Topological representations of motion groups and mapping class groups -- a unified functorial construction
aa r X i v : . [ m a t h . A T ] O c t A unified functorial construction of homologicalrepresentations of families of groups
Martin Palmer and Arthur Soulié29 th October 2019
Abstract
The families of braid groups, surface braid groups, mapping class groups and loop braidgroups have a representation theory of “wild type”, so it is very useful to be able to constructsuch representations topologically , so that they may be understood by topological or geomet-ric methods. For the braid groups B n , Lawrence and Bigelow have constructed families ofrepresentations starting from actions of B n on the twisted homology of configuration spaces.These were then used by Bigelow and Krammer to prove that the braid groups are linear.We develop a general underlying procedure to build homological representations of familiesof groups, encompassing all of the above-mentioned families and in principle many more, suchas families of general motion groups. Moreover, these families of representations are coherent ,in the sense that they extend to a functor on a larger category, whose automorphism groupsare the family of groups under consideration and whose richer structure may be used (i) toorganise the representation theory of the family of groups and (ii) to prove twisted homologicalstability results — both via the notion of polynomiality . We prove polynomiality for many such homological functors , including those (which we construct) extending the Lawrence-Bigelowrepresentations.This helps to unify previously-known constructions and to produce new families of repre-sentations — we do this for the loop braid groups, surface braid groups and mapping classgroups. In particular, for the loop braid groups, we construct three analogues of the Lawrence-Bigelow representations (of the classical braid groups), which appear to be new. Contents §1. Introduction 2§2. The general construction 5§2.1 Overview 5§2.2 Twisted module homomorphisms and twisted homology 6§2.3 Lifting actions to covering spaces 11§2.4 The lifting functor 14§2.5 Summary 15§3. Categorical framework for families of groups 16§3.1 Background on Quillen’s bracket construction 17§3.2 A topological enrichment 19§3.3 Semi-monoidal categories and semicategories 27§3.4 The input categories 28§4. Topological construction of representations 38§4.1 Classical braid groups 39§4.2 Surface braid groups 43§4.3 Mapping class groups 47
Mathematics Subject Classification : 57M07, 57N05, 57N65, 55R99, 55U99, 18B30, 18B40, 20C07.
Key words and phrases : Homological representations, mapping class groups, surface braid groups, loop braid groups,configuration spaces, polynomial functors.
Homological representations of families of groups.
A family of groups, namely a set of groups { G n } n ∈ N equipped with injections γ n : G n ֒ −→ G n +1 , is said to have wild representation theory if the indecomposable representations occur in familieswith at least two parameters. There is no classification schema in these cases. Such families ofgroups naturally arise in many situations in connection with topology: for instance the familiesof braid groups , mapping class groups and loop braid groups all have wild representation theory.Finding geometric or topological constructions of linear representations for these groups thus helpsto understand and organise their representation theory, since one then has the tools of topologyand geometry available.For the family of braid groups B n (where the inclusions are induced by adding a strand on the left),Lawrence [Law90] and Bigelow [Big01] constructed well-known families of linear representations,called the Lawrence-Bigelow representations , following different methods. They may be definedvia actions on twisted homology groups of configuration spaces of unordered points in a punctured2-disc. The most famous families among them are the family of (reduced)
Burau representations originally introduced in [Bur35] and the family of
Lawrence-Bigelow representations that Bigelow[Big01] and Krammer [Kra02] independently proved to be faithful (thus proving that the braidgroups are linear ). The keystone for constructing the Lawrence-Bigelow representations is actuallybased on a much more general underlying method. The first aim of this paper is to develop thisgeneral method in a larger context, as a procedure for constructing homological representations forany family of groups.
Coherent families of representations.
A natural goal is to extend these constructions so as todeal with representations satisfying some compatibility, or coherence, conditions. More precisely,rather than considering a representation just for one group G n , we are interested in collections oflinear representations { ̺ n : G n → GL R ( M n ) } n ∈ N ̺ n +1 to G n , with respect to somepreferred maps m n : M n → M n +1 , is ̺ n . Then we say that the representations { M n } n ∈ N form a family of linear representations of the groups { G n } n ∈ N . This notion can be encoded in a functorialway. Let G be the groupoid with objects indexed by natural numbers, with the groups { G n } n ∈ N asautomorphism groups and with no morphisms between distinct objects. For instance, we considerthe braid groupoid β to deal with braid groups and the decorated surfaces groupoid M for themapping class groups of surfaces (see §3). We assume the existence of a category C G containing G as its underlying groupoid and with a preferred morphism n → n + 1 for each object n . In allthe examples addressed in this paper, such a category C G is constructed through Quillen’s bracketconstruction using a monoidal structure on the groupoid G (see §3). Then, denoting by R -Modthe category of R -modules (for R a commutative ring), considering a functor from C G to R -Modis equivalent to considering a family of representations of the family of groups { G n } n ∈ N . Functorial homological constructions via topological categories.
Our general procedurefor defining homological representations is summarised in the diagram below. π C G C t G Cov Q Top k [ Q ] Top R R -Mod F Lift ⊗ M H i L i ( F ; M ) (1.1)The desired output is the diagonal functor C G → R -Mod, a (coherent) family of representations of { G n } n ∈ N . This is constructed in five steps: • Constructing a topological category C t G whose π recovers C G . The category C G is typicallyconstructed via Quillen’s bracket construction from a braided monoidal groupoid, and weexplain in §3.2 how this construction may be lifted to topological categories to produce anappropriate C t G . Its morphism spaces are typically embedding spaces between manifolds.This makes the next step very natural to define: • The key geometric input for the construction is a choice of functor F , defined on C t G andtaking values in Cov Q , the category of topological spaces equipped with regular coveringswith fixed deck transformation group Q . • The remaining steps encode in a general setting the idea of taking twisted homology ofcovering spaces: the functor Lift takes a regular covering with deck transformation group Q to the corresponding bundle of k [ Q ]-modules, the functor ⊗ M takes the fibrewise tensorproduct with a ( k [ Q ] , R )-bimodule M (“specialising the coefficients”), producing a bundle of R -modules, and finally H i is simply the twisted homology functor in degree i .There are also variants of this construction for reduced homology (where we work with categoriesof pairs of spaces) and for Borel-Moore homology (where we restrict to categories of spaces andproper maps). In particular, using Borel-Moore homology is especially interesting when the imageof the functor F consists of configuration spaces of points in a surface : a general result then givesfree generating sets for the resulting twisted Borel-Moore homology groups and a fortiori a betterunderstanding of the constructed representations (see §6).Many of these representations have been defined and studied before (at least at the level of indi-vidual groups, i.e., when restricted to the individual automorphism groups of C G ) — and indeedone purpose of describing this general procedure for constructing homological representations is togive a unified description for various different representations appearing in the literature, as wellas suggesting new constructions by comparing representations coming from different settings inthis unified context.In §5 we discuss the representations that are recovered as part of this general homological con-struction. This includes the Lawrence-Bigelow representations for braid groups, the
Moriyamarepresentations [Mor07] of the mapping class groups of the smooth connected compact orientablesurface of genus g and with one boundary component { Σ g, } g ∈ N , and the Long-Moody construction [Lon94]. We prove:
Theorem 1.1
Let m > be a fixed non-negative integer. There exist homological functors: LB BMm encoding the m -th Lawrence-Bigelow representations, defined on a category having β as its underlying groupoid; • Mor m encoding the m -th Moriyama representations, defined on a category having M as itsunderlying groupoid; • encoding each generalised Long-Moody functor introduced in [Sou18] . We also construct new families of representations for the surface braid groups and the loop braidgroups analogous to those of Lawrence-Bigelow for classical braid groups, and new families ofrepresentations of mapping class groups of surfaces (see §4). We highlight here three analogues ofthe Lawrence-Bigelow representations for the loop braid groups (see §4.5 for more details).
Theorem 1.2 (see Theorems 4.24, 4.25 and 4.26)
For any integers m > and i > , we constructhomological functors L i ( F αm ) : U L β −→ Z [ Q αm ] − Mod ,L i ( F βm ) : U ( L β ext ) −→ Z [ Q βm ] − Mod ,L i ( F γm ) : U ( L β ext ) −→ Z [ Q γm ] − Mod , defined over the group-rings of: • Q α = Z , Q β = ( Z / Z ) , Q γ = Z / Z , and • Q αm = Z ⊕ Z / Z , Q βm = ( Z / Z ) , Q γm = Z ⊕ ( Z / Z ) , for m > .In particular, these give coherent families of representations, over rings of Laurent polynomials, ofthe loop braid groups { LB n } n > ( for the first ) and of the extended loop braid groups { LB ext n } n > ( for the second and third ) . Polynomiality.
On another note, under some assumptions on C G , notions of polynomiality onthe objects in the category Fct ( C G , R - Mod ) are introduced and of particular importance. Thefirst notions of polynomial functors date back to Eilenberg and Mac Lane in [EM54]. Djament andVespa [DV19] introduce strong and weak polynomial functors for symmetric monoidal categoriesin which the monoidal unit is initial. These definitions are extended to pre-braided monoidalcategories in which the monoidal unit is initial in [Sou19; Sou18] and the notion of very strong polynomial functor in this context is introduced there. It is equivalent to the notion of coefficientsystems of finite degree of [RW17]. All these notions of polynomiality straightforwardly extend tothe slightly more general context of the present paper (see §7); various instances of these notionsare studied in [Pal17].
Applications.
The main motivation for our interest in very strong polynomial functors is theirhomological stability properties: Randal-Williams and Wahl [RW17] prove homological stabilityresults for certain families of groups with twisted coefficients given by very strong polynomialfunctors (see §8.3.1). In particular their results hold for surface braid groups, mapping classgroups of surfaces and loop braid groups. On the other hand, a first matter of interest in weakpolynomial functors is that weak polynomiality reflects more accurately the stable behaviour ofa given functor, than (very strong polynomiality). Also, contrary to (very) strong polynomialfunctors, weak polynomial functors of degree less or equal to some d ∈ N form a category P ol d ( C G )that is localizing in P ol d +1 ( C G ): this allows one to define quotient categories, which providean organizing tool for families of representations (see §8.3.2). The representation theories of thefamilies of groups that we study in this paper are wild and an active research topic (see for instance[BB05, Section 4.6], [Fun99], [Kor02] or [MR12]). Therefore the strong, very strong and weakpolynomial functors associated with these groups are not well-understood. Hence, Theorem 1.3offers a better understanding of these kinds functors and extends the scope of twisted homologicalstability to more sophisticated sequences of representations. Theorem 1.3
The m -th Lawrence-Bigelow functor LB BMm is both very strong and weak polynomialof degree m if m > , and LB BM is strong polynomial of degree but weak polynomial of degree .The Moriyama functor Mor m is both very strong and weak polynomial of degree m . utline. The paper is organised as follows. In §2, we introduce in detail the various tools todefine homological functors, as summarised in Diagram (1.1). In §3, we recall Quillen’s bracketconstruction, introduce its generalisation to deal with topological categories and then constructthe various categories associated with the families of groups which will be used to apply ourconstruction. We then apply the general construction of homological functors to the families of classical and surface braid groups , mapping class groups of surfaces and loop braid groups in §4.The next section §5 is devoted to recovering previously-known families of representations in theliterature from our homological construction. We then recall and slightly extend a key generalresult for Borel-Moore homology of configuration spaces in §6. In §7 we introduce the notionsof strong, very strong and weak polynomial functors in the present framework. Finally, we proveTheorem 1.3 and explain the applications of these polynomial properties in §8. The general construction of homological functors (also called homological representations when thesource is a group) that we explore in this paper can be summarised as follows: C = π ( C t ) C t Cov Q Top k [ Q ] Top R R -Mod , F Lift ⊗ M H i (2.1)where C is the category on which we are interested in defining a representation and C t is a topo-logical category recovering C on π . (See (2.26) in §2.5 for the full diagram.) Categories:
For a group Q , the objects of the category Cov Q are based, path-connected spaces X (admitting a universal cover) equipped with a surjection π ( X ) ։ Q , and morphisms are basedmaps commuting with these quotients. For a ring R , Top R denotes the category of topologicalspaces equipped with bundles of R -modules, and bundle maps. In each case, these are topological categories, using the compact-open topology for the mapping spaces. Functors:
The continuous functor Lift takes an object (
X, ϕ : π ( X ) ։ Q ) to the regular covering X ϕ → X to which it corresponds, viewed as a bundle of Q -sets, and then takes free k -modulesfibrewise, producing a bundle of k [ Q ]-modules over X . The functor ⊗ M is given by fibrewise tensorproduct with a ( k [ Q ] , R )-bimodule M and the right-hand functor is twisted homology in degree i .The idea is that, in each case of interest, we may construct interesting representations of C by:(a) choosing a topological category C t with π ( C t ) = C and a continuous functor F : C t → Cov Q ,via some geometric construction,(b) choosing “coefficients” (a ground ring k and a ( k [ Q ] , R )-bimodule M ) and the degree i inwhich to take twisted homology.This concentrates the interesting geometrical part of the construction, and automates the construc-tion of representations of C based on this, for any choice of coefficients and homological degree. Twisted homomorphisms:
In many interesting cases, there are interesting geometrically-definedcontinuous functors of the form F : C t → g Cov Q , where g Cov Q is the larger category whose objectsare ( X, ϕ : π ( X ) ։ Q ) as before, but whose morphisms ( X, ϕ ) → ( Y, ψ ) are based maps f : X → Y satisfying the weaker condition that f ∗ (ker( ϕ )) ⊆ ker( ψ ). We will also consider a more generalconstruction, taking as input such functors, and resulting in a representation of C on R -Mod tw ,the category of R -modules and twisted homomorphisms (also called crossed homomorphisms ). Outline:
In this section, we define precisely the (topological) categories and (continuous) functorsin diagram (2.1) and its enlargement involving g Cov Q . This defines step (b) of the general procedureoutlined above. In more detail, in §2.2 we discuss categories of bundles of modules and twistedbundle maps, and the fibrewise tensor product and twisted homology functors in this setting5for a brief summary of this structure, see Remark 2.5). In §2.3 we then describe carefully thefunctor Lift : Cov Q → Top k [ Q ] and its twisted version. In §3.4 and §4 we then perform the moregeometric step (a) in a variety of interesting examples, where C is a category corresponding to aninteresting family of groups, including surface braid groups, mapping class groups of surfaces andloop braid groups. In §3.4 we construct the appropriate topological categories C t , and in §4 weconstruct, geometrically, the functors F : C t → Cov Q . Applying the above procedure, we thereforeobtain interesting “homological representations” of the categories C . In §5, we review the alreadyexisting homological representations at the level of individual groups which are recovered using thisfunctorial approach. Therefore this general construction allows us to extend known representationsof a family of groups to a functor of an interesting category containing all of the groups in thefamily. The polynomial properties of this functor are discussed in §8. In this section, we define precisely the categories of “spaces equipped with bundles of modulesover bundles of algebras” that will appear in the intermediate steps of the more general version ofthe construction (2.1) – see Definition 2.1. We fix a (unital, commutative) ground ring k (whichis typically Z in our applications) and use the viewpoint that bundles of k -modules over X arefunctors from the fundamental groupoid of X to the category of k -modules. Categories and functors.
Let us write SMCat for the category of symmetric monoidal functors,and recall that we write Cat and Top for the categories of small categories and of topological spacesrespectively. Write U : SMCat −→ Catfor the obvious forgetful functor. For a fixed
C ∈
Obj (SMCat) there is a functorLoc C : Top op −→ SMCatgiven on objects by X Fct (Π ( X ) , C ), where Π ( X ) is the fundamental groupoid of a space X .We think of this as the category of C -local sysetms on X . This is motivated by the fact, mentionedin footnote 1 on page 6, that when C is the category of modules over a commutative ring k (withmonoidal structure given by the tensor product over k ) and the path-components of X admituniversal covers, then the objects of Loc C ( X ) are in natural bijection with bundles of k -modulesover X . Monoid and module objects.
Given a symmetric monoidal category C , objects in C equippedwith a certain structure, for example monoid objects, may be thought of as symmetric monoidalfunctors into C from appropriate finitely-presented symmetric monoidal categories.For example, consider the symmetric monoidal category Monoid defined by the following presen-tation: it has one generating object a and two generating morphisms µ : a ⊗ a → a and u : 1 → a ,where 1 denotes the empty monoidal product, with defining relations:(id ⊗ m ) ◦ m = ( m ⊗ id) ◦ m m ◦ (id ⊗ u ) = id m ◦ ( u ⊗ id) = id . (2.2)There is then an endofunctor Mon : SMCat −→ SMCat (2.3)defined by
C 7−→
Fct ⊗ ( Monoid , C ), where Fct ⊗ denotes the category of strict symmetric monoidalfunctors, which we think of as the “category of monoid objects in C ”. This can be taken as a definition of “bundles of k -modules”, and, when X is locally path-connected and semi-locally simply-connected (i.e. when each path-component of X admits a univeral cover), it is equivalent to the moreusual geometric definition, via the correspondence that associates to a bundle of k -modules (in the usual sense) over X the functor Π ( X ) → k -Mod defined using unique path-lifting (which holds since a bundle of k -modules is inparticular a covering space). Module with the following presen-tation: it has two generating objects a and b and three generating morphisms l : a ⊗ b → b , µ : a ⊗ a → a and u : 1 → a , with defining relations (2.2) together with: l ◦ (id a ⊗ l ) = l ◦ ( µ ⊗ id b ) . (2.4)There is then an endofunctor Mod : SMCat −→ SMCat (2.5)defined by
C 7−→
Fct ⊗ ( Module , C ), which we think of as the “category of monoid objects in C together with a module object”. Categories of structured bundles.
