗ ( g − , this implies that k ′ ◦ [(id Y ∗ ⊗ θ X ∗∗ ⊗ P m ⊗ id Z ) − (id Y ∗ ⊗ θ X ∗∗ ⊗ id P m ⊗ Z )]= k ′ ◦ [(id Y ∗ ⊗ X ∗∗ ⊗ d m +1 ⊗ id Z ) ◦ (id Y ∗ ⊗ h m ( X ∗∗ ) ⊗ id Z )+ (id Y ∗ ⊗ h m − ( X ∗∗ ) ⊗ id Z ) ◦ (id Y ∗ ⊗ X ∗∗ ⊗ d m ⊗ id Z )]In view of the definition of h ′ m ( X, Y ) and the naturality of the adjunction isomor-phism, this yields f ′′ mX,Y ( k ) − f ′ mX,Y ( k ) = h ′ m +1 ( X, Y )( k ◦ ( d m +1 ⊗ id Z )) + h ′ m ( X, Y )( k ) ◦ ( d m ⊗ id Z )for all morphisms k : P m ⊗ Z → X ∗ ⊗ Y ⊗ L ⊗ ( g − . If we set X = Y and composewith ι X ⊗ id L ⊗ ( g − , this equation becomes Z ( t ′′ ) ◦ ( ι X ⊗ id L ⊗ ( g − ) ◦ k − Z ( t ′ ) ◦ ( ι X ⊗ id L ⊗ ( g − ) ◦ k = h ′ m +1 (( ι X ⊗ id L ⊗ ( g − ) ◦ k ◦ ( d m +1 ⊗ id Z )) + h ′ m (( ι X ⊗ id L ⊗ ( g − ) ◦ k ) ◦ ( d m ⊗ id Z )64oth sides of this equation define a dinatural transformation from the bifunctor C op × C → V , ( X, Y ) Hom C ( P m ⊗ Z, X ∗ ⊗ Y ⊗ L ⊗ ( g − )already considered above to its coend Hom C ( P m ⊗ Z, L ⊗ g ). Our dinatural trans-formation factors over this coend, and the corresponding homomorphism, which isunique, can be read off directly from both the left and the right-hand side of theequation above. We get Z ( t ′′ ) ◦ k ′′ − Z ( t ′ ) ◦ k ′′ = h ′ m +1 ( k ′′ ◦ ( d m +1 ⊗ id Z )) + h ′ m ( k ′′ ) ◦ ( d m ⊗ id Z )for all k ′′ ∈ Hom C ( P m ⊗ Z, L ⊗ g ). Therefore, the family ( h ′ m ) constitutes a cochainhomotopy between the cochain maps induced by Z ( t ′ ) and Z ( t ′′ ). When saying that,it should be noted that these maps are only determined up to a scalar; we have takenhere for Z ( t ′ ) the representative that corresponds to the dinatural transformation k ( ι X ⊗ id L ⊗ ( g − ) ◦ f ′ mX,X ( k ) via the universal property of the coend of our bifunctor,and have made a similar choice for the representative of Z ( t ′′ ) using f ′′ mX,X .By putting these auxiliary results together, we can now associate with a mappingclass in Γ g,n a projective class of morphisms not only between the original blockspaces, but rather between the derived block spaces. As already stated above, wedo this by choosing a preimage under the epimorphism D n +1 defined at the end ofParagraph 1.9. The key fact that we need to prove is therefore the following: Theorem 3.5.