Denoting by Cat / C the slice category of Cat over C , recallthat the Grothendieck construction can be viewed as a functor R : Fct ( C op , Cat) −→ Cat / C , which, on objects, takes a functor F : C op → Cat to the functor R F → C , where Obj ( R F ) = { ( c, a ) | c ∈ Obj ( C ) , a ∈ Obj ( F ( c )) } and a morphism in R F from ( c, a ) to ( c ′ , a ′ ) is a morphism f : c → c ′ in C and a morphism g : a → F ( f )( a ′ ) in F ( c ). The functor R F → C simply takes ( c, a ) to c on objects and ( f, g ) to f on morphisms.Applying this to the functor U ◦ Loc k -Mod : Top op −→ Cat , we obtain a category R U ◦ Loc k -Mod equipped with a forgetful functor to the category of topologicalspaces – this may be thought of as the category of topological spaces equipped with a bundle of k -modules . We may equally well apply this to the functor U ◦ E ◦ Loc k -Mod : Top op −→ Cat , where E is any endofunctor of SMCat. When E = Mon we obtain the category R U ◦ Mon ◦ Loc k -Mod of topological spaces equipped with a bundle of k -algebras , and when E = Mod we instead obtain thecategory R U ◦ Mod ◦ Loc k -Mod of topological spaces equipped with a bundle of k -algebras togetherwith a bundle of modules over this bundle of k -algebras . To be more concise, we abbreviate thelatter to topological spaces equipped with a bundle of twisted k -modules . These are each equippedwith a forgetful functor to Top that remembers just the underlying space of the bundle(s).The constructions and discussion of this section so far may be summarised as follows. Definition 2.1 ( Categories of structured bundles. ) Let
Strc be a symmetric monoidal categoryrepresenting a certain structure (such as the finitely-presented
Monoid or Module described above).Then the Grothendieck construction R U ◦ Fct ⊗ ( Strc , − ) ◦ Loc k -Mod is the category of topological spaces equipped with a bundle of k -modules equipped with the structureStrc . In particular, for the structure Strc = Module we defineTop ( k ) := R U ◦ Mod ◦ Loc k -Mod , (2.6)and call this the category of topological spaces equipped with bundles of twisted k -modules . Ex-panding the definition, this may also be written asTop ( k ) = R Fct ⊗ ( Module , Fct (Π ( − ) , k -Mod)) . In particular, Top ( Z ) is the category of spaces equipped with bundles of twisted Z -modules , i.e. bun-dles of rings together with bundles of modules over that ring-bundle. When restricting to the subcategory of topological spaces whose path-components each admit a universalcover, this is literally true. In general, we simply choose to view a bundle of k -modules over a space X as a functorΠ ( X ) → k -Mod instead of its more usual definition. This is reasonable, since, in our applications, we will onlyever deal with spaces admitting universal covers. estricted subcategories. The intermediate categories used in the construction of homologi-cal functors will be certain subcategories of Top ( k ) , in which we fix either an underlying space ora trivial bundle of k -modules. In this subsection, we define these subcategories and discuss thestructures on these categories that we will use, namely a functor ( twisted homology ) to a cate-gory of modules and a symmetric monoidal structure ( fibrewise tensor product ). The diagram ofsubcategories is summarised in (2.7). Definition 2.2 ( Restricted categories of twisted k -modules. ) We define subcategories of Top ( k ) given (a) by fixing the underlying space, and considering pairs of bundles over a fixed space X , (b)by fixing a k -algebra R and requiring the bundle of k -algebras to be the trivial bundle with fibre R and (c) fixing both the space X and the k -algebra R and allowing only the bundle of R -modulesover X to vary. (a) For any functor F : C →
Cat and object c of C , there is a canonical inclusion of categories F ( c ) ֒ −→ R F given on objects by a ( c, a ). In the case of Top ( k ) , this means that, for any space X , there is acanonical inclusion of categoriesTop ( k ); X := Mod (Loc k -Mod ( X )) ֒ −→ Top ( k ) . Note that Top ( k ); X is a symmetric monoidal category. (b) To define the subcategory Top ( k ); R ⊂ Top ( k ) for a given k -algebra R , we first define a subfunctor F R : Top op → SMCat of the functor
Mod ◦ Loc k -Mod , where by subfunctor we mean that F R ( X )is a subcategory of Mod (Loc k -Mod ( X )) for every space X and that F R ( f ) is the restriction of thefunctor Mod (Loc k -Mod ( f )) to the subcategory F R ( X ) for every continuous map f : X → Y .To define this, we need to choose a subcategory F R ( X ) of Mod (Loc k -Mod ( X )) = Fct ⊗ ( Module , Fct (Π ( X ) , k -Mod))for each space X . We specify this category to be the full subcategory of those strict symmetricmonoidal functors Module → Fct (Π ( X ) , k -Mod) that take the object a of Module to the constantfunctor Π ( X ) → k -Mod at the k -module R . Note that: • An object of F R ( X ) is a bundle of R -modules over X . • A morphism of F R ( X ) from E to E is an endomorphism θ of R (as a k -module) togetherwith a morphism E → E of bundles of k -modules, that also respects the action of R on E and E , modulo the endomorphism θ . • The category of bundles of R -modules over X (and morphisms of bundles of R -modules) istherefore the subcategory of F R ( X ) consisting of those morphisms for which θ = id R .To make the last point precise, let Top X,R = Fct (Π ( X ) , R -Mod) be the category of bundles of R -modules over the space X , and form the Grothendieck construction Top R = R Fct (Π ( − ) , R -Mod)to define the category of bundles of R -modules over spaces. Then Top X,R ⊂ F R ( X ) is the subcat-egory described in the last point above.We now define Top ( k ); R := R U ◦ F R . This contains Top R as the subcategory (on the same objects) of bundles of R -modules, with all untwisted morphisms. (c) Finally, we define Top ( k ); X,R := F R ( X ) . This is by definition a subcategory of Top ( k ); X = Mod (Loc k -Mod ( X )), and it also has a canonicalembedding into Top ( k ); R = R U ◦ F R , by the general properties of the Grothendieck constructionmentioned above. It also contains the subcategory Top X,R , as mentioned above.8ummarising the constructions above, we have a diagram of subcategories:Top ( k ) Top ( k ); R Top ( k ); X Top ( k ); X,R
Top R Top
X,R (2.7)
Remark 2.3
Intuitively, the objects of Top ( k ) may be thought of as “parametrised families ofmodules over k -algebras”. The three subcategories in the left-hand square above correspond tofixing either the k -algebra R , or the space X parametrising the family, or both. However, westill allow all morphisms of Top ( k ) (they are full subcategories), so for example a morphism inTop ( k ); R between families of R -modules may involve a non-identity k -linear endomorphism of R .If we restrict this category further to allow only those morphisms that act by the identity on R ,we obtain the categories Top R and Top X,R on the right-hand side of the diagram above.
Remark 2.4
We also consider two variations of diagram (2.7).(A) Each of the categories in the top row of the diagram is the Grothendieck construction of afunctor Top op → Cat. If we precompose each of these functors with the forgetful functor(
X, A ) X from the category of pairs of spaces, we obtain categories of pairs ( X, A ) equippedwith bundles of twisted k -modules over X . We denote these categories as in (2.7), but witha superscript ( ) . (The bottom row is unchanged.)(B) We may also restrict each of the categories in (2.7) to their subcategories of spaces and proper maps. In this case we add a superscript ( ) pr to the notation. Remark 2.5 ( Twisted homology and fibrewise tensor product ) The structure on these categoriesthat we use in the second and third steps of the general construction of homological functors is a fibrewise tensor product and twisted homology . Twisted homology (in degree i , say) may be thoughtof as a functor Top ( k ) → k -Mod, which, on the subcategory Top ( k ); R , may be upgraded to a twistedhomology functor taking values in R -Mod (related to the previous one via diagram (2.11) below).We have a symmetric monoidal structure ⊗ k on the category Top ( k ); X , which we think of as a“fibrewise tensor product over k ”. Moreover, for objects of the subcategory Top ( k ); X,R , we mayalso take their fibrewise tensor product over R , and this operation in particular yields a functorTop ( k ); X,R −→ Top ( k ); X,S for each (
R, S )-bimodule. We summarise this in more detail in the rest of this subsection, includingthe corresponding functors for the variations of (2.7) described in Remark 2.4 (twisted homologyof pairs and twisted Borel-Moore homology respectively).
Fibrewise tensor product.
By definition,Top ( k ); X = Mod (Loc k -Mod ( X )) = Fct ⊗ ( Module , Fct (Π ( X ) , k -Mod))is a symmetric monoidal category, with the objectwise monoidal product induced by the tensorproduct of k -modules in k -Mod. We write this as ⊗ k and – using the viewpoint of objects ofTop ( k ); X as pairs of bundles of k -modules over X – we call this the fibrewise tensor product of(pairs of) bundles of k -modules over X . Remark 2.6
An alternative description of the fibrewise tensor product of two bundles of k -modules, using the viewpoint of bundles of k -modules directly instead of interpreting them asfunctors from the fundamental groupoid to the category of k -modules, is as follows. Let α : A → X and β : B → X be two bundles of k -modules; in other words, α is a fibre bundle with fibre a k -module M and structure group Aut k ( M ) and β is a fibre bundle with fibre another k -module N and structure group Aut k ( N ). Their fibrewise tensor product should be a fibre bundle over X with fibre M ⊗ k N and structure group Aut k ( M ⊗ k N ). Let U be an open cover of X thattrivialises both α and β . The fibre bundle α is determined by a collection of continuous maps σ U,V : U ∩ V → Aut k ( M ), as U and V vary over U , satisfying the cocycle condition on triple9ntersections. (Note that the maps σ U,V are in fact locally constant, since their codomains arediscrete.) Similarly, β is determined by a collection of continuous maps τ U,V : U ∩ V → Aut k ( N )satisfying the cocycle condition on triple intersections. To define the fibrewise tensor product α ⊗ k β , it suffices to specify a collection of maps υ U,V : U ∩ V → Aut k ( M ⊗ k N ) that satisfy thecocycle condition on triple intersections. Now, the monoidal structure on k -Mod induces a naturalgroup homomorphism ι : Aut k ( M ) × Aut k ( N ) −→ Aut k ( M ⊗ k N ) , and we may define υ U,V ( x ) = ι ( σ U,V ( x ) , τ U,V ( x )).In general, this monoidal structure on Top ( k ); X does not restrict to a monoidal structure on thesubcategory Top ( k ); X,R , since the fibrewise tensor product (over k ) of two R -modules is naturallynot an R -module, but an R ⊗ k R -module. On the other hand, there is of course another monoidalstructure defined on Top X,R = Fct (Π ( X ) , R -Mod), which we write as ⊗ R , namely the objectwisemonoidal product induced by the tensor product of R -modules in R -Mod. Remark 2.7
The relation between these two tensor products is as follows. Let M be a bundle of R -modules. Then we have that M ⊗ R M = ( M ⊗ k M ) ⊗ R e R, where R e = R ⊗ k R op .Moreover, for any ( R, S )-bimodule M (for k -algebras R and S ), there is a functor − ⊗ R M : R -Mod −→ S -Mod , (2.8)which we may apply fibrewise to bundles of R -modules, and more generally to objects of Top ( k ); X,R .Summarising and applying this, we have:
Proposition 2.8
For each k -module R , the monoidal structure ⊗ R on Top
X,R extends to one on
Top ( k ); X,R . The monoidal structure ⊗ k on Top ( k ); X and these monoidal structures ⊗ R on eachsubcategory Top ( k ); X,R are related as described in Remark 2.7 above. For k -algebras R and S andany ( R, S ) -bimodule M there is a functor − ⊗ R M : Top ( k ); X,R −→ Top ( k ); X,S , (2.9) for each X , and these assemble to give a functor − ⊗ R M : Top ( k ); R −→ Top ( k ); S . (2.10) Homology.
Recall that homology with local coefficients (in a fixed degree i , over a ring R ) is afunctor taking as input a space equipped with a bundle of R -modules and producing an R -moduleas output. More precisely, it is a continuous functor H i : Top R → R -Mod. (This is recalled inmore detail in [Pal18, §5.1], following [MS93].) We note here that it extends also to categories ofbundles of R -modules over bundles of k -algebras in a compatible way, as follows. Proposition 2.9
For any k -module R , the continuous functor H i : Top R → R - Mod extends to acommutative diagram of continuous functors
Top ( k ) k - ModTop ( k ); R R - Mod tw Top R R - Mod , H i H i H i (2.11) where R - Mod tw is the category of R -modules and twisted , or crossed , homomorphisms. roof sketch. One first checks that the usual twisted homology functor Top k → k -Mod extendsto Top ( k ) ( cf . . [Pal18, §5.1]). Unwinding the definitions, we then observe that, restricted toTop ( k ); R , the homology retains its R -module structure (which is not used in the definition of the i th homology k -module, but it is preserved at each step in the construction) – which gives us themiddle horizontal functor and the commutativity of the diagram. Remark 2.10 ( Relative homology ) Exactly the same statement holds for the categories of pairsof spaces equipped with twisted bundles of k -modules (variation (A) of Remark 2.4): there aretwisted relative homology functors fitting into a diagram like (2.11), with the superscript ( ) added to each category in the left-hand column. Borel-Moore homology.
Recall that the i th Borel-Moore homology group of a locally compactspace Y with local coefficients L , thought of as a bundle of R -modules may be defined by thefollowing inverse limit of relative (ordinary) homology groups H BMi ( Y, L ) = lim ←− A ∈ Cpt( Y ) H i ( Y, Y \ A ; L ) , (2.12)where Cpt ( Y ) is the poset of all compact subsets of Y . Alternatively, it is the same as the homologyof locally finite chains on Y (with coefficients in L ). Also, if Y is a non-compact Hausdorffspace, Borel-Moore homology groups can be defined via the relative homology of the one-pointcompactification , in other words H ∗ ( Y + , ∗ ; L ) where Y + denotes the one-point compactificationand ∗ the complement point of X . In particular if Y homeomorphic to the complement of aclosed subcomplex S in a finite CW -complex X , the Borel-Moore homology group H BM ∗ ( Y, L ) isisomorphic to the relative homology H BM ∗ ( X, S ; L ). We refer the reader to [Bre97, Chapter V] fora detailed introduction to Borel-Moore homology. It forms a continuous functor H BMi : Top pr R −→ R -Moddefined on the subcategory Top pr R ⊂ Top R of spaces equipped with bundles of R -modules andmorphisms of such whose underlying map of spaces is proper , i.e. where preimages of compactsets are compact. Applying the construction (2.12) to the constructions above and using Remark2.10, we obtain the following, where we recall that the superscript ( − ) pr denotes in each case thesubcategory on the same objects, restricting to those morphisms whose underlying map of spacesis proper. Proposition 2.11
For any k -module R , the continuous functor H BMi : Top pr R → R - Mod extendsto a commutative diagram of continuous functors
Top pr( k ) k - ModTop pr( k ); R R - Mod tw Top pr R R - Mod . H BMi H BMi H BMi (2.13)
This section is an interlude on the general question of when a (continuous) group action on a space X lifts to a given covering e X of X , and, if so, when the lifted action commutes with the action ofthe deck transformation group. This will be encoded in §2.4 into a “lifting functor”. The considered topological spaces:
Let X be a path-connected, locally path-connected andsemi-locally simply connected topological space and let x ∈ X be a basepoint.11et G be a locally path-connected topological group. We recall that this means that G is atopological space and an abstract group such that the multiplication operation · : G × G → G and the inverse operation ( − ) − : G → G are continuous with respect to the topology of G . Wedenote by ToGr the category of topological groups and ToGr lpc its subcategory of locally path-connected topological groups, both with continuous homomorphisms as its morphisms. The set ofpath-components of a topological group inherits it a group structure and thus induces a functor: π : ToGr → Gr . Recall that any object of Gr can be considered as a topological group using the discrete topology.Hence, sending a point of a (locally path-connected) topological group to the path componentthat it lies in induces a functor π − : ToGr lpc → Gr, where π H : H → π ( H ) is a continuousgroup homomorphism. Note that it is necessary to restrict to ToGr lpc , otherwise π H is not alwayscontinuous and a fortiori π − is not well-defined on morphisms.Finally, we recall the following fact: Lemma 2.12
Let G ′ be a discrete group ϕ ∈ Hom
ToGr ( G, G ′ ) . Then, ϕ = π ( ϕ ) ◦ π G : G → π ( G ) → π ( G ′ ) = G ′ .Proof. This follows from the definition of the functor π − and the fact that, as G ′ is discrete, π G ′ : G ′ → π ( G ′ ) = G ′ is the identity. Two particular group homomorphisms:
First, let φ : π ( X, x ) → Q be a surjective grouphomomorphism. By [Hat02, Propositions 1.38 and 1.39], ker ( φ ) corresponds to a regular path-connected covering space ξ : X φ → X so that Im ( ξ ∗ ) = ker ( φ ) (where ξ ∗ denotes the mapinduced by ξ for the fundamental group) and with deck transformations group D ( ξ ) ∼ = Q . For k anatural number, this gives the homology group H k (cid:0) X φ , Z (cid:1) a Z [ Q ]-module structure. Recall thatif ker ( φ ) = 0, then X φ is isomorphic to the universal covering e X of X .Secondly, let θ : G → Homeo x ( X ) be a continuous group homomorphism, where Homeo x ( X )is given the subspace topology induced from the compact-open topology on Map ( X, X ). Thisinduces a group homomorphism θ π : G → Aut ( π ( X, x )), defined by θ π ( g ) ([ γ ]) = [ θ ( g ) ( γ )]with g ∈ G and [ γ ] ∈ π ( X, x ). Also, we make the following assumption: Assumption 2.13
The group homomorphism θ π preserves the subgroup ker ( φ ).Therefore, there are well-defined actions θ rπ : G → Aut (ker ( φ )) and ¯ θ π : G → Aut ( Q ) , such that: ¯ θ π ( g ) ◦ φ = φ ◦ θ π ( g ) for all g ∈ G . Also, we deduce that: Proposition 2.14
Under Assumption 2.13, we have the following properties:1. Let x φ ∈ X φ such that p (cid:16) x φ (cid:17) = x . There is a unique group homomorphism θ φ : G → Homeo x φ (cid:0) X φ (cid:1) such that ξ ◦ θ φ ( g ) = θ ( g ) ◦ ξ for all g ∈ G .2. For any deck transformation ψ ∈ D ( ξ ) ∼ = Q we have ¯ θ π ( g ) ( ψ ) ◦ θ φ ( g ) = θ φ ( g ) ◦ ψ for all g ∈ G .Proof. As θ π preserves the subgroup Im ( ξ ∗ ) , the existence of θ φ is a consequence of [Hat02,Propositions 1.33] and its unicity of [Hat02, Propositions 1.34].12ote that ξ (cid:0) ¯ θ π ( g ) ( ψ ) (cid:0) θ φ ( g ) ( y ) (cid:1)(cid:1) = θ ( g ) ◦ p ( y )= ξ (cid:0) θ φ ( g ) ( ψ ( y )) (cid:1) , for all y ∈ X φ . As X φ is path-connected, it is enough to prove that the second equality is truefor one point of X φ (see [Hat02, Proposition 1.34]). Let be a loop in X based at x such that φ ([ γ ]) = ψ and we denote by γ φ for a path based at x φ lifting γ in X φ . On the one hand, φ ([ θ ( g ) ( γ )]) (cid:16) x φ (cid:17) is the unique lift of x φ which is the endpoint of the lift in X φ of the path θ ( g ) ( γ ) . On the other hand, θ φ ( g ) (cid:16) φ ([ γ ]) (cid:16) x φ (cid:17)(cid:17) is the unique lift of x φ given by the endpointof the path obtained applying θ φ ( g ) to γ φ : as ξ ◦ θ φ ( g ) = θ ( g ) ◦ ξ , this is the same point as theendpoint of the lift in X φ of the path θ ( g ) ( γ ) . We deduce that: ¯ θ π ( g ) ( ψ ) (cid:16) θ φ ( g ) (cid:16) x φ (cid:17)(cid:17) = φ ([ θ ( g ) ( γ )]) (cid:16) x φ (cid:17) = θ φ ( g ) (cid:16) φ ([ γ ]) (cid:16) x φ (cid:17)(cid:17) = θ φ ( g ) (cid:16) ψ (cid:16) x φ (cid:17)(cid:17) . The construction:
Finally, we use ordinary homology and
Borel-Moore homology to define lin-ear representations of the group G . Moreover, if the topological space X has boundary components,we also use homology relative to the boundary and reduced homology (in other words, homologyrelative to a point on the boundary) to construct linear representations of the group G .Hence, using the induced action on homology, we obtain from the first point of Proposition 2.14 awell-defined action of G on the homology groups of X φ : Definition 2.15
The morphisms θ : G → Homeo x ( X ) and φ : π ( X, x ) → Q induce represen-tations L k ( F θ,φ ) : G → Aut (cid:0) H k (cid:0) X φ , Z (cid:1)(cid:1) and L k ( F θ,φ ) BM : G → Aut (cid:0) H BMk (cid:0) X φ , Z (cid:1)(cid:1) . If X has boundary ∂X or a basepoint, the morphisms θ and φ also induce representations L k ( F θ,φ ) ∂ : G → Aut (cid:0) H k (cid:0) X φ , ∂X φ ; Z (cid:1)(cid:1) and L k ( F θ,φ ) red : G → Aut (cid:0) H red k (cid:0) X φ , Z (cid:1)(cid:1) . In addition, let us now make the following assumption:
Assumption 2.16
The induced action ¯ θ π : G → Aut ( Q ) of G on Q is trivial.Hence, by the second property of Proposition 2.14, θ φ ( g ) commutes with all deck transformationsfor all g ∈ G . A fortiori, the induced representations L k ( F θ,φ ) and L k ( F θ,φ ) BM commute with the Z [ Q ] -module structure of the homology groups H k (cid:0) X φ , Z (cid:1) , H BMk (cid:0) X φ , Z (cid:1) , H k (cid:0) X φ , ∂X φ ; Z (cid:1) and H red k (cid:0) X φ , Z (cid:1) Furthermore, since G is locally path-connected, and since π ( X, x ) and the automorphism groupsof the various homology groups are discrete, it follows from Lemma 2.12 that the induced actions θ π and θ φ factor through π ( G ) . Note that it is a fortiori enough to check that Assumption 2.16is satisfied just for one point in each path-component of G . We deduce that: Definition 2.17
The representations L k ( F θ,φ ) etc. of G of Definition 2.15 induce representations π ( L k ( F θ,φ )) : π ( G ) −→ Aut Z [ Q ] (cid:0) H k (cid:0) X φ , Z (cid:1)(cid:1) , and similarly for the other versions, called respectively the ordinary, Borel-Moore, relative to theboundary and reduced homological representations of π ( G ) induced by θ : G → Homeo x ( X ) and φ : π ( X, x ) → Q . 13 .4 The lifting functor In this section we define a continuous functor
Lift : Cov Q −→ Top k [ Q ] , (2.14)which encodes the key construction of lifting group actions to covering spaces, while commutingwith the action of the deck transformation group of the covering. In fact, we will extend this to afunctor Lift : g Cov Q −→ Top ( k ); k [ Q ] , (2.15)which encodes the construction of lifting (more general) group actions, while commuting with theaction of the deck transformation group only up to an induced action on the deck transformationgroup itself. Definition 2.18
If, in §2.2, we replace the symmetric monoidal category k - Mod with the sym-metric monoidal category of sets and functions, under disjoint union, the constructions go throughidentically, and yield a diagram
Top ( · ) Top ( · ); M Top ( · ); X Top ( · ); X,M
Top M Top
X,M (2.16)analogous to (2.7), where the subscript ( ) ( · ) is notation indicating that this is the non-linearised version of the construction, and M is a monoid (playing a role analogous to that of the k -algebra R in (2.7)). There is a morphism of diagrams of categories from (2.16) to (2.7) (with R = k [ M ] , themonoid- k -algebra of M ) given by (fibrewise) applying the free strict symmetric monoidal functor Set → k - Mod .We therefore just need to define a functor
Lift : g Cov Q −→ Top ( · ); Q , (2.17)for any group Q , which we will then compose with the functor (2.16) → (2.7) of Definition 2.18 inthe top-middle position of the diagram (with M = Q ). Definition 2.19 ( The lifting functor ) The lifting functor (2.17) is defined on objects as follows.Let ( X, x , ϕ : π ( X, x ) → Q ) be an object of g Cov Q . The kernel of ϕ is a normal subgroup of π ( X, x ) , and therefore corresponds to a regular covering of X with deck transformation group Q , which is in particular a bundle of Q -sets over X (whose fibres happen to all be isomorphic to Q itself considered as a Q -set). To be slightly more careful (in order to specify a bundle of Q -sets,and not just an isomorphism class of such), we take the universal cover e X of X (specifically, thecanonical model for e X consisting of endpoint-preserving homotopy classes of paths in X startingat x ), which is equipped with an action of π ( X, x ) , and then take its quotient by the action ofthe subgroup ker( ϕ ) . Denote this covering by ξ ϕ : X ϕ → X . This defines (2.17) on objects: Lift(
X, x , ϕ ) = ( X, ξ ϕ ) . In order to define (2.17) on morphisms, we first note that, although we did not need it to define thefunctor on objects, the bundle of Q -sets associated to ( X, x , ϕ ) comes equipped with a particularchoice of basepoint, covering the basepoint x of X . This is because the standard construction ofthe universal cover e X has a canonical basepoint (the constant path at x ), and therefore so doesits quotient X ϕ by the action of Q . Let us denote this basepoint by e x ∈ X ϕ .Suppose we are given a morphism ( X, x , ϕ ) → ( Y, y , ψ ) of g Cov Q , that is, a continuous map f : X −→ Y f ( x ) = y and f ∗ (ker( ϕ )) ⊆ ker( ψ ) . By the basic theory of covering spaces, this impliesthat, for each e y ∈ ξ − ψ ( y ) , there is a unique continuous map X ϕ → Y ψ that lifts the composition f ◦ ξ ϕ : X ϕ → Y and that takes e x to e y . We therefore obtain a uniquely-determined lift e f : X ϕ −→ Y ψ by requiring e f ( e x ) = e y . Now, the fact that f ∗ (ker( ϕ )) ⊆ ker( ψ ) implies also that there is a uniqueendomorphism f Q of the group Q such that f Q ◦ ϕ = ψ ◦ f ∗ . (2.18)One may now check (cf. the discussion of the previous subsection) that, for any point e x ∈ X ϕ andany element q ∈ Q , we have that ( f Q ( q )) ♯ ( e f ( e x )) = e f ( q ♯ ( e x )) , where ( ) ♯ denotes the action of Q by deck transformations on X ϕ and on Y ψ .The triple ( f, e f , f Q ) is therefore a morphism in Top ( · ); Q from ( X, ξ ϕ ) to ( Y, ξ ψ ) , i.e. a morphism Lift(
X, x , ϕ ) −→ Lift(
Y, y , ψ ) , namely a map of covering spaces (the pair ( f, e f ) ) together with an endomorphism f Q of Q , suchthat the map e f commutes with bundle-of- Q -sets structure on X ϕ and Y ψ up to this endomorphism.In other words, the triple ( f, e f , f Q ) is a twisted (or crossed ) morphism of bundles of Q -sets. Thiscompletes the definition of (2.17) on morphisms: Lift( f ) = ( f, e f , f Q ) . Finally, we compose (2.17) with the functor (2.16) → (2.7) of Definition 2.18 in the top-middleposition of the diagram (with M = Q ), to obtain a functor Lift : g Cov Q −→ Top ( k ); k [ Q ] , (2.19)which is the promised functor (2.15). Remark 2.20
In the above definition, if the morphism f has the stronger property that ψ ◦ f ∗ = ϕ ,where f ∗ denotes the induced homomorphism on π — in other words, if it lies in the subcategory Cov Q ⊆ g Cov Q — then the unique endomorphism f Q satisfying (2.18) must necessarily be theidentity. The morphism ( f, e f , f Q ) = ( f, e f , id Q ) therefore lies in the subcategory Top Q ⊆ Top ( · ); Q .Hence the functor (2.17) restricts to a functor Lift : Cov Q −→ Top Q . (2.20)Composing this with the functor (2.16) → (2.7) of Definition 2.18 in the top-right position of thediagram (with M = Q ), we therefore obtain a functor Lift : Cov Q −→ Top k [ Q ] , (2.21)which is the promised functor (2.14). Definition 2.21 ( The homological representation construction, untwisted ) Fix a topological cat-egory C t with π ( C t ) = C and a ground ring k . Assume that we have as input: • A continuous functor F : C t → Cov Q , where Q is a group. • A ( k [ Q ] , R ) -bimodule M , where R is a k -algebra. • A non-negative integer i . 15he corresponding homological representation of C is obtained as follows. We compose the con-tinuous functors F , (2.21), (2.10) and (2.11) to obtain a functor C t → R - Mod . Since R - Mod is adiscrete category, this factors (uniquely) through a functor L i ( F ; M ) : C −→ R - Mod . (2.22)This is the homological representation of C associated to the continuous functor F : C t → Cov Q , indegree i and with coefficients M . In the case when we take R = k [ Q ] as a ring and M = k [ Q ] as abimodule over itself (i.e., when we do not twist the coefficients), we denote this simply as L i ( F ) : C −→ R - Mod . (2.23)The definition of the twisted version of the homological representation construction follows Defini-tion 2.21 almost verbatim: Definition 2.22 ( The homological representation construction, twisted ) Fix a topological category C t with π ( C t ) = C and a ground ring k . Assume that we have as input: • A continuous functor F : C t → g Cov Q , where Q is a group. • A ( k [ Q ] , R ) -bimodule M , where R is a k -algebra. • A non-negative integer i .The corresponding twisted homological representation of C is obtained as follows. We composethe continuous functors F , (2.19), (2.10) and (2.11) to obtain a functor C t → R - Mod tw . Since R - Mod tw is a discrete category, this factors (uniquely) through a functor e L i ( F ; M ) : C −→ R - Mod tw . (2.24)This is the twisted homological representation of C associated to the continuous functor F : C t → g Cov Q , in degree i and with coefficients M . Again, in the case when we take R = k [ Q ] as a ringand M = k [ Q ] as a bimodule over itself (i.e., when we do not twist the coefficients), we denote thissimply as e L i ( F ) : C −→ R - Mod tw . (2.25)The two constructions above may be summarised in the following extension of diagram (2.1). C = π ( C t ) C t Cov Q Top k [ Q ] Top R R - Mod g Cov Q Top ( k ); k [ Q ] Top ( k ); R R - Mod tw F Lift − ⊗
M H i e L i ( F ; M ) L i ( F ; M ) (2.26) Remark 2.23
This construction may easily be adapted to deal with pairs of spaces and with
Borel-Moore homology (in the latter case we must restrict to categories of spaces and proper continuousmaps everywhere). The resulting homological representations are then denoted similarly, adding asuperscript ( ) or ( ) BM to the notation. The aim of this section is to introduce the categorical framework that is central to this paper tohandle families of groups. We first recall notions and properties of Quillen’s bracket construc-tion introduced in [Gra76, p.219] and pre-braided monoidal categories. Then we describe each16ategorical setting associated with the families of groups we deal with to apply the homologicalrepresentation constructions introduced in §2.
Throughout this section, we fix a (small) strict monoidal groupoid ( G , ♮, and a (small)left-module ( M , ♮ ) over ( G , ♮, . The following definition is a particular case of a more generalconstruction of [Gra76].
Definition 3.1
Quillen’s bracket construction hG , Mi on the left-module ( M , ♮ ) over the groupoid ( G , ♮, is the category with the same objects as M and the morphisms are given by:Hom hG , Mi ( X, Y ) = colim G [ Hom M ( − ♮X, Y )] . Thus, a morphism from X to Y in hG , Mi is denoted by [ A, ϕ ] : X → Y : it is an equivalenceclass of pairs ( A, ϕ ) where A is an object of G and ϕ : A♮X → Y is a morphism in M . Also,for two morphisms [ A, ϕ ] : X → Y and [ B, ψ ] : Y → Z in hG , Mi , the composition is defined by [ B, ψ ] ◦ [ A, ϕ ] = [
B, Y ♮A, ψ ◦ ( id B ♮ϕ )] . Remark 3.2
Let φ be an element of Hom M ( X, Y ) . Then, as an element of Hom hG , Mi ( X, Y ) ,we will abuse the notation and write φ for [0 , φ ] . This comes from the (faithful) canonical functor C hG , Mi : M ֒ → hG , Mi defined as the identity on objects and sending φ ∈ Hom M ( X, Y ) to [0 , φ ] .Actually, for all the examples discussed in this paper, the category M is a groupoid (see §3.4).Then, a natural question is the relationship between the automorphisms of the groupoid M andthose of its associated Quillen bracket construction hG , Mi . Recall that the monoidal groupoid ( G , ♮, is said to have no zero divisors if, for all objects A and B of G , A♮B ∼ = 0 if and only if A ∼ = B ∼ = 0 . Then, following mutatis mutandis from the proof of [RW17, Proposition 1.7], we provethat: Proposition 3.3
If the strict monoidal groupoid ( G , ♮, has no zero divisors, if Aut G (0) = { id } and if M is a groupoid, then M = G r ( hG , Mi ) . Henceforth, we assume that M is a groupoid, that the strict monoidal groupoid ( G , ♮, has no zero divisors and that Aut G (0) = { id } .Remark 3.4 We say that G and M have the cancellation property if A♮X ∼ = B♮X then A ∼ = B and the injection property if the morphism Aut G ( A ) → Aut M ( A♮X ) sending f ∈ Aut G ( A ) to f ♮id X is injective, for all objects A and B of G and X of M . If the groupoids G and M sat-isfy these two properties, then following [RW17, Theorem 1.10], for all objects A of G and X of M , Hom hG , Mi ( A, X ) is a set on which the group Aut M ( X ) acts by post-composition transi-tively and the image of the map Aut G ( A ) → Aut M ( A♮X ) s sending f ∈ Aut G ( A ) to f ♮id X is { φ ∈ Aut M ( A♮X ) | φ ◦ ( ι A ♮id X ) = ι A ♮id X } . A fortiori, we deduce the set isomorphismHom hG , Mi ( X, A♮X ) ∼ = Aut M ( A♮X ) / Aut G ( A ) . A natural question is to wonder when an object of
Fct ( M , C ) extends to an object of Fct ( hG , Mi , C ) for a category C , which is the aim of the following lemma, which proof follows mutatis mutandisfrom the one of [Sou18, Lemma 1.6]. Analogous statements can be found in [RW17, Proposition2.4]. Lemma 3.5
Let C be a category and F an object of Fct ( M , C ) . Assume that for all A ∈ Ob ( G ) and X ∈ Ob ( M ) , there exist assignments F ([ A, id
A♮X ]) : F ( X ) → F ( A♮X ) such that for all B ∈ Ob ( G ) : F ([ B, id
B♮A♮X ]) ◦ F ([ A, id
A♮X ]) = F ([ B♮A, id
B♮A♮X ]) . (3.1) Then, the assignments F ([ A, γ ]) = F ( γ ) ◦ F ([ A, id
A♮B ]) for all [ A, γ ] ∈ Hom hG , Mi ( X, A♮X ) definea functor F : hG , Mi → C if and only if for all A ∈ Ob ( G ) and X ∈ Ob ( M ) , for all γ ′′ ∈ Aut M ( X ) and all γ ′ ∈ Aut G ( A ) : F ([ A, id
A♮X ]) ◦ F ( γ ′′ ) = F ( γ ′ ♮γ ′′ ) ◦ F ([ A, id
A♮X ]) . (3.2)17imilarly, we can find a criterion for extending a morphism in the category Fct ( M , C ) to a mor-phism in the category Fct ( hG , Mi , C ) , the proof being a slight mutatis mutandis adaptation ofthe one of [Sou18, Lemma 1.7]. Lemma 3.6
Let C be a category, F and G be objects of Fct ( hG , Mi , C ) and η : F → G anatural transformation in Fct ( M , C ) . The restriction Fct ( hG , Mi , C ) → Fct ( M , C ) is obtained byprecomposing by the canonical inclusion C hG , Mi of Remark 3.2. Then, η is a natural transformationin the category Fct ( hG , Mi , C ) if and only if for all X, Y ∈ Ob ( M ) such that Y ∼ = A♮X with A ∈ Ob ( G ) : η Y ◦ F ([ A, id Y ]) = G ([ A, id Y ]) ◦ η X . (3.3)Finally, if the strict monoidal groupoid ( G , ♮, is braided, Quillen’s bracket construction hG , Mi inherits a monoidal product. Beforehand, we present the notion of a pre-braided monoidal category,introduced in [RW17]. This is a generalisation of that of a braided monoidal category. Definition 3.7 [RW17, Definition 1.5] Let ( C , ♮, be a strict monoidal category such that theunit is initial. Recall that ι X : 0 → B denotes the unique morphism from to an object X of C .We say that the monoidal category ( C , ♮, is pre-braided if its maximal subgroupoid G r ( C , ♮, isbraided monoidal (the monoidal structure being induced by that of ( C , ♮, ) and if the braiding b C A,B : A♮B → B♮A satisfies b C A,B ◦ ( id A ♮ι B ) = ι B ♮id A : A → B♮A for all
A, B ∈ Obj ( C ) .The following key property describes the application of Quillen’s bracket construction on a left-module ( M , ♮ ) over strict braided monoidal groupoid (cid:0) G , ♮, , b G− , − (cid:1) . It is a mutatis mutandisgeneralisation of [RW17, Proposition 1.8], the proof being therefore omitted. Proposition 3.8
If the groupoid ( G , ♮, is braided, then the definition of the monoidal product ♮ extends to hG , Mi by letting for [ X, ϕ ] ∈ Hom hG , Mi ( A, B ) and [ Y, ψ ] ∈ Hom hG , Mi ( C, D ) : [ X, ϕ ] ♮ [ Y, ψ ] = (cid:20)
X♮Y, ( ϕ♮ψ ) ◦ (cid:18) id X ♮ (cid:16) b G A,Y (cid:17) − ♮id C (cid:19)(cid:21) . Moreover, if we consider M = G , then the category ( hG , Gi , ♮, is pre-braided monoidal andthe unit of the monoidal structure is an initial object in the category hG , Gi . If, in addition, (cid:0) G , ♮, , b G− , − (cid:1) is symmetric monoidal, then the category (cid:0) hG , Gi , ♮, , b G− , − (cid:1) is symmetric monoidal. Remark 3.9
If the category M is not a groupoid, then slightly more general analogous results asthose of Lemmas 3.5 and 3.6 and the first part of Proposition 3.8 could be stated. However, thisis not the kind of situation we deal with in this paper. Induced framework for automorphism subgroups.
For all A ∈ Obj ( G ) , let H A be a sub-group of the automorphism group Aut G ( A ) . We denote by G ′ the subcategory of G with the sameobjects and with morphisms Hom G ′ ( A, B ) = ( H A if A ∼ = B in G ; ∅ otherwise.A natural question is to wonder when the monoidal structure of G restricts to G ′ : this is useful forsome situations of §3.4 to set a categorical framework for families of subgroups of automorphismgroups of some monoidal groupoid. Lemma 3.10
The (strict) braided monoidal structure (cid:0) G , ♮, , b G− , − (cid:1) restricts to G ′ if and only if,for all A, A ′ ∈ Obj ( G ) , ϕ ′ ♮ϕ ∈ H A ′ ♮A for all ϕ ′ ∈ H A ′ and ϕ ∈ H A .Proof. Note that the coherence conditions and the braiding of the induced braided monoidal struc-ture for G ′ automatically follow from those for G . Hence the monoidal product ♮ defines a (strict)braided monoidal structure (cid:0) G ′ , ♮, , b G− , − (cid:1) if and only if it restricts to a bifunctor ♮ : G ′ × G ′ → G ′ :this is equivalent to the fact that the monoidal structure ♮ defines group morphisms ♮ : H A ′ × H A → H A ′ ♮A for all A ′ , A ∈ Obj ( G ) . 18 .2 A topological enrichment Suppose now that G is a topological monoidal groupoid and M is a topological category with acontinuous left-action of G . (Recall that, by topological category , we mean a category enriched overthe symmetric monoidal category of topological spaces.) Definition 3.1 may be extended directlyto this setting, as follows. Definition 3.11
The category hG , Mi is defined to have the same objects as M , and for objects X, Y of M , we define Hom hG , Mi ( X, Y ) to be the quotient space " G A ∈ Ob( G ) Hom M ( A♮X, Y ) / ∼ , where ∼ is the equivalence relation given by ( A, ϕ ) ∼ ( A ′ , ϕ ′ ) if and only if ϕ = ϕ ′ ◦ ( σ♮ id X ) forsome σ ∈ Hom G ( A, A ′ ) . Note that this may also be written as a colimit, as in Definition 3.1. Remark 3.12
A topological version of Quillen’s bracket construction is mentioned briefly in Re-mark 2.10 of [Kra17], although there the categories are topological in the sense of being categoriesinternal to the category of topological spaces, rather than topologically-enriched categories. Lemma3.13 below is stated for topologically-enriched categories, but it is likely that it has an analoguefor categories internal to the category of topological spaces, in which case Lemma 2.11 of [Kra17]would be a particular case of this analogue.
Lemma 3.13
Let G be a topological monoidal groupoid and M a topological category with a con-tinuous left-action of G . Assume that, for each object A of G and each pair of objects X, Y of M ,the quotient map Hom M ( A♮X, Y ) −→ Hom M ( A♮X, Y ) / Aut G ( A ) (3.4) is a Serre fibration. Then there is a canonical isomorphism of categories π ( hG , Mi ) ∼ = h π ( G ) , π ( M ) i . (3.5) Proof.