Suppose that [ ψ ] , [ ψ ′ ] ∈ Γ g,n +1 ( n + 1) satisfy D n +1 ([ ψ ]) = D n +1 ([ ψ ′ ]).Then we have [ Z m ( ψ )] = [ Z m ( ψ ′ )]. Proof. (1) Clearly, [ ψ ] and [ ψ ′ ] differ by an element in the kernel of D n +1 , whichis in particular an element in the pure mapping class group PΓ g,n +1 . It thereforesuffices to show that [ Z m ( ψ )] = [id] for each mapping class [ ψ ] in the kernel of D n +1 .Recall that D n +1 is the compositionΓ g,n +1 ( n + 1) C n +1 −→ Γ g,n ( y ) F y −→ Γ g,n of the capping homomorphism, which arises from gluing a punctured disk in placeof the missing disk, and the forgetful map F y that, depending on the perspective,either fills this puncture or ‘forgets’ that the point was marked. According to Propo-sition 1.8, the kernel of C n +1 is generated by the Dehn twist d n +1 . But by Lemma 3.2above, we have [ Z m ( d n +1 )] = [id]. This implies that the assignment [ ψ ] [ Z m ( ψ )]is well-defined for [ ψ ] ∈ Γ g,n ( y ), not only for [ ψ ] ∈ Γ g,n +1 ( n + 1).652) To treat the second morphism F y in the above composition, we use the Birmansequence from Paragraph 1.10. In order to do this, we proceed as in Paragraph 1.12and first replace the additional puncture y by the base point x of the fundamentalgroup that comes from the polygon model of the surface as the identification of allthe vertices of the polygon, except for those that correspond to the marked pointson the boundary components. The map that forgets the base point x instead of thepuncture will be denoted by F x instead of F y . The Birman sequence then takes theform π (Σ g,n , x ) P x −→ Γ g,n ( x ) F x −→ Γ g,n −→ , where, as before, P x denotes the pushing map. To complete the proof, we thereforehave to show that [ Z m ( P x ([ γ ]))] = [id] for all homotopy classes [ γ ] ∈ π (Σ g,n , x ).Because P x is a group antihomomorphism, it suffices to prove this for all [ γ ] in agenerating set of the fundamental group.(3) As discussed in Paragraph 1.2, the fundamental group π (Σ g,n , x ) is generatedby the homotopy classes of the simple closed curves α , β , . . . , α g , β g , all of which arenonseparating, together with the curves δ , . . . , δ n , all of which are separating. Froma formula stated at the end of Paragraph 1.10, we know that P x ([ α ]) = [ d α ′ d − α ′′ ].Therefore Lemma 3.4 yields that [ Z m ( P x ([ α ]))] = [id].(4) For any of the other generators that correspond to nonseparating curves, say β i ,we use the change of coordinates principle (cf. [FM, Par. 1.3.1, p. 37]) to obtain a dif-feomorphism ϕ : Σ g,n → Σ g,n satisfying ϕ ( α ) = β i . The discussion in [FM, loc. cit.]shows that we can assume that ϕ is orientation-preserving and satisfies ϕ ( x ) = x .From Paragraph 1.10, we know that [ P x ([ β i ])] = [ ϕ ][ P x ([ α ])][ ϕ − ]. Now the com-patibility with composition described in Paragraph 2.6 implies that[ Z m ( P x ([ β i ]))] = [ Z m ( ϕ )][ Z m ( P x ([ α ]))][ Z m ( ϕ − )] = [ Z m ( ϕ )][ Z m ( ϕ − )] = [id](5) For the separating curves δ , . . . , δ n , the formula already mentioned above yieldsthat P x ([ δ j ]) = [ d δ ′ j d − δ ′′ j ]. We have C n +1 ([ d n ]) = [ d δ ′ n ] and C n +1 ([ d n,n +1 ]) = [ d δ ′′ n ].Therefore, Lemma 3.3 shows that [ Z m ( P x ([ δ n ]))] = [id]. If j = n , we proceed asabove and use the general change of coordinate principle (cf. [FM, loc. cit.]) to obtaina diffeomorphism ϕ : Σ g,n → Σ g,n with ϕ ( δ n ) = δ j that is orientation-preserving andsatisfies ϕ ( x ) = x . As above, we get [ P x ([ δ j ])] = [ ϕ ][ P x ([ δ n ])][ ϕ − ], and a very similarcomputation as there then yields [ Z m ( P x ([ δ j ]))] = [id].66y construction, a mapping class that permutes the boundary components does ingeneral not carry a derived block space into itself. However, the pure mapping classgroup preserves these spaces: Corollary 3.6.