First note that π ( hG , Mi ) and h π ( G ) , π ( M ) i have the same object set, by the definitionof the discrete and topologically-enriched Quillen bracket constructions, and the functor π . Specif-ically, their common object set is ob( M ) . It therefore remains to show that, for objects X and Y of M , there is a natural bijection between π (Hom hG , Mi ( X, Y )) and Hom h π ( G ) ,π ( M ) i ( X, Y ) . Set Φ = G A ∈ ob( G ) Hom M ( A♮X, Y ) . Unravelling the definitions, what we need to prove is that there is a natural bijection π (Φ / ∼ t ) ∼ = π (Φ) / ∼ h , where ∼ t is the equivalence relation given by ( A, ϕ ) ∼ t ( A ′ , ϕ ′ ) if and only if there is a morphism σ ∈ Hom G ( A, A ′ ) such that ϕ = ϕ ′ ◦ ( σ♮ id) , and ∼ h is the equivalence relation given by ( A, [ ϕ ]) ∼ h ( A ′ , [ ϕ ′ ]) if and only if there is a morphism σ ∈ Hom G ( A, A ′ ) such that ϕ ≃ ϕ ′ ◦ ( σ♮ id) . Note thatthe only difference between these definitions is that the equality is replaced by a homotopy in thedefinition of ∼ h .As sets, these are both quotients of (the underlying set of) Φ , so we just need to show that, giventwo elements ( A, ϕ ) and ( A ′ , ϕ ′ ) of Φ , they have the same image in π (Φ / ∼ t ) if and only if theyhave the same image in π (Φ) / ∼ h .(a) Suppose first that ( A, ϕ ) and ( A ′ , ϕ ′ ) have the same image in π (Φ) / ∼ h . This means that thereis a morphism σ ∈ Hom G ( A, A ′ ) and a path γ : [0 , −→ Hom M ( A♮X, Y ) ⊆ Φ with γ (0) = ( A, ϕ ) and γ (1) = ( A, ϕ ′ ◦ ( σ♮ id)) . Composing with the projection Φ → Φ / ∼ t andwriting [ − ] t for the equivalence classes with respect to ∼ t , we obtain a path in Φ / ∼ t from [( A, ϕ )] t to [( A, ϕ ′ ◦ ( σ♮ id))] t = [( A ′ , ϕ ′ )] t . Hence ( A, ϕ ) and ( A ′ , ϕ ′ ) have the same image in π (Φ / ∼ t ) .19b) To prove the converse, we first make an assumption, which we will justify later. Namely, weassume that that quotient map q : Φ −→ Φ / ∼ t is a Serre fibration. Now assume that ( A, ϕ ) and ( A ′ , ϕ ′ ) have the same image in π (Φ / ∼ t ) , so thereis a path δ : [0 , → Φ / ∼ t with δ (0) = [( A, ϕ )] t and δ (1) = [( A ′ , ϕ ′ )] t . By our assumption that q isa Serre fibration, we may lift this to a path ε : [0 , → Φ with ε (0) = ( A, ϕ ) and ε (1) ∼ t ( A ′ , ϕ ′ ) .Its image ε ([0 , is path-connected, so it must lie in Hom M ( A♮X, Y ) ⊆ Φ . Hence we have a path ε : [0 , −→ Hom M ( A♮X, Y ) with ε (0) = ( A, ϕ ) and ε (1) = ( A, ϕ ′′ ) ∼ t ( A ′ , ϕ ′ ) , for some ϕ ′′ ∈ Hom M ( A♮X, Y ) . The relation ( A, ϕ ′′ ) ∼ t ( A ′ , ϕ ′ ) means that there is a morphism σ ∈ Hom G ( A, A ′ ) such that ϕ ′′ = ϕ ′ ◦ ( σ♮ id) .Hence ε is a homotopy witnessing that ϕ ≃ ϕ ′ ◦ ( σ♮ id) , so we have shown that ( A, [ ϕ ]) ∼ h ( A ′ , [ ϕ ′ ]) ,in other words, ( A, ϕ ) and ( A ′ , ϕ ′ ) have the same image in π (Φ) / ∼ h .(c) It now just remains to prove our earlier assumption that q is a Serre fibration. Directly fromthe definition, one may easily verify the following two facts: • F i f i : F i E i → B is a Serre fibration if and only if each f i : E i → B is a Serre fibration. • f : E → B is a Serre fibration if and only if (i) f ( E ) is a union of path-components of B and(ii) f : E → f ( E ) is a Serre fibration.It therefore suffices to prove that(i) q (Hom M ( A♮X, Y )) is a union of path-components of Φ / ∼ t for each A ∈ ob( G ) ,(ii) Hom M ( A♮X, Y ) → q (Hom M ( A♮X, Y )) is a Serre fibration for each A ∈ ob( G ) .Let us partition ob( G ) into equivalence classes O α , under the equivalence relation where two objects A, A ′ of G are equivalent if and only if there is a morphism A → A ′ in G . (This is an equivalencerelation since G is a groupoid.) we may then write Φ = F α Φ α , where Φ α = G A ∈O α Hom M ( A♮X, Y ) . The equivalence relation ∼ t on Φ clearly preserves the topological disjoint union F α Φ α , so wehave Φ / ∼ t = G α (Φ α / ∼ t ) . Also note that, for any two objects
A, A ′ ∈ O α (for fixed α ), we have q (Hom M ( A♮X, Y )) = q (Hom M ( A ′ ♮X, Y )) . So, if we make a choice of object A α ∈ O α for each α , we have a decomposition of Φ / ∼ t as atopological disjoint union: Φ / ∼ t = G α q (Hom M ( A α ♮X, Y )) . This immediately implies point (i) above.For point (ii), note that two elements ϕ, ϕ ′ ∈ Hom M ( A♮X, Y ) have the same image under q if andonly if they are ∼ t -equivalent, which is equivalent to saying that they lie in the same orbit of the Aut G ( A ) -action on Hom M ( A♮X, Y ) . Hence the map q A : Hom M ( A♮X, Y ) −→ q (Hom M ( A♮X, Y )) (3.6)is isomorphic to (3.4), at least on underlying sets. If we can show that they are isomorphic alsoas continuous maps of spaces, then we will be done, since we know by hypothesis that (3.4) is aSerre fibration. Since (3.4) and (3.6) are surjective continuous maps with the same domain andthe same point-fibres, and we know moreover that (3.4) is a quotient map, it will suffice to provethat (3.6) is also a quotient map.Let U ⊆ q (Hom M ( A♮X, Y )) be a subset such that q − A ( U ) is open in Hom M ( A♮X, Y ) . We needto show that U is open in q (Hom M ( A♮X, Y )) . To see this, let A ∈ O α and note that, by the fact20iscussed above that the equivalence relation ∼ t preserves the decomposition of Φ into a topologicaldisjoint union, the restriction q α = q | Φ α : Φ α −→ q (Φ α ) = q (Hom M ( A♮X, Y )) is a quotient map. So it will suffice to show that q − α ( U ) is open in Φ α . Now, from the definitions,we observe the following description of the subset q − α ( U ) ⊆ Φ α = G A ′ ∈O α Hom M ( A ′ ♮X, Y ) . For each object A ′ ∈ O α , choose an isomorphism σ A ′ : A ′ → A in G . This induces a homeomorphism Υ A ′ = − ◦ ( σ A ′ ♮ id) : Hom M ( A♮X, Y ) −→ Hom M ( A ′ ♮X, Y ) . Then we have q − α ( U ) = G A ′ ∈O α Υ A ′ ( q − A ( U )) . Since q − A ( U ) is open in Hom M ( A♮X, Y ) , it follows that Υ A ′ ( q − A ( U )) is open in Hom M ( A ′ ♮X, Y ) for each A ′ ∈ O α . Thus q − α ( U ) is open in Φ α , as required. Verifying the condition.
In order to verify the condition (3.4) in each of our examples in §3.4below, we will use the following proposition. Let L and M be smooth, connected d -manifolds,each equipped with a distinguished boundary-component, denoted ∂ L resp. ∂ M , and let L♮M betheir boundary connected sum, which then also has an obvious distinguished boundary-component ∂ ( L♮M ) . Let A ⊂ L be a (possibly empty) closed submanifold of the interior of L and let B ⊂ M be a (possibly empty) closed submanifold of the interior of M . Definition 3.14
Define
Diff ∂ ( L♮M ; A ⊔ B ) to be the group of diffeomorphisms of L♮M that fix A ⊔ B as a subset and that restrict to the identityon a neighbourhood of ∂ ( L♮M ) . We define Diff ∂ ( L ; A ) and Diff ∂ ( M ; B ) similarly. However, forsimplicity of notation, we will henceforth drop the A and B and write simply Diff ∂ ( L♮M ) , etc.,unless there is any ambiguity about what the submanifolds A and B could be.These groups are topologised as follows (we describe this explicitly for Diff ∂ ( L ) , and for the othertwo cases it is exactly analogous). Let C be a closed neighbourhood of ∂ L in L and let Diff C ( L ) be the subgroup of Diff ∂ ( L ) consisting of diffeomorphisms that restrict to the identity on C .We then give Diff( L ) the Whitney topology, each Diff C ( L ) the subspace topology inherited fromthe Whitney topology on Diff( L ) , and we give Diff ∂ ( L ) the final topology with respect to thecollection of subsets Diff C ( L ) as C varies. Namely, a subset U of Diff ∂ ( L ) is open if and only if itsintersection with Diff C ( L ) is open in Diff C ( L ) for all C . In other words, we are viewing Diff ∂ ( L ) as the colimit Diff ∂ ( L ) = colim C (Diff C ( L )) . Note that this may differ from the subspace topology that
Diff ∂ ( L ) inherits directly from theWhitney topology on Diff( L ) (the colimit topology may be finer). However, these two topologieson Diff ∂ ( L ) are weakly equivalent. In particular, they have the same π .We have a quotient map Ψ : Diff ∂ ( L♮M ) −→ Diff ∂ ( L♮M ) / Diff ∂ ( L ) . (3.7)since Diff ∂ ( L ) acts on Diff ∂ ( L♮M ) on the right by ϕ · ψ = ϕ ◦ ( ψ♮ id M ) . Proposition 3.15
The quotient map (3.7) is a Serre fibration. emark 3.16 This is related to results of Cerf [Cer61, Corollaire 2, §II.2.2.2, page 294], Palais[Pal60, Theorem B] and Lima [Lim63], but we were not able to find an instance of their resultsthat covers the setting that we require here. We therefore give a complete proof of Proposition3.15, using key results of Cerf [Cer61, Lemme II.2.1.2, page 291] and of Palais [Pal60, Theorem A]as an input.
Proof of Proposition 3.15.
First of all, we smoothly construct the boundary connected sum
L♮M .To do this, choose an embedding e : D d − × [ − , ֒ → L such that • e − ( ∂L ) = ( ∂D d − × [ − , ∪ ( D d − × { } ) , • this intersection with ∂L is contained in the distinguished boundary-component of L , • the image of e is disjoint from the submanifold A ⊂ int( L ) , • e is a smooth embedding away from ∂D d − × { } . Similarly, choose an embedding f : D d − × [0 , ֒ → M satisfying similar conditions, in particular • f − ( ∂M ) = ( ∂D d − × [0 , ∪ ( D d − × { } ) .We then define: L♮M := L ∪ e ( D d − × [ − , ∪ f M, which has an obvious induced smooth structure. For − t < , define M t = ( D d − × [ t, ∪ f M ,a submanifold-with-corners of L♮M . Choose an embedding c : ∂ ( L♮M ) × [0 , ] ֒ −→ L♮M such that • c ( x,
0) = x for all x ∈ ∂ ( L♮M ) , • c ( x, t ) = ((1 − t ) y, s ) for all x = ( y, s ) ∈ ∂D d − × [ − , and t ∈ [0 , ] , • the image of c is disjoint from the submanifold A ⊔ B ⊂ int( L♮M ) .This is all illustrated in Figure 3.1.For each < ǫ , the image C ǫ = c ( ∂ ( L♮M ) × [0 , ǫ ]) is a closed neighbourhood of ∂ ( L♮M ) in L♮M . Recall from Definition 3.14 above that
Diff C ǫ ( L♮M ) is the group of diffeomorphisms ϕ of L♮M such that ϕ ( x ) = x for all x ∈ C ǫ and ϕ ( A ⊔ B ) = A ⊔ B , equipped with the Whitneytopology. Since the collection { C ǫ } is cofinal in the directed set of all closed neighbourhoods of ∂ ( L♮M ) , it follows from our definition of the topology on Diff ∂ ( L♮M ) that: Diff ∂ ( L♮M ) ∼ = colim ǫ → (Diff C ǫ ( L♮M )) . (3.8)We also define Diff C ǫ ( L♮M ; M t ) to be the group of diffeomorphisms ϕ of L♮M such that ϕ ( x ) = x for all x ∈ C ǫ ∪ M t and ϕ ( A ⊔ B ) = A ⊔ B , equipped with the Whitney topology.For each − t < and < ǫ we have a quotient map Ψ ǫ,t : Diff C ǫ ( L♮M ) −→ Diff C ǫ ( L♮M ) / Diff C ǫ ( L♮M ; M t ) . For any − t t ′ < and < ǫ ′ ǫ there are natural maps Diff C ǫ ( L♮M ) / Diff C ǫ ( L♮M ; M t ) −→ Diff C ǫ ′ ( L♮M ) / Diff C ǫ ′ ( L♮M ; M t ′ ) , so we may take the directed colimit of the maps Ψ ǫ,t to obtain colim ǫ,t → (Ψ ǫ,t ) : Diff ∂ ( L♮M ) −→ colim ǫ,t → (Diff C ǫ ( L♮M ) / Diff C ǫ ( L♮M ; M t )) , where we have used the identification (3.8) for the domain. Since each Ψ ǫ,t is a quotient map, itfollows from general facts about colimits in the category of topological spaces that colim ǫ,t → (Ψ ǫ,t ) isalso a quotient map. The map Ψ : Diff ∂ ( L♮M ) −→ Diff ∂ ( L♮M ) / Diff ∂ ( L ) , On ∂D d − ×{ } , it cannot be smooth, since this is a codimension-2 face of the cylinder D d − × [ − , L has only codimension-0 faces (its interior) and codimension-1 faces (its boundary). L = M = ∂ ( L♮M )= L ∩ im( c )= M ∩ im( c ) im( e ) im( f ) − t M t D d − × [ − , Figure 3.1
The construction of
L♮M , the submanifold M t and the collar neighbourhood c of ∂ ( L♮M ). i.e., the map (3.7) that we would like to show is a Serre fibration, is also a quotient map, with thesame domain. Observe that { ( C ǫ ∪ M t ) ∩ L } is cofinal in the directed set of all closed neighbourhoodsof ∂ L in L . This means that two diffeomorphisms of Diff ∂ ( L♮M ) have the same image under Ψ if and only if they have the same image under colim ǫ,t → (Ψ ǫ,t ) . As they are quotient maps of the samespace, it follows that Ψ ∼ = colim ǫ,t → (Ψ ǫ,t ) .We will prove below that each Ψ ǫ,t is a fibre bundle (and hence a Serre fibration), and then deducethat Ψ is a Serre fibration using the following general fact. ( ∗ ) Any filtered colimit of based Serre fibrations between compactly-generated weak-Hausdorffspaces is again a Serre fibration.For a reference for this fact, see Proposition 1.2.3.5(1) of [TV08], which states that a filteredcolimit of fibrations is a fibration in any compactly generated model category. The classical modelcategory of based compactly-generated weak-Hausdorff spaces, with its Quillen model structure inwhich the fibrations are the Serre fibrations, is compactly generated (see for example Proposition6.3 of [MMSS01]).To apply ( ∗ ) in our situation, first note that we are taking a directed colimit, which is in particulara filtered colimit. We then need to check that the diffeomorphism groups Diff C ǫ ( L♮M ) and theirquotients are compactly-generated weak-Hausdorff spaces. Diffeomorphism groups of manifolds, inthe Whitney topology, are always first-countable and Hausdorff, and thus compactly-generated andweak-Hausdorff. Moreover, the property of being compactly-generated is preserved when takingquotients. The property of being weak Hausdorff is not preserved when taking quotients; however,in the process of proving that each Ψ ǫ,t is a fibre bundle below, we will also show that its targetspace Diff C ǫ ( L♮M ) / Diff C ǫ ( L♮M ; M t ) is Hausdorff.It therefore remains to show that each Ψ ǫ,t is a fibre bundle (and its target space is Hausdorff).Write Emb C ǫ ( M t , L♮M ) for the space of smooth embeddings ϕ : M t → L♮M such that ϕ ( x ) = x for all x ∈ C ǫ ∩ M t and ϕ ( B ) ⊆ A ⊔ B . There is a restriction map Φ ǫ,t : Diff C ǫ ( L♮M ) −→ Emb C ǫ ( M t , L♮M ) , which is equivariant with respect to the left-action of Diff C ǫ ( L♮M ) by post-composition. Note thatthis factors through the quotient map Ψ ǫ,t , so we have an induced map23 iff C ǫ ( L♮M ) / Diff C ǫ ( L♮M ; M t )Diff C ǫ ( L♮M ) Emb C ǫ ( M t , L♮M ) . Ψ ǫ,t Φ ǫ,t b Φ ǫ,t Moreover, if two diffeomorphisms of
Diff C ǫ ( L♮M ) have the same image under Φ ǫ,t , their differencelies in Diff C ǫ ( L♮M ; M t ) , so the induced map b Φ ǫ,t is injective. We will prove in the next paragraphsthat, after restricting its codomain Emb C ǫ ( M t , L♮M ) to its image, the map Φ ǫ,t is a fibre bundle.Hence, restricting Emb C ǫ ( M t , L♮M ) to im(Φ ǫ,t ) in the above diagram, we obtain a diagram Diff C ǫ ( L♮M ) / Diff C ǫ ( L♮M ; M t )Diff C ǫ ( L♮M ) im(Φ ǫ,t ) Ψ ǫ,t Φ ǫ,t b Φ ǫ,t in which the vertical and horizontal maps are quotient maps (since surjective fibre bundles arealways quotient maps) and the diagonal map b Φ ǫ,t is a bijection. This implies that b Φ ǫ,t is in fact ahomeomorphism, and hence Ψ ǫ,t = b Φ − ǫ,t ◦ Φ ǫ,t is a fibre bundle, as required. Moreover, the targetspace of Ψ ǫ,t is homeomorphic to a subspace of Emb C ǫ ( M t , L♮M ) , which is Hausdorff, so we havealso incidentally shown that the target space of Ψ ǫ,t is Hausdorff.It finally remains to prove that Φ ǫ,t : Diff C ǫ ( L♮M ) → im(Φ ǫ,t ) ⊆ Emb C ǫ ( M t , L♮M ) is a fibre bundle.Since it is equivariant with respect to the left-action of Diff C ǫ ( L♮M ) , it will suffice to prove thatthe action of Diff C ǫ ( L♮M ) on im(Φ ǫ,t ) is locally retractile ( ≡ admits local cross-sections ). This isbecause, by [Pal60, Theorem A], any G -equivariant map into a G -locally retractile space is a fibrebundle.Thus, we have to prove the following statement: given an embedding e ∈ im(Φ ǫ,t ) ⊆ Emb C ǫ ( M t , L♮M ) ,we may find an open neighbourhood U of e and a continuous map γ : U →
Diff C ǫ ( L♮M ) such that γ ( e ) = id and γ ( f ) ◦ e = f for any f ∈ U . Since Diff C ǫ ( L♮M ) acts transitively on im(Φ ǫ,t ) , it willsuffice to prove this for just one such e , which we take to be the inclusion M t ֒ → L♮M .To prove this, we apply a result of Cerf [Cer61, Lemme II.2.1.2, page 291], which we first recall.Let X be a manifold-with-corners. This means in particular that X has a stratification into faces (for example, if X is a connected manifold with boundary, but no higher-codimension corners, thenits set of faces is π ( ∂X ) ⊔ { X } ). Each point x ∈ X may lie in many faces, but it has a unique smallest face (according to inclusion) in which it lies, which we denote by F X ( x ) . Now if Y is anysubmanifold-with-corners of X , we define C ∞ face ( Y, X ) = { smooth maps ϕ : Y → X such that F X ( ϕ ( x )) = F X ( x ) for each x ∈ Y } , equipped with the Whitney topology. The Extension Lemma
II.2.1.2 of [Cer61] says that, if Y isclosed in X and V is any neighbourhood of Y in X , then the restriction map C ∞ face ( X, X ) −→ C ∞ face ( Y, X ) admits a section s defined on an open neighbourhood V of the inclusion in C ∞ face ( Y, X ) , such that s (incl) = id and s ( f )( x ) = x for all f ∈ V and x ∈ X r V . Step 1.