There is a projective action of PΓ g,n on Z m (Σ X ,...,X n g,n ) = Ext m ( X n ⊗ · · · ⊗ X , L ⊗ g )that agrees with the original action on Z (Σ X ,...,X n g,n ) = Hom C ( X n ⊗ · · · ⊗ X , L ⊗ g )if m = 0.As in Paragraph 2.6, the entire mapping class acts projectively on the direct sum M τ ∈ S n Z m (Σ X τ (1) ,...,X τ ( n ) g,n )of derived block spaces. If all boundary components have the same label X , themapping class group indeed acts projectively on a single derived block space: Corollary 3.7.
For X ∈ C , there is a projective action of Γ g,n on Ext m ( X ⊗ n , L ⊗ g )that agrees with the original action on Hom C ( X ⊗ n , L ⊗ g ) if m = 0. The construction of the mapping class group representations in the preceding para-graph was not based on generators and relations for the mapping class group, as sucha presentation of the mapping class group is notoriously difficult. In the case g = 0,however, we gave such a presentation in Paragraph 1.13. It is instructive to see whythe defining relations are satisfied in this case. In order to verify these relations, wewill need a couple of lemmas. As stated in Paragraph 2.1, we are assuming that ourcategory is strict. Lemma 3.8.
Suppose that X , . . . , X n are objects of C . Then we have1. c X ⊗···⊗ X n − ,X n = ( c X ,X n ⊗ id X ⊗···⊗ X n − ) ◦ (id X ⊗ c X ,X n ⊗ id X ⊗···⊗ X n − ) ◦ · · · ◦ (id X ⊗···⊗ X n − ⊗ c X n − ,X n )2. c X ,X ⊗···⊗ X n = (id X ⊗···⊗ X n − ⊗ c X ,X n ) ◦ · · · ◦ (id X ⊗ c X ,X ⊗ id X ⊗···⊗ X n ) ◦ ( c X ,X ⊗ id X ⊗···⊗ X n )67. c X n ,X ⊗···⊗ X n − ◦· · ·◦ c X ,X ⊗···⊗ X n ⊗ X ◦ c X ,X ⊗···⊗ X n = θ X ⊗···⊗ X n ◦ ( θ − X ⊗· · ·⊗ θ − X n )4. c X ⊗···⊗ X n ,X ◦ c X ⊗···⊗ X n ⊗ X ,X ◦· · ·◦ c X ⊗···⊗ X n − ,X n = θ X ⊗···⊗ X n ◦ ( θ − X ⊗· · ·⊗ θ − X n ) Proof. (1) The first assertion is clearly correct for n = 2 and then follows induc-tively from the equation c X ⊗···⊗ X n − ,X n = ( c X ⊗···⊗ X n − ,X n ⊗ id X n − ) ◦ (id X ⊗···⊗ X n − ⊗ c X n − ,X n ) . (2) Associated with every braiding is another braiding in which c X,Y is replacedby c − Y,X . If we apply the first assertion to this braiding instead, take inverses, andpermute X , . . . , X n cyclically, we obtain the second assertion.(3) To prove the third assertion, we need the auxiliary statement that c X k ,X k +1 ⊗···⊗ X n ⊗ X ⊗···⊗ X k − ◦ · · · ◦ c X ,X ⊗···⊗ X n ⊗ X ◦ c X ,X ⊗···⊗ X n = c X ⊗···⊗ X k ,X k +1 ⊗···⊗ X n ◦ ( θ X ⊗···⊗ X k ⊗ id X k +1 ⊗···⊗ X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X k ⊗ id X k +1 ⊗···⊗ X n )for k = 1 , . . . , n −
1. For k = 1, this is obvious. For k ≤ n −
2, we have inductively c X k +1 ,X k +2 ⊗···⊗ X n ⊗ X ⊗···⊗ X k ◦ c X ⊗···⊗ X k ,X k +1 ⊗···⊗ X n ◦ ( θ X ⊗···⊗ X k ⊗ id X k +1 ⊗···⊗ X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X k ⊗ id X k +1 ⊗···⊗ X n )= (id X k +2 ⊗···⊗ X n ⊗ c X k +1 ,X ⊗···⊗ X k ) ◦ ( c X k +1 ,X k +2 ⊗···⊗ X n ⊗ id X ⊗···⊗ X k ) ◦ (id X k +1 ⊗ c X ⊗···⊗ X k ,X k +2 ⊗···⊗ X n ) ◦ ( c X ⊗···⊗ X k ,X k +1 ⊗ id X k +2 ⊗···⊗ X n ) ◦ ( θ X ⊗···⊗ X k ⊗ θ X k +1 ⊗ id X k +2 ⊗···⊗ X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X k +1 ⊗ id X k +2 ⊗···⊗ X n ) . By using the Yang-Baxter equation (cf. [Ka, Thm. XIII.1.3, p. 317]) on the first threeterms, this expression becomes( c X ⊗···⊗ X k ,X k +2 ⊗···⊗ X n ⊗ id X k +1 ) ◦ (id X ⊗···⊗ X k ⊗ c X k +1 ,X k +2 ⊗···⊗ X n ) ◦ ( c X k +1 ,X ⊗···⊗ X k ⊗ id X k +2 ⊗···⊗ X n ) ◦ ( c X ⊗···⊗ X k ,X k +1 ⊗ id X k +2 ⊗···⊗ X n ) ◦ ( θ X ⊗···⊗ X k ⊗ θ X k +1 ⊗ id X k +2 ⊗···⊗ X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X k +1 ⊗ id X k +2 ⊗···⊗ X n ) , which by the defining property of a ribbon structure is equal to( c X ⊗···⊗ X k ,X k +2 ⊗···⊗ X n ⊗ id X k +1 ) ◦ (id X ⊗···⊗ X k ⊗ c X k +1 ,X k +2 ⊗···⊗ X n ) ◦ ( θ X ⊗···⊗ X k ⊗ X k +1 ⊗ id X k +2 ⊗···⊗ X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X k +1 ⊗ id X k +2 ⊗···⊗ X n )= c X ⊗···⊗ X k +1 ,X k +2 ⊗···⊗ X n ◦ ( θ X ⊗···⊗ X k +1 ⊗ id X k +2 ⊗···⊗ X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X k +1 ⊗ id X k +2 ⊗···⊗ X n ) , completing our inductive step. 684) To derive the third assertion, we now use the case k = n − c X n ,X ⊗···⊗ X n − to obtain c X n ,X ⊗···⊗ X n − ◦ c X n − ,X n ⊗ X ⊗···⊗ X n − ◦ · · · ◦ c X ,X ⊗···⊗ X n ⊗ X ◦ c X ,X ⊗···⊗ X n = c X n ,X ⊗···⊗ X n − ◦ c X ⊗···⊗ X n − ,X n ◦ ( θ X ⊗···⊗ X n − ⊗ id X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X n − ⊗ id X n )= c X n ,X ⊗···⊗ X n − ◦ c X ⊗···⊗ X n − ,X n ◦ ( θ X ⊗···⊗ X n − ⊗ θ X n ) ◦ ( θ − X ⊗ · · · ⊗ θ − X n )= θ X ⊗···⊗ X n ◦ ( θ − X ⊗ · · · ⊗ θ − X n )again by the defining property of a ribbon structure.(5) A ribbon structure for the braiding c − Y,X considered in the proof of the secondassertion is given by the morphisms θ − X : X → X . If we apply the third assertion tothis braiding and this ribbon structure instead and then take inverses, we obtain thefourth assertion.Another lemma that we will need concerns precomposition with the twist: Lemma 3.9.