Note that, since each embedding f ∈ im(Φ ǫ,t ) extends to a diffeomorphism of L♮M , itrestricts to an embedding of ∂M r ∂ M into ( ∂M r ∂ M ) ⊔ ( ∂L r ∂ L ) , and hence it induces aninjection f ∂ : π ( ∂M r ∂ M ) → π ( ∂M r ∂ M ) ⊔ π ( ∂L r ∂ L ) . By definition of the embeddingspace Emb C ǫ ( M t , L♮M ) , f also sends B into A ⊔ B , so it also induces an injection f ♯ : π ( B ) → π ( B ) ⊔ π ( A ) . The function f ( f ∂ , f ♯ ) is locally constant, so its fibres are open. Let U ′ be theopen subset of im(Φ ǫ,t ) consisting of all f such that f ∂ is the inclusion and f ♯ ( π ( B )) = π ( B ) .24 L t r C ǫ = L t ∩ C ǫ = M t L t := L r ( D d − × ( t, − t Figure 3.2
Extending an embedding from M t to M t ∪ C ǫ and then to all of L♮M . Note that the second condition implies that f ( B ) = B , since f is an embedding and B is a closedmanifold. Step 2.
Each embedding f ∈ U ′ restricts to the identity on M t ∩ C ǫ , so we may extend it to asmooth map from the manifold-with-corners M t ∪ C ǫ into L♮M by defining it to be the identityalso on the rest of C ǫ . (See Figure 3.2 for a schematic picture.) This extension is continuous in theinput f , meaning that we have defined a continuous map γ ′ : U ′ → C ∞ ( M t ∪ C ǫ , L♮M ) . Moreover,for each f ∈ U ′ , the extension γ ′ ( f ) lies in the subspace C ∞ face ( M t ∪ C ǫ , L♮M ) , since it sends pointsof int( L♮M ) into int( L♮M ) and, for any boundary-component P of L♮M lying in M t ∪ C ǫ , it sends P into itself (this is because f ∂ = incl ). Thus, we have a continuous map γ ′ : U ′ −→ C ∞ face ( M t ∪ C ǫ , L♮M ) such that γ ′ (incl) = incl and γ ′ ( f ) | M t = f for all f ∈ U ′ . Step 3.
Now set X = L♮M and Y = M t ∪ C ǫ in the Extension Lemma of Cerf above, and choose V to be any open neighbourhood of M t ∪ C ǫ in L♮M that is disjoint from the submanifold A ⊂ int( L ) .Composing the local section s obtained from the Extension Lemma with γ ′ , we have a continuousmap γ ′′ = s ◦ γ ′ : U ′′ = ( γ ′ ) − ( V ) −→ C ∞ face ( L♮M, L♮M ) such that γ ′′ (incl) = id and for any f ∈ U ′′ we have γ ′′ ( f ) | M t = f and γ ′′ ( f )( A ) = A . Moreover,by construction, we also know that γ ′′ ( f )( x ) = x for all x ∈ C ǫ and γ ′′ ( f )( B ) = B . Step 4.
Finally, note that
Diff(
L♮M ) is open in C ∞ ( L♮M, L♮M ) , so U = ( γ ′′ ) − ( C ∞ face ( L♮M, L♮M ) ∩ Diff(
L♮M )) is an open neighbourhood of the inclusion in im(Φ ǫ,t ) . For each f ∈ U , the diffeomorphism γ ′′ ( f ) of L♮M fixes each point of C ǫ and sends A ⊔ B onto itself, so it is an element of Diff C ǫ ( L♮M ) . Sowe have a continuous map γ = γ ′′ | U : U −→
Diff C ǫ ( L♮M ) such that γ (incl) = id and, for all f ∈ U , we have γ ( f ) ◦ incl = f . This completes the proof.For future convenience, we record a useful corollary of the proof of Proposition 3.15. Definition 3.17
Let N be a smooth, connected d -manifold with non-empty boundary and let T bea compact, connected, proper ( d − -submanifold of N with non-empty boundary ∂T containedin a single boundary-component ∂ N of N . Assume that N r T has two components and denotetheir closures by N and N , which are manifolds with corners. In this context, proper means that the interior of T lies in the interior of N , the boundary of T lies in theboundary of N and, moreover, near the boundary, T ⊂ N is modelled on R d − × { } × [0 , ∞ ) ⊂ R d − × [0 , ∞ ). N = N ∪ N and T = N ∩ N . The prototypical example of this setting is N = L♮M and T = D d − × { } (see Figure 3.1), with N = L and N = M . Define Emb
Diff ∂ ( N , N ) to be the set of smooth embeddings ϕ : N → N , equipped with a germ of an extension ¯ ϕ to someneighbourhood of N in N , such that • there exists some neighbourhood U of ∂ N such that ¯ ϕ ( x ) = x for all x ∈ U ∩ domain( ¯ ϕ ) , • there exists an extension of ¯ ϕ to a diffeomorphism of N that acts by the identity on U .This is topologised as follows. Let U be a neighbourhood of ∂ N in N and let V be a neighbourhoodof N in N . Let Emb
Diff U ( V, N ) be the set of smooth embeddings ϕ : V → N such that ϕ ( x ) = x for all x ∈ U ∩ V and there exists an extension of ϕ to a diffeomorphism of N that acts by theidentity on U . This is given the subspace topology induced by the Whitney topology on the spaceof smooth maps C ∞ ( V, N ) . Note that, if U ′ ⊆ U and V ′ ⊆ V are neighbourhoods as above, thereare continuous restriction maps Emb
Diff U ( V, N ) −→ Emb
Diff U ′ ( V ′ , N ) . The set
Emb
Diff ∂ ( N , N ) is the colimit of the underlying sets of this diagram of spaces, so we maytopologise it by defining Emb
Diff ∂ ( N , N ) = colim U,V (Emb
Diff U ( V, N )) . Lemma 3.18
Under the conditions of Proposition 3.15, there is a natural homeomorphism
Diff ∂ ( L♮M ) / Diff ∂ ( L ) ∼ = Emb Diff ∂ ( M, L♮M ) . Proof.
This follows from the sequence of homeomorphisms:
Diff ∂ ( L♮M ) / Diff ∂ ( L ) ∼ = colim ǫ,t → (Diff C ǫ ( L♮M ) / Diff C ǫ ( L♮M ; M t )) ∼ = colim ǫ,t → (Emb Diff C ǫ ( M t , L♮M )) ∼ = Emb Diff ∂ ( M, L♮M ) . The first homeomorphism comes from the fact that we showed, during the proof of Proposition3.15, that the maps Ψ and colim ǫ,t → (Ψ ǫ,t ) are homeomorphic, so in particular their target spaces arehomeomorphic.The second homeomorphism is the homeomorphism b Φ ǫ,t from the proof of Proposition 3.15. Thethird homeomorphism is by definition of the topology on Emb
Diff ∂ ( M, L♮M ) and the facts that { C ǫ } is a cofinal family of neighbourhoods playing the role of U in the definition and { M t } is a cofinalfamily of neighbourhoods playing the role of V . A variant for decorated manifolds.
In practice, we will mainly use a slight variant of thissetup, where diffeomorphisms are only assumed to be the identity on a neighbourhood of twodisjoint discs in the boundary.
Definition 3.19 ( Decorated manifolds and their boundary connected sum ) First, let us say that a boundary-cylinder for a smooth d -manifold M is a map e : D d − × [0 , −→ M such that e − ( ∂M ) = ( ∂D d − × [0 , ∪ ( D d − × { } ) and e is a smooth embedding away fromthe sphere ∂D d − × { } . A decorated manifold is then a smooth d -manifold M , equipped witha closed submanifold A ⊂ int( M ) and an ordered pair ( e , e ) of disjoint boundary-cylinders for M r A . Given two decorated manifolds, one may form their boundary connected sum, as in thebeginning of the proof of Proposition 3.15, by gluing image( e ) of the first manifold to image( e ) of the second manifold. 26 efinition 3.20 ( Diffeomorphisms and embeddings of decorated manifolds ) For any decoratedmanifold ( M, A, e , e ) , we define Diff dec ( M ) to be the group of diffeomorphisms of M that send A onto itself and restrict to the identity on someneighbourhood of e ( D d − × { } ) ⊔ e ( D d − × { } ) . This is topologised as in Definition 3.14, as acolimit of Whitney topologies. Note that, for a pair of decorated manifold L and M , the topologicalgroup Diff dec ( L ) may be viewed as a subgroup of Diff dec ( L♮M ) , by extending diffeomorphisms bythe identity on M .Similarly, if ( L, A, e , e ) and ( M, B, f , f ) are two decorated manifolds, we define Emb
Diffdec ( M, L♮M ) to be the set of smooth embeddings ϕ : M → L♮M , equipped with a germ ¯ ϕ of an extension of ϕ to a neighbourhood of M in L♮M , such that ϕ restricts to the identity on a neighbourhoodof f ( D d − × { } ) and ¯ ϕ extends to a diffeomorphism of L♮M that restricts to the identity on aneighbourhood of e ( D d − × { } ) . This is also topologised as a colimit of Whitney topologies, asin Definition 3.17.The proofs of Proposition 3.15 and Lemma 3.18 may easily be adapted to prove the followinganalogue. Proposition 3.21
For any two decorated manifolds L and M , the quotient map Diff dec ( L♮M ) −→ Diff dec ( L♮M ) / Diff dec ( L ) (3.9) is a Serre fibration and its target space is homeomorphic to Emb
Diffdec ( M, L♮M ) . Remark 3.22
We note that all of the above may be adapted to the setting where the closedsubmanifolds A ⊂ int( L ) and B ⊂ int( M ) are equipped with orientations, and all diffeomorphismsand embeddings in Definitions 3.14 and 3.17 are required to preserve these orientations. Proposition3.15, Lemma 3.18 and Proposition 3.21 generalise immediately to this setting. All of the examples of categories C for which we would like topologically to construct representationswill be of the form hG , Mi , where G is a braided monoidal category and M is a left-module of G . We therefore need to find a topological monoidal groupoid G t and a topological category M t with a left-action of G t , satisfying condition (3.4) of Lemma 3.13 and such that π ( G t ) ∼ = G and π ( M t ) ∼ = M . Given this, a continuous functor hG , Mi → g Cov Q will induce a functor C = hG , Mi ∼ = π ( hG t , M t i ) = π ( C t ) −→ R - Mod tw , via the construction summarised in the diagram (2.26). Note that there is no need for G t tobe braided, since this structure is not needed in order to form the topological Quillen bracketconstruction, or for Lemma 3.13. In fact, it will be convenient for our examples to drop even morestructure from G , and assume only that it is a semi-monoidal category . (Recall that this is definedanalogously to a monoidal category, but without any of the structure or conditions involving leftor right units.) This is because it will typically be easy to lift the monoidal structure of G to anassociative binary operation on G t , but it is often not possible to make this lifted operation unital in a natural way. Remark 3.23 If G is a topological semi-monoidal groupoid and M is a topological category witha continuous left-action of G , then Definition 3.11 generalises directly to this setting, and producesa semicategory hG , Mi . (The associator of G is used to define composition in hG , Mi and thepentagon condition for the associator implies associativity of this composition.) Moreover, Lemma3.13 also generalises to this setting, and implies, under the same hypotheses, that there is anisomorphism of semicategories π ( hG , Mi ) ∼ = h π ( G ) , π ( M ) i . More precisely, a binary operation admitting an associator that satisfies the pentagon condition. G t , such that π ( G t ) ∼ = G as semi-monoidal groupoids and which satisfies condition (3.4) of Lemma3.13. Then hG t , M t i is a topological semicategory, and the input for the topological constructionwill be a continuous semifunctor hG t , M t i → g Cov Q . Via Lemma 3.13, Remark 3.23 and diagram(2.26) we then obtain a semifunctor hG , Mi → R - Mod tw . However, the source and target ofthis semifunctor are both categories (since G is a monoidal category, not just a semi-monoidalcategory), and so we may ask whether this semifunctor is in fact a functor. The final (small) stepof the topological construction will then be to verify that it does in fact preserve identities, and istherefore a functor. Remark 3.24
In practice, in our examples that we construct in §§4.4 and 4.5 using this framework,we will ignore this issue of a lack of identities in hG t , M t i , and proceed as if it were a topologicalcategory (with identities), to avoid unnecessary extra complications. However, formally, one shouldmodify that procedure as described in the paragraph above. The families of groups for which it is natural to define the first homological functors are the braidgroups of surfaces, mapping class groups of surfaces and loop-braid groups. Before doing this(in §4), we first introduce the suitable categorical framework to deal with the application of theconstruction of §2 to these families of groups.In this section, we first recollect the various definitions and properties of these families of groups.Then we present an appropriate groupoid encoding the considered family of groups in each situa-tion. These groupoids will systematically be braided monoidal or modules over a braided monoidalcategory: this allows one to apply Quillen’s bracket construction (see §3.1) in each case, the result-ing category being “richer” in the sense that it has more morphisms. This is done in §§3.4.1–3.4.3for (surface) braid groups and mapping class groups of surfaces, and in §3.4.4 for the loop-braidgroups.In §3.4.5, we construct topological versions of all of these groupoids, recovering each of the discretegroupoids — and hence their Quillen bracket constructions (using the results of §3.2) — aftertaking π . We recall that the (classical) braid group on n > strings denoted by B n is the group generated by σ , ..., σ n − satisfying the relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 for all i ∈ { , . . . , n − } and σ i σ j = σ j σ i for all i, j ∈ { , . . . , n − } such that | i − j | > . B and B both are the trivial group. The familyof braid groups is associated with the braid groupoid β , with objects the natural numbers n ∈ N and morphisms (for n, m ∈ N ): Hom β ( n, m ) = ( B n if n = m ∅ if n = m. The composition of morphisms ◦ in the groupoid β corresponds to the group operation of the braidgroups. So we identify the composition in σ ◦ σ ′ with the group multiplication σσ ′ in B n (with theconvention that we read from the right to the left for the group operation).We recall from [Mac98, Chapter XI, Section 4] that a monoidal product ♮ : β × β → β is definedby the usual addition for the objects and laying two braids side by side for the morphisms. Theobject is the unit of this monoidal product. The strict monoidal groupoid ( β , ♮, is braided:the braiding is defined for all natural numbers n and m such that n + m > by b β n,m = ( σ m ◦ · · · ◦ σ ◦ σ ) ◦ · · · ◦ ( σ n + m − ◦ · · · ◦ σ n ◦ σ n − ) ◦ ( σ n + m − ◦ · · · ◦ σ n +1 ◦ σ n ) where { σ i } i ∈{ ,...,n + m − } denote the Artin generators of the braid group B n + m .28 .4.2 Mapping class groups of surfaces The following suitable category to consider the mapping class groups of surfaces for our work isintroduced in [RW17, Section 5.6]. The decorated surfaces groupoid M is defined by: • Objects: decorated surfaces ( S, I ) , where S is a smooth, connected, compact surface withat least one boundary component, together with a parametrised interval I : [ − , ֒ → ∂ S in the boundary. Hence there is a distinguished boundary component decorated by I anddenoted by ∂ S . When there is no ambiguity, we omit the parametrised interval I from thenotation. • Morphisms: isotopy classes of diffeomorphisms of surfaces which restrict to the identity on aneighbourhood of their parametrised intervals I . Note that the non-distinguished boundarycomponents may be freely moved by the mapping classes. The automorphism group of S forms the mapping class group of S which is denoted by π Diffeo I ( S ) .Recall that a diffeomorphism of a surface which fixes an interval in a boundary component isisotopic to a diffeomorphism which fixes pointwise the boundary component ∂ S of the surface.When the surface S is orientable, the orientation on S is induced by the orientation of I . Theisotopy classes of diffeomorphisms then automatically preserve that orientation as they restrict tothe identity on a neighbourhood of I . Remark 3.25
Instead of boundary components, we could have equivalently considered surfaceswith punctures . Namely, for each object S of M we associate ¯ S the surface obtained by gluing adisc with one puncture on all the boundary components but ∂ S . We denote by P the correspondingfinite set of punctures. Let Diff I ( S, P ) the group of diffeomorphisms of ¯ S including points fillingin the punctures (called marked points ), which restrict to the identity on a neighbourhood of theparametrised interval I and which send the set P of marked points to itself (i.e. permuting themarked points). When P is the empty set, we omit it from the notation. We denote by ¯ M thecategory associated with this alternative. Since π ( Diff I ( S )) ∼ = π Diff I (cid:0) ¯ S, P (cid:1) , the categories ¯ M and M are equivalent. Also the mapping class group of S identifies with the group of isotopyclasses of homeomorphisms of S (see for example [FM12, Section 1.4.2]). Hence we representmapping classes using diffeomorphisms of surfaces with boundaries, but sometimes we considerhomeomorphisms instead of diffeomorphisms and punctures instead of boundaries when they aremore convenient to consider.We denote by D the unit -disc. Let Σ , denote the cylinder S × [0 , (which can be thoughtof as the disc D with a smaller disc is the interior which is removed), Σ , denote the torus withone boundary component (cid:0) S × S \ Int (cid:0) D (cid:1)(cid:1) and N , denote a Möbius band. For S an objectof the groupoid M , by the classification of surfaces, there exist g, s, c ∈ N such that there is adiffeomorphism: S ≃ (cid:18) ♮ s Σ , (cid:19) ♮ (cid:18) ♮ g Σ , (cid:19) ♮ (cid:18) ♮ c N , (cid:19) . Notation 3.26 If c = 0 , then g and s are unique, we denote by Σ sg, the boundary connected sum (cid:0) Σ , (cid:1) ♮s ♮ Σ ♮g , and by Γ sg, the mapping class group π Diff I (cid:0) Σ sg, (cid:1) . If g = 0 , we denote by N sc, theboundary connected sum (cid:0) Σ , (cid:1) ♮s ♮ N ♮c , and by N sc, the mapping class group π Diff I (cid:0) N sc, (cid:1) . Inboth cases, when s = 0 , we omit it most of the time from the notation.If c = g = 0 , then the boundary connected sum (cid:0) Σ , (cid:1) ♮s is diffeomorphic to the unit -disc where n interior discs are removed D s = D \ { p , . . . , p s } : we will abuse the notation and denote thesetwo objects in the same way.The groupoid M has a monoidal structure induced by gluing; for completeness, the definition isoutlined below (see [RW17, Section 5.6.1] for technical details). For two decorated surfaces ( S , I ) and ( S , I ) , the boundary connected sum ( S , I ) ♮ ( S , I ) = ( S ♮S , I ♮I ) is defined with S ♮S the surface obtained from gluing S and S along the half-interval I +1 and the half-interval I − , and29 ♮I = I − S I +2 . The diffeomorphisms being the identity on a neighbourhood of the parametrisedintervals I and I , we canonically extend the diffeomorphisms of S and S to S ♮S . The braidingof the monoidal structure b M ( S ,I ) , ( S ,I ) : ( S , I ) ♮ ( S , I ) → ( S , I ) ♮ ( S , I ) is given by doing halfa Dehn twist in a pair of pants neighbourhood of ∂S and ∂S . By [RW17, Proposition 5.18], theboundary connected sum ♮ induces a strict braided monoidal structure (cid:16) M , ♮, (cid:0) D , I (cid:1) , b M − , − (cid:17) .There are no zero divisors in the category M and Aut M (cid:0) D (cid:1) = { id D } by Alexander’s trick.Let M +2 (respectively M − ) be the full subgroupoids of M with objects the orientable surfaces(respectively the non-orientable or genus 0 surfaces). The monoidal structure ( M , ♮, restrictsto a braided monoidal structure both on the subgroupoids M +2 and M − , denoted in the same way (cid:0) M +2 , ♮, (cid:1) and (cid:0) M − , ♮, (cid:1) . Torelli groups:
Let M + , gen2 be the full subgroupoid of M +2 with objects the orientable surfacessuch that ∂S = ∂ S . The monoidal product ♮ restricts to a braided monoidal structure on M + , gen2 .We denote by ab the category of finitely generated abelian groups. The direct sum ⊕ inducesa strict symmetric monoidal structure ( ab , ⊕ , Gr ) , the symmetry being given by the canonicalpermutation of the free product.Recall that the isotopy classes of the diffeomorphisms of an object Σ g, of M + , gen2 act naturallyon its first homology group H (Σ g, , Z ) : the first homology groups H ( − , Z ) thus define a functorfrom the category M to the category ab. Using Van Kampen’s theorem, this functor is strongmonoidal with respect to the structures (cid:0) M + , gen2 , ♮, (cid:1) and ( ab , ⊕ , Gr ) . We recall that the Torelligroup I g, of the surface Σ g, is the kernel of the action Γ g, → Aut ( H (Σ g, ; Z )) . Let M I bethe subcategory of M + , gen2 with the same objects and restricting to the Torelli groups for theautomorphism groups.For g and g ′ two natural numbers, let ϕ be an element of I g, and ϕ ′ be an element of I g ′ , . Sincethe first homology functor H ( − , Z ) : M I → ab is strong monoidal, it follows from the universalproperty of the kernel that ϕ♮ϕ ′ belongs to I g ′ + g, . By Lemma 3.10, the boundary connected suminduces a strict braided monoidal structure (cid:0) M I , ♮, D (cid:1) . There are several ways to introduce the surface braid groups: first, they can be defined as thefundamental groups of some configuration spaces on surfaces; secondly, there are also explicit pre-sentations of these groups by generators and relations; finally, they can be seen as normal subgroupsof the mapping class groups of surfaces with punctures. For completeness, these definitions are out-lined below and we refer the reader to [Bel04] and [GJ15] for a detailed and complete introductionto these groups.