For all f ∈ Hom C ( X ⊗ Y, ), we have f ◦ ( θ X ⊗ id Y ) = f ◦ (id X ⊗ θ Y ).More generally, precomposition with θ X ⊗ id Y and precomposition with id X ⊗ θ Y induce the same homomorphism on Ext m ( X ⊗ Y, ). Proof. (1) As we recorded in Paragraph 2.1, the functor – ⊗ Y ∗ is the right adjointof the functor – ⊗ Y . The adjunction is given by η : Hom C ( X ⊗ Y, ) → Hom C ( X, Y ∗ ) , f ( f ⊗ id Y ∗ ) ◦ (id X ⊗ coev Y ) . Therefore, the diagramHom C ( X ⊗ Y, ) Hom C ( X ⊗ Y, )Hom C ( X, Y ∗ ) Hom C ( X, Y ∗ ) ◦ ( θ X ⊗ id Y ) η η ◦ θ X commutes. On the other hand, also the diagramHom C ( X ⊗ Y, ) Hom C ( X ⊗ Y, )Hom C ( X, Y ∗ ) Hom C ( X, Y ∗ ) ◦ (id X ⊗ θ Y ) η η ( θ Y ∗ ) ◦ η ( f ◦ (id X ⊗ θ Y )) = ( f ⊗ id Y ∗ ) ◦ (id X ⊗ θ Y ⊗ id Y ∗ ) ◦ (id X ⊗ coev Y )= ( f ⊗ id Y ∗ ) ◦ (id X ⊗ id Y ⊗ θ ∗ Y ) ◦ (id X ⊗ coev Y )= ( f ⊗ θ ∗ Y ) ◦ (id X ⊗ coev Y ) = θ ∗ Y ◦ ( f ⊗ id Y ∗ ) ◦ (id X ⊗ coev Y )= θ ∗ Y ◦ η ( f ) = θ Y ∗ ◦ η ( f ) . But by the naturality of the twist, we have θ Y ∗ ◦ g = g ◦ θ X for g ∈ Hom C ( X, Y ∗ ), sothat the bottom rows in the two diagrams are equal. By comparing the two diagrams,we obtain the first assertion.(2) From our projective resolution ←−− P ←−− P ←−− P ←−− · · · of the unit object, we get a projective resolution of X by tensoring with X , i.e., bydefining Q m := P m ⊗ X , and even a projective resolution of X ⊗ Y by tensoringagain with Y . Then X ⊗ Y Q ⊗ Y Q ⊗ Y · · · X ⊗ Y Q ⊗ Y Q ⊗ Y · · · id X ⊗ θ Y id Q ⊗ θ Y id Q ⊗ θ Y is a lift of id X ⊗ θ Y to this projective resolution. On the other hand, a lift of θ X ⊗ id Y is given by X ⊗ Y Q ⊗ Y Q ⊗ Y · · · X ⊗ Y Q ⊗ Y Q ⊗ Y · · · θ X ⊗ id Y θ Q ⊗ id Y θ Q ⊗ id Y So the action of id X ⊗ θ Y on Ext m ( X ⊗ Y, ) is induced by precomposition withid Q m ⊗ θ Y on Hom C ( Q m ⊗ Y, ), while the action of θ X ⊗ id Y is induced by pre-composition with θ Q m ⊗ id Y on this space. But by the preceding step, these twoprecomposition maps are equal. 70he results of Paragraph 3.2 yield in the case g = 0 that the projective action ofΓ ,n +1 ( n + 1) on the direct sum M τ ∈ S n Ext m ( X τ ( n ) ⊗ · · · ⊗ X τ (1) , )descends to a projective action of Γ ,n . In fact, the action is in this case not onlyprojective, but rather an ordinary linear action. As we will explain now, we can seethis explicitly from the presentations of these groups that we gave in Paragraph 1.13.According to Proposition 1.9, we haveΓ ,n +1 ( n + 1) ∼ = Z n ⋊ B n It is not difficult to check that the defining relations of Z n ⋊ B n discussed there arestrictly, not only projectively, satisfied on the direct sum M τ ∈ S n Hom C ( P m ⊗ X τ ( n ) ⊗ · · · ⊗ X τ (1) , ) . To show that the action descends to an action of Γ ,n , we have to show that theadditional relations b , b , · · · b n − ,n · · · b , b , = d − and ( b , b , · · · b n − ,n ) n = d − d − · · · d − n hold on this space. It is sufficient to verifythe relations on a single summand of the direct sum: Proposition 3.10.