Surface braid groups as configuration spaces.
Let S be an object of the decorated surfacesgroupoid M . Let F n ( S ) be the configuration spaces of n ordered points in S : F n ( S ) := (cid:8) ( x , . . . , x n ) ∈ S × n | x i = x j if i = j (cid:9) . Let C n ( S ) be the configuration spaces of n unordered points in S , induced by the natural actionby permutation of coordinates of the symmetric group S n on F n ( S ) : C n ( S ) := (cid:8) ( x , . . . , x n ) ∈ S × n | x i = x j if i = j (cid:9) / S n . The braid group on the surface S on n strings is the fundamental group of this unordered con-figuration space B n ( S ) = π ( C n ( S ) , c ) , where c = ( z , . . . , z n ) with z i pairwise-distinct pointson the boundary component ∂ S for each i ∈ { , . . . , n } . We recall that the braid groups on the -disc D are the classical braid groups of §3.4.1 and we therefore omit D from the notations inthis situation.Moreover, for another natural number m , the preimage of the product S m × S n under the canonicalprojection B m + n ( S ) ։ S m + n is called the intertwining ( m, n ) -braid group B m,n ( S ) . Namely, it30s the fundamental group of the configuration space F m + n ( S ) / ( S m × S n ) . In addition, the map F m + n ( S ) / ( S m × S n ) → F n ( S ) / ( S n ) defined by forgetting the first m coordinates is a locallytrivial fibration with fibre F m (cid:0) S ( n ) (cid:1) / ( S m ) , where S ( n ) denotes the surface (cid:0) Σ , (cid:1) ♮n ♮S . The longexact sequence in homotopy of this fibration gives the following split short exact sequence: / / B m (cid:0) S ( n ) (cid:1) / / B m,n ( S ) Λ Sm,n / / B n ( S ) / / . (3.10)The splitting B n ( S ) ֒ → B m,n ( S ) is the map induced by the inclusion F n ( S ) ֒ → F m + n (cid:0) D ♮S (cid:1) defined by arbitrary fixing m points p , . . . , p m in the interior of D and sending ( x , . . . , x n ) to ( p , . . . , p m , x , . . . , x n ) . Therefore the intertwining braid group B m,n ( S ) is isomorphic to thesemidirect product B m (cid:0) S ( n ) (cid:1) ⋊ B n ( S ) : in particular, the natural action of B n ( S ) on B m (cid:0) S ( n ) (cid:1) is thus equivalent to the conjugate action in B m,n ( S ) if we regard these two groups as subgroupsof B m,n ( S ) . We refer the reader to [GJ15, Section 3.1] for further details. Presentations of surface braid groups. [Bel04] gives a presentation by generators and rela-tions of surface braid groups. As of now, we fix three natural numbers s > , g > and c > ,and consider the surfaces Σ sg, and N sc, described in §3.4.2. Throughout this work, we use thefollowing presentation: Proposition 3.27 ([Bel04, Theorems 1.1 and A.2])
The braid group on n points on the orientablesurface Σ sg, , denoted by B n (cid:0) Σ sg, (cid:1) with g > , admits the following presentation: • Generators: S = { σ i } i ∈{ ,...,n − } , A = { a i } i ∈{ ,...,g } , B = { b i } i ∈{ ,...,g } and X = { ξ i } i ∈{ ,...,s } ; • Relations: – Braid relations: σ i σ i +1 σ i = σ i +1 σ i σ i +1 and σ k σ j = σ j σ k for all i, k, j ∈ { , . . . , n − } so that | k − j | > ; – Mixed relations: ( R cσ i = σ i c and [ cσ n − c, σ n − ] = 1 for all c ∈ A ∪ B ∪ X and i ∈ { , . . . , n − } ; ( R a j σ n − b j = σ n − b j σ n − a j σ n − , (cid:2) σ − n − a k σ n − , a l (cid:3) = 1 , (cid:2) σ − n − b k σ n − , b l (cid:3) = 1 , (cid:2) σ − n − a k σ n − , b l (cid:3) = 1 , (cid:2) σ − n − b k σ n − , a l (cid:3) = 1 for all j, k, l ∈ { , . . . , g } so that k < l ; ( R (cid:2) σ − n − ξ j σ n − , c (cid:3) = 1 and (cid:2) σ − n − ξ k σ n − , ξ l (cid:3) = 1 for all c ∈ A ∪ B and j, k, l ∈{ , . . . , s } so that k < l .The braid group on n points on the non-orientable surface N sc, , denoted by B n (cid:0) N sc, (cid:1) with c > ,admits the following presentation: • Generators: S = { σ i } i ∈{ ,...,n − } , C = { c i } i ∈{ ,...,c } and X = { ξ i } i ∈{ ,...,s } ; • Relations: – Braid relations: σ i σ i +1 σ i = σ i +1 σ i σ i +1 and σ k σ j = σ j σ k for all i, k, j ∈ { , . . . , n − } so that | k − j | > ; – Mixed relations: ( R eσ i = σ i e for all e ∈ C ∪ X and i ∈ { , . . . , n − } ; ( R σ n − c j σ n − c j σ n − = c j σ n − c j , (cid:2) σ − n − c k σ n − , c l (cid:3) = 1 for all j, k, l ∈ { , . . . , c } sothat k < l ; ( R (cid:2) σ − n − ξ j σ n − , e (cid:3) = 1 , (cid:2) σ − n − xσ − n − , x (cid:3) = 1 and (cid:2) σ − n − ξ k σ n − , ξ l (cid:3) = 1 for all e ∈ C , x ∈ X and j, k, l ∈ { , . . . , s } so that k < l . Remark 3.28
In each case, the generators of the sets A , B , C and X are actually given by thegenerators of the fundamental group of the considered surface. Also, we use an opposite conventionto the one of [Bel04]: namely our numbering is the converse of the one chosen there so that therespective roles σ and σ n − in the mixed relations in the above presentation are switched comparedto those of [Bel04]. 31 urface braid groups as mapping classes. Finally the surface braid groups B n (Σ g, ) and B n ( N c, ) are respectively isomorphic to the kernels of the homomorphisms f Γ g,n : Γ ng, → Γ g, and f N c,n : N nc, → N c, , induced by the map which glues a disc on all the boundary components whichare freely moved. Namely, we have the following short exact sequences / / B n (Σ g, ) / / Γ ng, f Γ g,n / / Γ g, / / (3.11) / / B n ( N c, ) / / N nc, f N c,n / / N c, / / . (3.12)They are constructed as follows: the parametrized isotopy extension theorem of [Cer61, II, 2.2.2Corollaire 2] provides the locally-trivial fibre bundles Diff I (cid:0) Σ ng, (cid:1) → Diff I (Σ g, ) → C n (Σ g, ) and Diff I (cid:0) N nc, (cid:1) → Diff I ( N c, ) → C n ( N c, ) , where the left hand maps are defined by gluing a disc onall the boundary components of S but ∂ S ; then we consider the associated long exact sequenceof homotopy groups and use contractibility results of [Gra73, Théorème 1] for the component ofthe diffeomorphism groups. We refer the reader to [Bir74] or [GJ15, Section 2.4] for more details.This last point of view is the most convenient for the categorical framework we intend to set up.Let B be the subgroupoid of M with the same objects and with morphisms those of M thatbecome trivial after capping the non-parametrised boundary components. The monoidal structure ( M , ♮, restricts to a braided monoidal structure on the subgroupoid B , denoted in the sameway ( B , ♮, . We refer the reader to [RW17, Section 5.6.1] for further technical details.Let B g, +2 and B c, − be the full subcategories of B on the objects (cid:8) Σ ng, (cid:9) n ∈ N and (cid:8) N nc, (cid:9) n ∈ N respec-tively. Note that B , +2 and B , − both are equivalent to the braid groupoid β introduced in §3.4.1.Moreover, in this case, the braided monoidal structure ♮ of B restricts to the braided monoidalstructure of the braid groupoid ( β , ♮, described in §3.4.1.Then, the monoidal structure of B induces left β -module structures on the groupoids B g, +2 and B c, − . More precisely, the associative unital functors ♮ : β × B g, +2 → B g, +2 and ♮ : β × B c, − → B c, − are defined by the restriction of the monoidal product ♮ : B × B → B to the subcategories β × B g, +2 and β × B c, − . Hence we may apply Quillen’s bracket construction and define (cid:10) β , B g, +2 (cid:11) and (cid:10) β , B c, − (cid:11) . We now focus on (extended and non-extended) loop braid groups. We review in this section variousways to define these groups and refer to [Dam17] for a complete and unified presentation of thevarious definitions of these groups.
Loop braid groups as mapping class groups.
Loop braid groups may be defined in termsof motion groups of circles in a -disc. This is the setting that we shall use to construct suitable topological categories for the loop braid groups. We denote by D the unit -disc. Let C ( n ) := C ∐ · · · ∐ C n be a collection of n disjoint, unknotted, oriented circles, that form a trivial link of n components in the interior of D . Let Diff ∂ (cid:0) D , C ( n ) (cid:1) be the group of self-diffeomorphisms of D that that fix ∂ D pointwise and fix C ( n ) as a subset. We denote by Diff ∂ (cid:0) D , C +( n ) (cid:1) the subgroupof Diff ∂ (cid:0) D , C ( n ) (cid:1) of elements that also preserve the orientation of C ( n ) .The extended loop braid group LB ext n is the group of isotopy classes of Diff ∂ (cid:0) D , C ( n ) (cid:1) . The (non-extended) loop braid group LB n is the group of isotopy classes of Diff ∂ (cid:0) D , C +( n ) (cid:1) . Remark 3.29
The usual definition of loop braid groups as isotopy classes is in terms of self-homeomorphisms instead of self-diffeomorphisms. However, as pointed out in [Dam17, Remark3.7], it follows from [Wat72, Lemma 1.4 and Lemma 2.4] that the two definitions coincide.
Loop braid groups via configuration spaces.
Let C n S (cid:0) D (cid:1) be the space of configurationsof n unordered, disjoint, unlinked circles S in D . The extended loop braid group on n circles32 B ext n is the fundamental group of C n S (cid:0) D (cid:1) . In other words, denoting by Emb( n S , D ) the spaceof embeddings of n disjoint circles S into the -disc D , LB ext n is the fundamental group of thepath-component of the quotient Emb( n S , D ) / Diff( n S ) consisting of unlinks.Denote by Diff + ( n S ) the subgroup of Diff( n S ) of those diffeomorphisms that preserve the orien-tation of each circle. Analogously, the (non-extended) loop braid group LB n is the fundamentalgroup of the path-component of the quotient Emb( n S , D ) / Diff + ( n S ) consisting of oriented un-links. Alternative definitions.
There are several other ways to define loop braid groups: we brieflyreview two equivalent definitions of these groups that will be useful for our work. First, we haveexplicit presentations of extended and non-extended loop braid groups by generators and relations.Namely, the loop braid group LB n admits a presentation given by generators { σ i , τ i | i ∈ { , . . . , n − }} ; the generators { σ i | i ∈ { , . . . , n − }} satisfy the relations of the classical braid group B n (see§3.4.1), the generators { τ i | i ∈ { , . . . , n − }} satisfy the relations of the symmetric group S n andwe have three additional mixed relations (see [Dam17, Proposition 3.14]). In other words, LB n ∼ = ( B n × S n ) / ( Mixed relations ) . The extended loop braid group LB ext n admits a presentation given by generators { σ i , τ i | i ∈ { , . . . , n − }} ⊔ { ρ i | i ∈ { , . . . , n }} ; the generators { σ i | i ∈ { , . . . , n − }} satisfy the relations of the classical braid group B n (see§3.4.1), the generators { τ i | i ∈ { , . . . , n − }} satisfy the relations of the symmetric group S n ,the generators { ρ i | i ∈ { , . . . , n }} satisfy the relations of the abelian group ( Z / Z ) n and we haveeight additional mixed relations (see [Dam17, Proposition 3.16]). In summary, we have: LB ext n ∼ = ( B n × S n × ( Z / Z ) n ) / ( Mixed relations ) . The original reference for these presentations is [BH13].Also we can identify loop braid groups as particular subgroups of the automorphisms of free groups.We denote by F n = h x , . . . , x n i the free group on n ∈ N generators. Then the group LB n identifieswith the subgroup of the automorphism group Aut ( F n ) of the automorphisms which map eachgenerator of F n = h x , . . . , x n i to a conjugate of some generator, and LB ext n with those which canalso be mapped to an inverse of some generator: LB n = (cid:8) ϕ ∈ Aut ( F n ) | ∀ i ∈ { , . . . , n } , ∃ s ∈ S n , ∃ w i ∈ F n , ϕ ( x i ) = w − i x s ( i ) w i (cid:9) ; LB ext n = n ϕ ∈ Aut ( F n ) | ∀ i ∈ { , . . . , n } , ∃ s ∈ S n , ∃ w i ∈ F n , ϕ ( x i ) = w − i x ± s ( i ) w i o . We now construct topological lifts of all of the monoidal groupoids constructed in this section. ViaLemma 3.13, Proposition 3.15 and Proposition 3.21, we will also obtain topological lifts of eachof the various categories that we constructed using the Quillen bracket construction, and we willdescribe their morphism spaces in terms of spaces of embeddings.Fix an integer d > . We will first construct a topological semi-monoidal groupoid D d of diffeo-morphisms of d -manifolds. Then we will show that π ( D ) contains M as a sub-(semi-monoidalgroupoid), and hence also all of its subgroupoids described above. Then we will use D and anoriented version D +3 in an analogous way for the (extended) loop braid groups.33 efinition 3.30 Let D d be the topological groupoid defined as follows. Its objects are all decoratedmanifolds ( M, A, e , e ) of dimension d , as in Definition 3.19. A morphism in D d from ( M, A, e , e ) to ( M ′ , A ′ , e ′ , e ′ ) is a diffeomorphism ϕ : M → M ′ such that ϕ ( A ) = A ′ and ϕ ( e i ( x, t )) = e ′ i ( x, t ) for all ( x, t ) ∈ D d − × [0 , ǫ ] and i ∈ { , } , for some ǫ > . These sets of morphisms are topologisedas colimits of Whitney topologies, as in Definition 3.14; composition is continuous with this topol-ogy, since composition of smooth, proper functions is continuous in the Whitney topology (anddiffeomorphisms are proper). Hence D d is a topological groupoid. Definition 3.31 ( Boundary connected sum ) The boundary connected sum of two decorated d -manifolds was described already in Definition 3.19; we now recall this and make all of the detailsexplicit. Let ( L, A, e , e ) and ( M, B, f , f ) be two decorated d -manifolds and define L♮M = ( L ⊔ M ) / ∼ , where ∼ is the equivalence relation generated by e ( x, ∼ f ( x, for all x ∈ D d − . We give thisa smooth structure as follows. There are obvious topological embeddings L ֒ −→ L♮M and
M ֒ −→ L♮M, and another topological embedding D d − × [ − , ֒ −→ L♮M given by ( x, t ) e ( x, − t ) for t and ( x, t ) f ( x, t ) for t > . We define a smooth structureon L♮M by declaring that these are all smooth embeddings. Finally, we define ( L, A, e , e ) ♮ ( M, B, f , f ) = ( L♮M, A ⊔ B, e , f ) . Definition 3.32 ( Semi-monoidal structure ) We define the functor ♮ : D d × D d −→ D d on objects via the boundary connected sum of Definition 3.31. Now suppose we have morphisms ϕ : ( L, A, e , e ) → ( L ′ , A ′ , e ′ , e ′ ) and ψ : ( M, B, f , f ) → ( M ′ , B ′ , f ′ , f ′ ) in D d . Recall that theseare just diffeomorphisms ϕ : L → L ′ and ψ : M → M ′ satisfying a certain property. This propertyimplies that ϕ and ψ glue to a well-defined diffeomorphism L♮M → L ′ ♮M ′ , which is moreover amorphism ( L, A, e , e ) ♮ ( M, B, f , f ) −→ ( L ′ , A ′ , e ′ , e ′ ) ♮ ( M ′ , B ′ , f ′ , f ′ ) . It is then easily checked that this gives D d the structure of a topological semi-monoidal groupoid. Definition 3.33
Let D + d be the topological groupoid whose objects are decorated d -manifolds ( M, A, e , e ) together with an orientation of A ⊂ int( M ) , and whose morphisms are diffeomor-phisms ϕ as in Definition 3.30 such that the restriction ϕ | A : A → A ′ is orientation-preserving. Theboundary connected sum for such decorated d -manifolds is defined exactly as in Definition 3.31,with the orientation for A ⊔ B being induced from those of A and B . This then extends, just as inDefinition 3.32, to a structure of a topological semi-monoidal groupoid on D + d . Lemma 3.34
Let G be any sub-(semi-monoidal groupoid) of D d and let M be any sub-groupoid of D d that is preserved under the left-action of G . Then the Serre fibration condition (3.4) of Lemma3.13 is satisfied for this G and M . The same holds when D d is replaced by D + d .Proof. This follows directly from Proposition 3.21 for D d , together with Remark 3.22 for D + d .34 ecovering the decorated surfaces groupoid. Let M t be the full subgroupoid of D on thosedecorated surfaces ( S, A, e , e ) where S is compact and connected, the intervals e ( D × { } ) and e ( D × { } ) lie on the same boundary-component of S and A = ∅ . This inherits a topologicalsemi-monoidal structure from D . It is not hard to check that π ( M t ) ∼ = M as semi-monoidal groupoids, since, for diffeomorphisms of surfaces, the condition of fixing (a neigh-bourhood of) an interval in a boundary-component is equivalent to the condition of fixing two(neighbourhoods of) intervals in that boundary-component. By Lemmas 3.34 and 3.13 (togetherwith Remark 3.23), we deduce that π ( U M t ) ∼ = U M as semicategories, where U G = hG , Gi for a (topological) semi-monoidal groupoid G .We now describe the morphism spaces of U M t in terms of embedding spaces. Let S and S ′ betwo objects of M t , i.e., compact, connected, smooth surfaces, each equipped with an ordered pairof boundary-cylinders on the same boundary-component. Lemma 3.35
There is a homeomorphism U M t ( S, S ′ ) ∼ = ( Emb
Diffdec ( S, S♮T ) if there exists an object T of M t such that S♮T ∼ = S ′ ∅ otherwise.Proof. The space U M t ( S, S ′ ) is the space Hom hG , Mi ( X, Y ) of Definition 3.11, where we write G = M = M t , X = S and Y = S ′ . In the notation of the proof of Lemma 3.13, this is the quotientspace Φ / ∼ t . In that proof, it is shown (a) that this splits as the topological disjoint union of certainspaces denoted q (Hom M ( A♮X, Y )) , as A runs over representatives of isomorphism classes of objects,and (b) that this space is homeomorphic to the quotient space Hom M ( A♮X, Y ) / Aut G ( A ) . In ourcase, M = G is a groupoid, so this quotient space is either empty (if A♮X = Y ) or homeomorphic tothe quotient space Aut G ( A♮X ) / Aut G ( A ) (if A♮X ∼ = Y ). Since the collection of objects of M t under ♮ satisfies cancellation , there is at most one isomorphism class of objects A such that A♮X ∼ = Y .Putting this all together, we have shown that there is a homeomorphism U M t ( S, S ′ ) ∼ = ( Diff dec ( S♮T ) / Diff dec ( T ) if there exists an object T of M t such that S♮T ∼ = S ′ ∅ otherwise.Applying the second part of Proposition 3.21 completes the proof. Remark 3.36 ( Subgroupoids of M ) This construction may be carried out just as easily forsubgroupoids of M . Let G be any sub-(monoidal groupoid) of M and let M be any subgroupoidof M that is preserved under the left-action of G . Write G t for the preimage of G under theprojection M t −→ π ( M t ) ∼ = M , and similarly M t . Applying Lemmas 3.34 and 3.13 (together with Remark 3.23), we deduce that π ( hG t , M t i ) ∼ = hG , Mi as semicategories. One may then find similar descriptions of the morphism spaces of the topologicalsemicategory hG t , M t i as in Lemma 3.35. Example 3.37 ( The braid groupoid β ) For example, β is the full subgroupoid of M whose objectsare decorated surfaces that are diffeomorphic to a punctured -disc , i.e., D minus a finite collectionof open subdiscs whose closures are pairwise disjoint. From Remark 3.36, we obtain a topologicalsemi-monoidal groupoid β t such that π ( β t ) ∼ = β and π ( U β t ) ∼ = U β , and the morphism spacesof U β t may be described as follows. Let D m and D n be an m -punctured disc and an n -punctureddisc respectively. Then U β t ( D m , D n ) ∼ = ( Emb
Diffdec ( D m , D n ) if m n ∅ if m > n, Emb
Diffdec ( D m , D n ) is the space of embeddings D m → D n fixing a neighbourhood of two disjointintervals in ∂ D and that may be extended to a diffeomorphism of D n . Remark 3.38 ( The partial and injective braid categories ) There are alternative categories to U β = h β , β i for encoding the family of braid groups. First let B (cid:0) R (cid:1) be the category with thenon-negative integers as its objects, and a morphism m → n is a choice of k min { m, n } and apath in the the (unordered) configuration space C n (cid:0) R (cid:1) from a k -element subset of { x , . . . , x m } to a k -element subset of { x , . . . , x n } up to endpoint-preserving homotopy. Composition of twomorphisms is defined by concatenating paths and deleting configuration points for which the con-catenated path is defined only half-way and the identity is given by a constant path. This is calledthe partial braid category . This category is used, for example, in [Pal18] to organise representationsof braid groups for twisted homological stability results.Then the injective braid category B f (cid:0) R (cid:1) is the subcategory of B (cid:0) R (cid:1) on the same objects butwhose morphisms m → n are those where k = m . We have a (faithful) inclusion functor B f (cid:0) R (cid:1) ֒ −→ B (cid:0) R (cid:1) . There is also a full functor h β , β i −→ B f (cid:0) R (cid:1) (3.13)defined by the identity on objects and sending a morphism [ m − n, σ ] to the concatenation of thetrivial braid starting from m points and ending at n points (where all the additional n − m pointsare on the same side) followed by the geometric braid corresponding to σ . Note that (3.13) isnot faithful: indeed Hom B f ( R ) (1 , has three elements whereas Hom h β , β i (1 , is isomorphic to B / B as a set, and therefore has infinitely many elements given by the pure braid group PB . Example 3.39 ( The setting for surface braid groups ) As another example, let B g, +2 and B c, − be thesubgroupoids of M defined in §3.4.3, whose objects are the collection of surfaces Σ ng, respectively N nc, for all n ∈ N , and whose morphisms are isotopy classes of embeddings that become isotopic tothe identity if we fill in the non-parametrised boundary-components with discs. (Recall that oneboundary-component is equipped a parametrised interval; the others are called non-parametrised .)Applying Remark 3.36, we obtain topological groupoids ( B g, +2 ) t and ( B c, − ) t , each equipped witha continuous left-action of β t , such that π (( B g, +2 ) t ) ∼ = B g, +2 and π (( B c, − ) t ) ∼ = B c, − , and moreover π ( h β t , ( B g, +2 ) t i ) ∼ = h β , B g, +2 i and π ( h β t , ( B c, − ) t i ) ∼ = h β , B c, − i . The morphism spaces of h β t , ( B g, +2 ) t i may be described as follows (there is an analogous descriptionof the morphism spaces of h β t , ( B c, − ) t i ): h β t , ( B g, +2 ) t i (Σ mg, , Σ ng, ) ∼ = ( Emb
Diff , iddec (Σ mg, , Σ ng, ) if m n ∅ if m > n, where Emb
Diff , iddec (Σ mg, , Σ ng, ) denotes the subspace of Emb
Diffdec (Σ mg, , Σ ng, ) of all (decorated) embed-dings Σ mg, ֒ → Σ ng, that admit an extension to a diffeomorphism of Σ ng, that becomes isotopic tothe identity after including Σ ng, ⊂ Σ g, . One may see this exactly as in the proof of Lemma 3.35:we first see that the morphism space is empty if m > n , and if m n it is homeomorphic to thequotient Aut ( B g, +2 ) t (Σ ng, ) / Aut β ( D n − m ) ∼ = Diff iddec (Σ ng, ) / Diff dec ( D n − m ) , where Diff iddec (Σ ng, ) is the group of all (decorated) diffeomorphisms of Σ ng, that become isotopic tothe identity after including Σ ng, ⊂ Σ g, . (Recall that β is a full subgroupoid of M — so β t is afull subgroupoid of D —, whereas ( B g, +2 ) t is not, which is why we have a smaller diffeomorphismgroup of Σ ng, .) By Proposition 3.21, this is a subspace of Diff dec (Σ ng, ) / Diff dec ( D n − m ) ∼ = Emb Diffdec (Σ mg, , Σ ng, ) , which one may easily check to be the subspace Emb
Diff , iddec (Σ mg, , Σ ng, ) described above.36 opological groupoids for the loop braid groups.Definition 3.40 ( The loop braid groupoids ) We define ( L β ext ) t ⊂ D to be the full subgroupoid on the collection of all decorated -manifolds ( M, A, e , e ) where M is diffeomorphic to the -disc and A is diffeomorphic to the disjoint union of a finite collection ofcircles, forming an unlink in M . Similarly, we define L β t ⊂ D +3 to be the full subgroupoid on the collection of all decorated -manifolds ( M, A, e , e ) where M is diffeomorphic to the -disc and A is diffeomorphic to the disjoint union of a finite collectionof oriented circles, forming an oriented unlink in M . These are the (extended and non-extended) topological loop braid groupoids . We define their discrete versions simply by: L β ext = π (( L β ext ) t ) and L β = π ( L β t ) . The topological groupoids ( L β ext ) t and L β t inherit semi-monoidal structures from D and D +3 respectively. Hence L β ext and L β are (discrete) semi-monoidal groupoids. Remark 3.41
The semi-monoidal groupoids L β ext and L β are in fact monoidal (a unit is givenby ( D , ∅ , e , e ) ), and moreover symmetric . Hence the monoidal categories U L β ext and U L β arealso symmetric. More generally, let us write D sph d for the full subgroupoid of D d on all decorated d -manifolds M whose two boundary-cylinders lie on the same boundary-component ∂ M ∼ = S d − .This inherits a topological semi-monoidal structure from D d . Then π ( D sph d ) is a braided monoidalgroupoid for d = 2 and a symmetric monoidal groupoid for d > . Hence U ( π ( D sph d )) is pre-braidedfor d = 2 and symmetric for d > . The same statements hold for the analogous subgroupoid D sph , + d of D + d . Remark 3.42
Applying Lemmas 3.34 and 3.13 (and Remark 3.23), we see that π ( U ( L β ext ) t ) ∼ = U L β ext and π ( U L β t ) ∼ = U L β . Similarly to Lemma 3.35, the morphism spaces of U ( L β ext ) t and U L β t may be described as follows: U ( L β ext ) t ( D m , D n ) ∼ = ( Emb
Diffdec ( D m , D n ) if m n ∅ if m > n, U L β t ( D m , D n ) ∼ = ( Emb
Diff , +dec ( D m , D n ) if m n ∅ if m > n, where D m denotes the decorated -manifold given by an oriented unlink with m components inthe -disc and Emb
Diffdec ( D m , D n ) is defined as in Definition 3.20 (roughly, this means: embeddings D → D that fix a -disc in the boundary of D , send the embedded m -component unlink into theembedded n -component unlink, and that may be extended to a self-diffeomorphism of D that fixesthe n -component unlink as a subset). Then Emb
Diff , +dec ( D m , D n ) is its subspace where embeddingsmust also preserve the given orientations of the unlinks.Finally, we justify the name loop braid groupoid . Lemma 3.43
There are isomorphisms
Aut L β ext ( D n ) ∼ = LB ext n and Aut L β ( D n ) ∼ = LB n . Proof.
By definition, the automorphism group of D n in L β ext is π (Diff dec ( D n )) , Diff dec ( D n ) is the topological group of diffeomorphisms of D that send the embedded n -component unlink onto itself and that restrict to the identity on a neighbourhood of two disjoint -discs in ∂ D . (See Definition 3.20.) By definition (cf. Remark 3.29), LB ext n is π of the topologicalgroup of diffeomorphisms of D that send the embedded n -component unlink onto itself and thatrestrict to the identity on ∂ D . It therefore suffices to show that, for isotopy classes of diffeomor-phisms of -manifolds M with a spherical boundary-component ∂ M , fixing two disjoint -discs in ∂ M is equivalent to fixing all of ∂ M . This is similar to the fact that we used for surfaces: thatfixing one interval in a boundary-component is equivalent, for isotopy classes of diffeomorphisms,to fixing two disjoint intervals in that boundary-component. However, for -manifolds it is a lesstrivial fact. To see this, we argue as follows. Let Diff(
M, ∂ M ) be the group of diffeomorphismsof M that send ∂ M to itself. The restriction map Diff(
M, ∂ M ) −→ Diff( ∂ M ) = Diff( S ) is a fibre bundle, by [Cer61, Corollaire 2, §II.2.2.2, page 294], and hence its restriction Diff D ⊔ D ( M ) −→ Diff D ⊔ D ( S ) = Diff ∂C ( C ) is also a fibre bundle, where the subscript D ⊔ D means that diffeomorphisms must restrict to theidentity on a given pair of disjoint discs in ∂ M = S , and C is the -dimensional cylinder S × [0 , .The fibre is Diff ∂ M ( M ) and we obtain an exact sequence · · · → π (Diff ∂C ( C )) −→ π (Diff ∂ M ( M )) ( ∗ ) −−−→ π (Diff D ⊔ D ( M )) −→ π (Diff ∂C ( C )) . Our aim is to show that ( ∗ ) is a bijection, so it suffices to know that π and π of Diff ∂C ( C ) aretrivial. But in fact Diff ∂C ( C ) is contractible, by a theorem of Gramain [Gra73, Théorème 1]. Remark 3.44
A similar fact (to the one used in the proof of Lemma 3.43) holds also for asingle -disc in a spherical boundary-component of a -manifold. Namely, for isotopy classes ofdiffeomorphisms of a -manifold M with a spherical boundary-component ∂ M , fixing a -disc in ∂ M is equivalent to fixing the whole of ∂ M . This follows, via a similar argument as in Lemma3.43, using the fact (due to Smale [Sma59]) that Diff ∂ D ( D ) is contractible. The same statementis in fact true for isotopy classes of diffeomorphisms of -manifolds with a spherical boundary-component, since Diff ∂ D ( D ) is also contractible ([Hat83]). However, in higher dimensions thisdoes not continue to hold: π (Diff ∂ D d − ( D d − )) is isomorphic to the group of exotic d -spheres,which is very often non-trivial in higher dimensions. Also, π (Diff ∂ D ( D )) has recently beenshown to be non-trivial [Wat18]. This section is devoted to the application of the general construction of §2 to the families of groupsintroduced in §3.4. In §§4.1–4.3, we apply the lifting construction of §2.3 to obtain representationsof (surface) braid groups and mapping class groups. In each case, we also extend these to functorsdefined on categories of the form hG , Mi (whose automorphism groups are one of the familiesof groups in question), where h− , −i is the Quillen bracket construction of §3.1. We do this byexplicitly writing down an extension to certain “generating” morphisms of this category, and thenverifying that conditions (3.1) and (3.2) of Lemma 3.5 are satisfied.In §4.4 we then reinterpret the constructions of §§4.1–4.3 using the functorial version of the liftingconstruction, summarised in §2.5. Using this construction, together with the topological categoriesconstructed in §3.4, we recover each of the functors of §§4.1–4.3, each being induced a certaincontinuous functor defined on one of the topological categories of §3.4.Then, in §4.5, we directly apply the functorial version of the lifting construction to constructfamilies of representations of the (extended and non-extended) loop braid groups. These appearto be new, and are analogues of the reduced Burau and Lawrence-Bigelow representations of theclassical braid groups. Note that the condition of fixing a neighbourhood of two discs in the boundary is clearly homotopy equivalentto fixing just the two discs. he tools for §§4.1–4.3. We briefly review the tools, steps and spirit of the construction usedfor §§4.1–4.3. (The more functorial version of the construction used in §§4.4–4.5 is summarised indetail in §2.5.) For a locally path-connected topological group G , we consider a space X on which G acts through a continuous group homomorphism θ : G −→ Homeo x ( X ) where x ∈ X . In order to use covering space theory, we assume that X is path-connected, locallypath-connected and semi-locally simply connected, so that it admits a universal cover. Moreover,we consider a surjective group homomorphism φ : π ( X, x ) ։ Q so that the induced action θ π of G on the fundamental group of X satisfies Assumption 2.13. These two morphisms θ and φ define afunctor F θ,φ : G → g Cov Q . (The category g Cov Q is defined in §2.1.) Then we define a representation L k ( F θ,φ ) of π ( G ) using the k -th integral homology group of the regular path-connected coveringspace of X associated with φ , denoted X φ : L k ( F θ,φ ) : π ( G ) −→ Aut Z (cid:0) H k (cid:0) X φ ; Z (cid:1)(cid:1) . If, in addition, the action θ π of G on π ( X, x ) satisfies Assumption 2.16, the action defined by L k ( F θ,φ ) commutes with the Z [ Q ] -module structure of H k (cid:0) X φ ; Z (cid:1) : L k ( F θ,φ ) : π ( G ) −→ Aut Z [ Q ] (cid:0) H k (cid:0) X φ ; Z (cid:1)(cid:1) . This first application of the general construction of §2 relies on considering the total number ofhalf-twists and the total winding number for an ordered configuration space of points. As proved in§5.1, it allows one to recover the well-known families of
Lawrence-Bigelow representations originallyintroduced by Ruth Lawrence [Law90].We fix m and n two natural numbers such that m > and n > . We recall that D n is the surface D r { d , . . . , d n } , for { d i } i ∈{ ,...,n } a collection of pairwise disjoint open discs in the interior of theclosed unit -disc D and that C m ( D n ) is the configuration space of m unordered points in D n .We consider the topological group Diff ∂ ( D n ) of self-diffeomorphisms of D n which restrict to theidentity on the boundary of D . We fix m distinct points { z i } i ∈{ ,...,m } in the boundary of D andtake the configuration c = { z , . . . , z m } as the basepoint of C m ( D n ) .Let θ m,n : Diff ∂ ( D n ) → Homeo c ( C m ( D n )) be the continuous group morphism giving the naturalaction of Diff ∂ ( D n ) on the coordinates of the configuration space C m ( D n ) , i.e. defined by ϕ ( { x , . . . , x m } 7−→ { ϕ ( x ) , . . . , ϕ ( x m ) } ) . In particular, θ m,n ( ϕ ) preserves the basepoint c since ϕ fixes pointwise the boundary of D .The choice of the quotient of the fundamental group π ( C m ( D n ) , c ) depends on m . Beforehand,we recall from §3.4.3 that π ( C m ( D n ) , c ) identifies with the surface braid group B m ( D n ) andthat γ denotes the abelianisation map of a group G . For m = 1 : Let Σ : Z n → Z be the sum map ( a , . . . , a n ) P i ∈{ ,...,n } a i . Then if m = 1 weconsider the composite φ = φ ,n : B ( D n ) γ −−−→→ Z n Σ −−−→→ Z . For m > : We first have to introduce two new homomorphisms. The inclusion of D n in D bygluing a disc on all the interior boundary components induces an inclusion map i : C m ( D n ) ֒ → C m (cid:0) D (cid:1) : a configuration in D n is in particular a configuration in D . We denote by i ∗ the inducedsurjective homomorphism on π and consider the composite with the abelianisation map T : B m ( D n ) i ∗ −−−→→ B m γ −−−→→ Z .