For f ∈ Hom C ( P m ⊗ X n ⊗ · · · ⊗ X , ), we have Z m ( b , b , · · · b n − ,n · · · b , b , )( ¯ f ) = Z m ( d − )( ¯ f )as well as Z m (( b , b , · · · b n − ,n ) n )( ¯ f ) = Z m ( d − d − · · · d − n )( ¯ f ) . Proof. (1) It follows from the second assertion in Lemma 3.8 that Z ( b n − ,n · · · b , b , )( f )= f ◦ (id P m ⊗ X n ⊗···⊗ X ⊗ c X ,X ) ◦ (id P m ⊗ X n ⊗···⊗ X ⊗ c X ,X ⊗ id X ) ◦ · · · ◦ (id P m ⊗ c X ,X n ⊗ id X n − ⊗···⊗ X )= f ◦ (id P m ⊗ c X ,X n ⊗···⊗ X ) 71nd then from the first assertion there that Z ( b , b , · · · b n − ,n · · · b , b , )( f )= f ◦ (id P m ⊗ c X ,X n ⊗···⊗ X ) ◦ (id P m ⊗ c X n ,X ⊗ id X n − ⊗···⊗ X ) ◦ · · · ◦ (id P m ⊗ X n ⊗···⊗ X ⊗ c X ,X ⊗ id X ) ◦ (id P m ⊗ X n ⊗···⊗ X ⊗ c X ,X )= f ◦ (id P m ⊗ c X ,X n ⊗···⊗ X ) ◦ (id P m ⊗ c X n ⊗···⊗ X ,X )= f ◦ (id P m ⊗ θ X n ⊗···⊗ X ) ◦ (id P m ⊗ θ − X n ⊗···⊗ X ⊗ θ − X ) , while Z ( d − )( f ) = f ◦ (id P m ⊗ id X n ⊗···⊗ X ⊗ θ − X ). So, if we introduce the abbreviations X := X n ⊗ · · · ⊗ X and Y := X , we have to show that the two cochain maps f f ◦ (id P m ⊗ θ X ⊗ Y ) ◦ (id P m ⊗ θ − X ⊗ θ − Y )and f f ◦ (id P m ⊗ id X ⊗ θ − Y ) of the cochain complex (Hom C ( P m ⊗ X ⊗ Y, )) inducethe same homomorphism in cohomology.By the first assertion in Lemma 3.9, the cochain map f f ◦ (id P m ⊗ θ X ⊗ Y ) isequal to the cochain map f f ◦ ( θ P m ⊗ id X ⊗ Y ), and we saw in Lemma 3.2 thatthis cochain map is homotopic to the identity. This shows that our assertion willfollow if we can show that the two cochain maps f f ◦ (id P m ⊗ θ − X ⊗ id Y ) and f f ◦ (id P m ⊗ id X ⊗ θ − Y ) induce the same map in cohomology. But this followsdirectly from the second assertion in Lemma 3.9 by taking inverses.(2) To prove the second relation, we note that the first assertion in Lemma 3.8 gives Z ( b , b , · · · b n − ,n )( f )= f ◦ (id P m ⊗ c X n − ,X n ⊗ id X n − ⊗···⊗ X ) ◦ · · ·◦ (id P m ⊗ id X n − ⊗···⊗ X ⊗ c X ,X n ⊗ id X ) ◦ (id P m ⊗ id X n − ⊗···⊗ X ⊗ c X ,X n )= f ◦ (id P m ⊗ c X n − ⊗···⊗ X ,X n )and therefore the fourth assertion of this lemma yields Z (( b , b , · · · b n − ,n ) n )( f )= f ◦ (id P m ⊗ c X n − ⊗···⊗ X ,X n ) ◦ (id P m ⊗ c X n − ⊗···⊗ X ⊗ X n ,X n − ) ◦ · · · ◦ (id P m ⊗ c X n ⊗···⊗ X ,X )= f ◦ (id P m ⊗ θ X n ⊗···⊗ X ) ◦ (id P m ⊗ θ − X n ⊗ · · · ⊗ θ − X )As in the case of the first relation, the cochain map f f ◦ (id P m ⊗ θ X n ⊗···⊗ X ) is equalto the cochain map f f ◦ ( θ P m ⊗ id X n ⊗···⊗ X ) by the first assertion in Lemma 3.9,72hich is homotopic to the identity by Lemma 3.2. Therefore, our cochain map ishomotopic to the cochain map Z ( d − d − · · · d − n )( f ) = f ◦ (id P m ⊗ θ − X n ⊗ · · · ⊗ θ − X )as required. In order to explain why Corollary 3.7 generalizes the main result of our previousarticle (cf. [LMSS1]), we now specialize the situation to the case where C is thecategory of left modules over the factorizable ribbon Hopf algebra A described inParagraph 2.2, and furthermore to the case where g = 1 and n = 0. From Para-graph 1.12, we know that Γ , ∼ = SL(2 , Z ) and Γ , ∼ = B . From Corollary 3.7, wethen obtain a projective action of Γ , ∼ = SL(2 , Z ) on Ext mA ( K, L ), which arises asa quotient of the projective action of Γ , ∼ = B on Hom A ( P m , L ) for a projectiveresolution K ←−− P ←−− P ←−− P ←−− · · · of the base field K , considered as a trivial A -module, as explained in Paragraph 2.2.That the projective action of Γ , indeed descends to a projective action of Γ , on the cohomology groups can be seen quite explicitly in this case: According toParagraph 2.6, the generators s and t of Γ , act by postcomposition with S and T , respectively, and we have seen in Paragraph 2.8 that they satisfy the identity S ◦ T ◦ S = ρ ( v ) T − ◦ S ◦ T − , which means that they satisfy the defining identityof Γ , ∼ = B projectively. We have also seen there that the 2-chain relation is satis-fied projectively, i.e., that the actions of s and d − agree up to a scalar. It thereforefollows from Lemma 3.2 that s acts on the cohomology groups as a scalar multi-ple of the identity, so that the defining relations of the modular group are satisfiedprojectively.In order to relate this result to Hochschild cohomology, we choose our projectiveresolution of the base field in a special way. We first choose a projective resolution A ←−− Q ←−− Q ←−− Q ←−− · · · of the algebra A in the category of left A ⊗ A op -modules, or equivalently the categoryof A -bimodules, where we require, as in [CE, Chap. IX, §
3, p. 167], that the left andthe right action of A on a bimodule become equal when restricted to K . By [CE,73hap. X, Thm. 2.1, p. 185], we can then set P m := Q m ⊗ A K to obtain a projectiveresolution of A ⊗ A K ∼ = K .Now the left A -module L can be considered as an A -bimodule via the trivial right A -action, i.e., the action ϕ.a := ε ( a ) ϕ for ϕ ∈ L = A ∗ . We denote L by L ε if it isconsidered as an A -bimodule in this way. With the help of the cochain mapHom A ⊗ A op ( Q m , L ε ) → Hom A ( Q m ⊗ A K, L ) , f ( q ⊗ λ λf ( q ))with inverseHom A ( Q m ⊗ A K, L ) → Hom A ⊗ A op ( Q m , L ε ) , g ( q g ( q ⊗ K ))we see that the cochain complexes Hom A ( Q m ⊗ A K, L ) and Hom A ⊗ A op ( Q m , L ε ) areisomorphic, so that also their cohomology groups Ext mA ( K, L ) and Ext mA ⊗ A op ( A, L ε )are isomorphic. But the latter cohomology groups are, by definition, the Hochschildcohomology groups HH m ( A, L ε ).Via this isomorphism of cochain complexes, we can transfer the action of Γ , to theHochschild cochain groups of the bimodule L ε . The generators s and t of Γ , thenclearly still act by postcomposition with the bimodule homomorphisms S and T ,while the Dehn twist d acts by precomposition with the endomorphism q v.q of Q m , or equivalently by postcomposition with the endomorphism ϕ v.ϕ of L ε .From this correspondence, we see that our Lemma 3.2 can be considered as ananalogue of Proposition 1.2 in our previous treatment [LMSS1].While the two results are analogous, it is not yet clear that the projective represen-tations of the modular group constructed here and in [LMSS1, Cor. 5.6, p. 419] areisomorphic; in fact, the action constructed above is on the space HH m ( A, L ε ), whilethe action constructed in [LMSS1] is on the space HH m ( A, A ). However, it turnsout that the two actions are indeed isomorphic:
Proposition 3.11.
The projective actions of SL(2 , Z ) on the spaces HH m ( A, L ε )and HH m ( A, A ) are isomorphic.
Proof. (1) In general, for an A -bimodule N , we can modify the bimodule structureby defining the new left action as a.n := n.S − ( a )and similarly the new right action as n.a := S − ( a ) .n . We denote N by N ′ if we con-sider it endowed with this bimodule structure. From the point of view of bimodulesas A ⊗ A op -modules, this operation is the pullback along the ring homomorphism A ⊗ A op → A ⊗ A op , a ⊗ b S − ( b ) ⊗ S − ( a ) . A ′ ←−− Q ′ ←−− Q ′ ←−− Q ′ ←−− · · · is a projective resolution of A ′ , and the cochain complex Hom A ⊗ A op ( Q ′ m , N ′ ) is notonly isomorphic to the cochain complex Hom A ⊗ A op ( Q m , N ), but even set-theoreticallyequal. We therefore have that Ext mA ⊗ A op ( A ′ , N ′ ) = Ext mA ⊗ A op ( A, N ).(3) The antipode yields a bimodule isomorphism S : A ′ → A , which by the compar-ison theorem lifts to a chain map A ′ Q ′ Q ′ · · · A Q Q · · · S S S By pulling back along the chain map ( S m ), we obtain an isomorphismExt mA ⊗ A op ( A, N ′ ) → Ext mA ⊗ A op ( A ′ , N ′ ) = Ext mA ⊗ A op ( A, N ) . In other words, we obtain an isomorphism between HH m ( A, N ′ ) and HH m ( A, N ).(4) We know from [LMSS1, Lem. 5.2, p. 417] and the subsequent discussion that theprojective action of SL(2 , Z ) on HH m ( A, A ) is isomorphic to the one on the coho-mology groups HH m ( A, ε A ad ), where the generators s and t act by postcompositionwith ˆ S and ˆ T , respectively. Here, ε A ad is the A -bimodule with A as underlying vec-tor space, trivial left action a.a ′ := ε ( a ) a ′ , and right action ad( a ⊗ a ′ ) = S ( a ′ (1) ) aa ′ (2) (cf. [LMSS1, Sec. 2, p. 406]). We note that the preceding discussion shows that ε A ad itself is not an SL(2 , Z )-module, but only a Γ , -module; however, the inducedprojective Γ , -action on HH m ( A, ε A ad ) descends to a projective SL(2 , Z )-action.If we set N = ε A ad , we find that N ′ is the bimodule with underlying vector space A ,left action given by a.a ′ = a (2) a ′ S − ( a (1) ), and trivial right action. From the way howwe constructed the isomorphism between HH m ( A, N ′ ) and HH m ( A, N ) in the pre-ceding step, we see that it becomes equivariant if we let the generators s and t also acton HH m ( A, N ′ ) by postcomposition with ˆ S and ˆ T , respectively. But now Lemma 2.4shows that ¯ ι : N ′ → L ε is a bimodule isomorphism that is also equivariant under theprojective Γ , -action. Therefore postcomposition with ¯ ι yields an isomorphism be-tween HH m ( A, N ′ ) and HH m ( A, L ε ) that is SL(2 , Z )-equivariant. Combining thiswith our isomorphism between HH m ( A, ε A ad ) and HH m ( A, N ′ ) already obtained,our assertion follows. 75n view of this proposition, we can indeed say that the mapping class group actionsobtained here in Corollary 3.7 generalize the projective SL(2 , Z )-representation onthe Hochschild cohomology groups HH m ( A, A ) obtained in [LMSS1, Cor. 5.6, p. 419].
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