39e may have the following geometrical interpretation of this morphism: for γ ∈ π ( C m ( D n ) , c ) and c γ a simple closed curve in C m ( D n ) representative of γ , one can think of T ( γ ) as countingthe total number of half-twists that occur in the path c γ of configurations of m points in D . Herea half-twist means doing half a Dehn twist in a tubular neighbourhood along the path c γ : namelythis is something like one of the standard generators of the braid group (consisting of a pair ofadjacent strands crossing each other, and no other crossings). This assumes an orientation on thecurve c γ and a fortiori half-twists with the opposite orientation count negatively.Furthermore, let j : C m ( D n ) ֒ → C m + n (cid:0) D (cid:1) be the map defined by { x , . . . , x m } 7−→ { x , . . . , x m , p , . . . , p n } , which glues a disc with a marked point p i onto each interior boundary component ∂d i of D n .The new points are thus considered as coordinates of the configuration space. We denote by j ∗ : B m ( D n ) ֒ → B m + n the induced injective homomorphism on π and consider the compositewith the abelianisation map R : B m ( D n ) j ∗ ֒ −−−→ B m + n γ −−−→→ Z . This morphism can geometrically be interpreted in the following way: for γ ∈ π ( C m ( D n ) , c ) and c γ as above, R ( γ ) counts the total number of half-twists that occur in a path c γ , and alsobetween configuration points and the additional marked points { p , . . . , p n } . In principle, it alsocounts half-twists between pairs of marked points, but of course these remain fixed, so there arezero of these. Hence R ( γ ) − T ( γ ) is the total number (counted with signs) of half-twists that occurbetween configuration points and the marked points in the path c γ of configurations. This is twicethe total number of times that a configuration point winds around a marked point: R ( γ ) − T ( γ ) is thus always even and corresponds to twice the total winding number of c γ . Hence we maydefine W : B m ( D n ) ։ Z to be the surjective morphism defined by the total winding number,i.e. γ ( R ( γ ) − T ( γ )) . These descriptions of the homomorphisms T , R and W come from[Bud05, Section 2].Finally we choose the product ( T × W ) ◦ ∆ : B m ( D n ) −→→ Z , defined by γ ( T ( γ ) , W ( γ )) , forthe choice φ = φ m,n of the quotient of the fundamental group. In addition, we have the followingproperty: Lemma 4.1
For each m > , the homomorphism φ m,n is invariant under the action ( θ m,n ) π of Diff ∂ ( D n ) .Proof. First, note that this statement is equivalent to saying that φ m,n is invariant under the action π (( θ m,n ) π ) of π (Diff ∂ ( D n )) ∼ = B n .For m = 1 , the morphism π (cid:0) ( θ ,n ) π (cid:1) corresponds to the Artin representation a n : B n → Aut ( F n ) :for each elementary braid σ i , the automorphism a n ( σ i ) sends the generator g j to g i +1 if j = i , to g − i +1 g i g i +1 if j = i + 1 and to g j if j / ∈ { i, i + 1 } . The result thus follows from the fact that φ ,n isthe composite of the abelianisation by the sum map.For m > , the result is a consequence of the more general facts that i ∗ and R are invariantunder the action of π (cid:0) ( θ m,n ) π (cid:1) . Indeed, let g ′ be an extension of an element g of Diff ∂ ( D n ) to Diff ∂ (cid:0) D (cid:1) . Then g ′ is isotopic to id D , since the space Diff ∂ (cid:0) D (cid:1) is path-connected (see [Mun60,Theorem 1.3], or [Sma59, Theorem B] for the stronger fact that it is contractible) and it makesthe following diagrams commutative: C m ( D n ) i / / θ m,n ( g ) (cid:15) (cid:15) C m (cid:0) D (cid:1) θ m, ( g ′ ) (cid:15) (cid:15) C m ( D n ) j / / θ m,n ( g ) (cid:15) (cid:15) C m + n (cid:0) D (cid:1) θ m + n, ( g ′ ) (cid:15) (cid:15) C m ( D n ) i / / C m (cid:0) D (cid:1) C m ( D n ) j / / C m + n (cid:0) D (cid:1) . Let H g ′ be an isotopy of D from g ′ to id D . Taking the product of m times H g ′ thus induces ahomotopy of C m (cid:0) D (cid:1) . We deduce that i ∗ ◦ ( θ m,n ) π ( g ) = i ∗ and a fortiori T ◦ ( θ m,n ) π ( g ) = T . On40he other hand, we claim that the action of ( θ m + n, ) π ( g ′ ) on B m + n corresponds (up to isotopy)to conjugation by some element σ ∈ B n ֒ → B m + n . Therefore γ ◦ ( θ m + n, ) π ( g ′ ) = γ and a fortori R ◦ ( θ m,n ) π ( g ) = R . The result thus follows from the definition of φ m,n .It remains to verify our claim that ( θ m + n, ) π ( g ′ ) acts on B m + n by conjugation by an element of B n ⊂ B m + n . To see this, let b be a geometric braid in D × [1 , on m + n strands, where m ofthem begin and end at a configuration in ∂ D and n of them begin and end at a configuration inthe interior of D . Extend this to a geometric braid in D × [0 , by gluing two trivial ( m + n ) -strand braids to the top and bottom. Then ( θ m + n, ) π ( g ′ ) ([ b ]) is represented by the geometricbraid G ( b ) , where G is the self-homeomorphism of D × [0 , given by G ( x, h ) = ( g ′ ( x ) , h ) . Recallthat we have an isotopy H g ′ of self-diffeomorphisms of D fixing ∂ D pointwise, with H g ′ (0) = g ′ and H g ′ (1) = id D . Using this, we define a homotopy K of self-homeomorphisms of D × [0 , by K ( t )( x, h ) = ( H g ′ ( ht )( x ) , h ) if h ∈ [0 , H g ′ ( t )( x ) , h ) if h ∈ [1 , H g ′ ((3 − h ) t )( x ) , h ) if h ∈ [2 , This induces a homotopy of geometric braids from G ( b ) = K (0)( b ) to K (1)( b ) , since, at all times t ,the self-homeomorphism acts on the top and bottom discs (corresponding to h = 0 , ) by g ′ , whichfixes the endpoints of b (setwise) by construction. Now note that the geometric braid K (1)( b ) isequal to b in the middle section D × [1 , . Moreover, the top section of K (1)( b ) (i.e. its intersectionwith D × [2 , ) is the inverse of the bottom section of K (1)( b ) (i.e. its intersection with D × [0 , ).Let σ denote the braid represented by the geometric braid K (1)( b ) ∩ ( D × [0 , . We have shownthat ( θ m + n, ) π ( g ′ ) ([ b ]) = [ G ( b )] = [ K (1)( b )] = σ ◦ [ b ] ◦ σ − . Finally, we need to check that σ lies in the subgroup B n ⊂ B m + n . But this is immediate from theconstruction, since K (1) fixes ∂ D × [0 , pointwise, and the trivial ( m + n ) -strand braid has m of its strands lying in the boundary of D .Therefore, the morphisms θ m,n and φ m,n satisfy Assumptions 2.13 and 2.16, for all natural numbers m . Following Definition 2.17, we thus define the representations L k (cid:0) F θ m,n ,φ m,n (cid:1) : B n −→ Aut R (cid:16) H k (cid:16) ( C m ( D n )) φ m,n ; Z (cid:17)(cid:17) for all integers m > and k, n > , where R = Z [ Z ] when m = 1 and R = Z [ Z ] when m > . Functoriality.
We fix a natural number m . All the homological representations L m (cid:0) F θ m,n ,φ m,n (cid:1) assemble to define a functor β → Z [ A m ] - Mod where A = Z and A m > = Z . We denote thisfunctor by LB m . The reason for this notation will be explained in §5.1.Actually, this functor extends to Quillen’s bracket construction: by Lemma 3.5, it is enough todefine properly LB m on the morphisms [ n ′ − n, id n ′ ] for all objects n and n ′ of β so that n ′ > n .Recall that the boundary connected sum ♮ defines an embedding ι D ♮id D n : D n ֒ → D ♮ D n for allnatural numbers n , adding an interior boundary component in D n . This embedding induces mapsfor the configuration spaces e m,n : C m ( D n ) → C m ( D n ) , which itself induces an injective mor-phism for the fundamental groups π ( e m,n ) : B m ( D n ) ֒ → B m ( D n ) . We denote the composite e m,n ′ − ◦ · · · ◦ e m, n ◦ e m,n by e m,n → n ′ : C m ( D n ) → C m ( D n ′ ) . From the functoriality of thefundamental groups with respect to the category of based topological spaces, the induced mor-phism for the fundamental groups π ( e m,n → n ′ ) : B m ( D n ) ֒ → B m ( D n ′ ) is equal to the composite π ( e m,n ′ − ) ◦ · · · ◦ π ( e m, n ) ◦ π ( e m,n ) .It follows from the above definitions that the restriction of the morphism φ m,n ′ to the braidgroup B m ( D n ) along π ( e m,n → n ′ ) is the morphism φ m,n : in other words, the following diagram is41ommutative π ( C m ( D n ) , c ) (cid:31) (cid:127) π ( e m,n → n ′ ) / / φ m,n ' ' ◆◆◆◆◆◆◆◆◆◆◆ π ( C m ( D n ′ ) , c ) φ m,n ′ w w ♣♣♣♣♣♣♣♣♣♣♣ A m . Hence there exists a unique based lift e φm,n → n ′ : C m ( D n ) φ → C m ( D n ′ ) φ so that the followingdiagram is commutative C m ( D n ) φ m,n e φm,n → n ′ / / ξ m,n (cid:15) (cid:15) C m ( D n ′ ) φ m,n ′ ξ m,n ′ (cid:15) (cid:15) C m ( D n ) e m,n → n ′ / / C m ( D n ′ ) . All these lifts are the key maps to define the functor LB m on the morphisms of the category h β , β i : Definition 4.2
For every natural numbers n and n ′ so that n > n ′ , let LB m ([ n ′ − n, id n ′ ]) bethe homomorphism induced by the lift e φm,n → n ′ for the standard homology. Proposition 4.3
The functor LB m extends to define objects of Fct ( h β , β i , Z [ A m ] - Mod) .Proof.
First of all it follows from the above definitions that LB m ([ n ′ − n, id n ′ ]) = LB m ([ n ′ , id n ′ − ]) ◦ · · · ◦ LB m ([1 , id n ]) . Following Lemma 3.5, we only have to check that Relation (3.2) is satisfied. For all σ ∈ B n andall σ ′ ∈ B n ′ , by the definition of θ m,n , the action of π (( θ m,n ′ + n ) π ) ( σ ′ ♮σ ) ∈ B n ′ + n on the n lastpoints of C m ( D n ′ + n ) is completely induced by σ since σ ′ is the identity on these n last points.Hence, the following diagram is commutative: C m ( D n ) e m,n → n ′ / / π ( θ m,n )( σ ) (cid:15) (cid:15) C m ( D n ′ + n ) π ( θ m,n ) ( σ ′ ♮σ ) (cid:15) (cid:15) C m ( D n ) e m,n → n ′ / / C m ( D n ′ + n ) . The unicity of the lifts with respect to the covering spaces n C m ( D i ) φ m,i o i ∈ N induced by themorphisms { φ m,i } i ∈ N implies that e φm,n → n ′ ◦ (cid:0) π (cid:0) θ φm,n (cid:1) ( σ ) (cid:1) = (cid:0) π (cid:0) θ φm,n (cid:1) ( σ ′ ♮σ ) (cid:1) ◦ e φm,n → n ′ . The result thus follows from the assignments for LB m , given by the maps for the standard andBorel-Moore homology (see (2.12)) induced by these lifts. Alternative using the abelianisation of mixed braid groups.
As pointed out in [BGG17],the morphisms φ m,n can be introduced using kernels of the abelianisation of some mixed braidgroups. We assume that n > . Recall that we denote by γ ( B m,n ) and γ ( B n ) the respec-tive canonical abelianisation morphisms of B m,n and B n , and that we omit the groups from thenotations where there is no ambiguity.They canonically induce a morphism Λ D m,n so that Λ D m,n ◦ γ ( B m,n ) = γ ( B n ) ◦ Λ D m,n , where Λ D m,n : B m,n (cid:0) D (cid:1) ։ B n (cid:0) D (cid:1) is the morphism introduced in §3.4.3 induced by forgetting the first m coordinates. We consider the kernel of Λ D m,n , which depends only on m since B m,n (cid:0) D (cid:1) /Γ (cid:0) B m,n (cid:0) D (cid:1)(cid:1) ∼ = ( Z ⊕ if m = 1 ; Z ⊕ if m > .42ence ker (cid:18) Λ D m,n (cid:19) ∼ = A m . Then φ m,n is the unique surjective morphism (given by the universalproperty of the kernel) so that the following diagram is commutative (and where the two lines areexact): / / B m ( D n ) / / φ m,n (cid:15) (cid:15) (cid:15) (cid:15) B m,n (cid:0) D (cid:1) Λ D m,n / / γ (cid:15) (cid:15) (cid:15) (cid:15) B n (cid:0) D (cid:1) / / γ (cid:15) (cid:15) (cid:15) (cid:15) / / A m / / B m,n (cid:0) D (cid:1) /Γ (cid:0) B m,n (cid:0) D (cid:1)(cid:1) Λ D m,n / / Z / / . An interesting application of this eqivalent definition of the morphisms φ m,n is the following alter-native purely algebraic proof of Lemma 4.1. Proof of Lemma 4.1.
We fix ψ ∈ B n , x ∈ A m and ˜ x ∈ B m ( D n ) so that φ ,m (˜ x ) = x . Recall thatthe map Λ D m,n admits a left inverse and therefore the upper short exact sequence of the abovediagram is split. Note that, by the multiplication rule of a semidirect product, the action of ψ on x is induced by the conjugation seen as elements of the mixed braid group: ( θ m,n ( ψ ) ( x ) , B n ) = (cid:0) B m ( D n ) , ψ − (cid:1) (˜ x, B n ) (cid:0) B m ( D n ) , ψ (cid:1) which proves that Assumption 2.13 is satisfied. We deducethat γ (cid:0) ψ − ˜ xψ (cid:1) = γ (˜ x ) . Then it follows from the commutativity of the above diagram that φ m,n (˜ x ) = x = ψ − xψ , which corresponds to the proof of Assumption 2.16. Remark 4.4
A natural idea would be to apply the same principle using the further lower centralquotient groups B m,n (cid:0) D (cid:1) /Γ k (cid:0) B m,n (cid:0) D (cid:1)(cid:1) for k > . However [BGG17, Corollary 3.8] provesthat Γ (cid:0) B m,n (cid:0) D (cid:1)(cid:1) = Γ (cid:0) B m,n (cid:0) D (cid:1)(cid:1) if m, n > , hence this idea is not relevant. It mightpotentially be relevant for m = 1 or m = 2 , but in this case the lower central quotients are not yetwell-understood. Taking inspiration from the situation for classical braid groups described in §4.1, the main idea toconstruct homological representations for the surface braid groups consists in using the quotientsdefined by the lower central series of some mixed surface braid groups. As for the classical braidgroups, the abelianisation gives an interesting family of representations (see §4.2.1) which, as faras the authors are aware, does not appear in the literature. However, in contrast to classical braidgroups, it is relevant to consider the further lower central quotients. In particular the third lowercentral quotients represent an interesting case and this case is detailed in §4.2.2. For orientablesurfaces, this option is actually a reinterpretation of Bellingeri, Godelle and Guaschi in [BGG17]of the work by An and Ko [AK10] to extend some homological representations from the classicalbraid groups to the surface braid groups (see §5.2 for further details on this point).For the remainder of §4.2, we fix two natural numbers g > and c > and consider a surface S which is either the orientable surface Σ g, or the non-orientable surface N c, , as denoted in §3.4.2.For all natural numbers n , we denote by S ( n ) the surface (cid:0) Σ , (cid:1) ♮n ♮S obtained using the boundaryconnected sum.Let n > and m > be two natural numbers. Let BrDiff I (cid:0) S ( n ) (cid:1) be the topological groupof diffeomorphisms of S ( n ) that fix a given interval I ⊂ ∂S pointwise and that become isotopicto the identity (fixing I throughout the isotopy) after capping the non-parametrised boundary-components of ∂S ( n ) (i.e. the n boundary-components corresponding to the n copies of ∂ Σ , ).Note that π (cid:0) BrDiff I (cid:0) S ( n ) (cid:1)(cid:1) is the automorphism group of S ( n ) in the groupoid B introduced in§3.4.3. In particular, π (cid:16) BrDiff I (cid:16) S ( n ) (cid:17)(cid:17) ∼ = B n ( S ) is the n -th surface braid group of S . We now fix m distinct points { z i } i ∈{ ,...,m } in the parametrisedinterval I in the boundary of S and choose the configuration c = { z , . . . , z m } as the basepoint of43 m (cid:0) S ( n ) (cid:1) .Let θ m (cid:0) S ( n ) (cid:1) : BrDiff I (cid:0) S ( n ) (cid:1) → Homeo c (cid:0) C m (cid:0) S ( n ) (cid:1)(cid:1) be the continuous group morphism givingthe natural action of BrDiff I (cid:0) S ( n ) (cid:1) on the coordinates of the configuration space C m (cid:0) S ( n ) (cid:1) , i.e.defined by ϕ ( { x , . . . , x m } 7−→ { ϕ ( x ) , . . . , ϕ ( x m ) } ) . In particular, θ m (cid:0) S ( n ) (cid:1) ( ϕ ) preserves the basepoint c since ϕ fixes pointwise the interval I con-taining c . Remark 4.5
In the general construction recalled at the beginning of this section, we are taking G = BrDiff I (cid:0) S ( n ) (cid:1) and X = C m ( S ( n ) ) . Note that π ( C m ( S ( n ) ) , c ) ∼ = B m ( S ( n ) ) . To complete theconstruction, we now define the relevant quotient of B m ( S ( n ) ) .We recall from §3.4.3 that Λ Sm,n : B m,n ( S ) ։ B n ( S ) denotes the split surjective morphisminduced by forgetting the first m coordinates and that B m,n ( S ) ∼ = B m (cid:0) S ( n ) (cid:1) ⋊ B n ( S ) . Hence thenatural action of B n ( S ) on B m (cid:0) S ( n ) (cid:1) induced by θ m (cid:0) S ( n ) (cid:1) is equivalent to the conjugate actionin B m,n ( S ) , if we regard these two groups as subgroups of B m,n ( S ) .Recall that we denote by γ l ( B m,n ( S )) and γ l ( B n ( S )) the respective canonical projections on thequotient by the l th term of the lower central series of B m,n ( S ) and B n ( S ) , and that we omitthe groups from the notations where there is no ambiguity. They canonically induce a morphism Λ Sl,m,n so that Λ Sl,m,n ◦ γ l ( B m,n ( S )) = γ l ( B n ( S )) ◦ Λ Sm,n .We consider the kernel of Λ Sl,m,n . By the universal property of the kernel, there exists a uniquesurjective morphism that we denote by φ Sl,m,n , so that the following diagram is commutative (andwhere the two lines are exact): / / B m (cid:0) S ( n ) (cid:1) / / φ Sl,m,n (cid:15) (cid:15) (cid:15) (cid:15) B m,n ( S ) Λ Sm,n / / γ l (cid:15) (cid:15) (cid:15) (cid:15) B n ( S ) / / γ l (cid:15) (cid:15) (cid:15) (cid:15) / / ker (cid:16) Λ Sl,m,n (cid:17) / / B m,n ( S ) /Γ l ( B m,n ( S )) Λ Sl,m,n / / B n ( S ) /Γ l ( B n ( S )) / / . (4.1)Then, we have the following key property. Proposition 4.6
For all integers m > , ker (cid:16) φ Sl,m,n (cid:17) is preserved by the action π (cid:0)(cid:0) θ m (cid:0) S ( n ) (cid:1)(cid:1) π (cid:1) of B n ( S ) on B m (cid:0) S ( n ) (cid:1) .Proof. We consider ψ ∈ B n ( S ) and x ∈ ker (cid:16) φ Sl,m,n (cid:17) . As elements of B m,n ( S ) , the action of ψ on x is defined by (cid:0) θ m,n ( ψ ) ( x ) , B n ( S ) (cid:1) = (cid:16) B m ( S ( n ) ) , ψ − (cid:17) (cid:0) x, B n ( S ) (cid:1) (cid:16) B m ( S ( n ) ) , ψ (cid:17) . By theuniversal property of the kernel, there exists a unique morphism ker (cid:16) φ Sl,m,n (cid:17) → Γ l ( B m,n ( S )) suchthat the following square is commutative ker (cid:16) φ Sl,m,n (cid:17) / / (cid:127) _ (cid:15) (cid:15) Γ l ( B m,n ( S )) (cid:127) _ (cid:15) (cid:15) B m (cid:0) S ( n ) (cid:1) (cid:31) (cid:127) / / B m,n ( S ) . Therefore (cid:0) x, B n ( S ) (cid:1) ∈ Γ l ( B m,n ( S )) . Since Γ l ( B m,n ( S )) is a normal subgroup of B m,n ( S ) ,it follows from the commutativity of the left-hand square of the diagram (4.1) that ψ − xψ ∈ ker (cid:16) φ Sl,m,n (cid:17) . 44 emark 4.7
In the current situation, we are only able to deduce Assumption 2.13 and not alsoAssumption 2.16, as we could in the proof of Lemma 4.1 above. Indeed the conjugation actionautomatically becomes trivial only for the abelianisation and not for the further lower centralquotients.Hence Assumption 2.13 is satisfied and then Definition 2.15 gives a homological representation, forall index k > : L k (cid:16) F θ m ,φ Sl,m,n (cid:17) : B n ( S ) −→ Aut Z (cid:18) H k (cid:18)(cid:16) C m (cid:16) S ( n ) (cid:17)(cid:17) φ Sl,m,n ; Z (cid:19)(cid:19) . Using the abelianisation of a mixed surface braid group B m,n ( S ) , the constructed homologicalrepresentation L k (cid:16) F θ m ,φ S ,m,n (cid:17) satisfies two additional interesting properties. First, the followinglemma shows that the quotient groups Λ S ,m,n does not depend on n if n > . Such property isinter alia crucial for stating some polynomiality results in §8. Lemma 4.8
For all m > and n > , there is an isomorphism ker (cid:16) Λ S ,m,n (cid:17) ∼ = ker (cid:16) Λ S ,m,n +1 (cid:17) .Proof. Either if S = Σ g, or if S = N c, , we recall from §3.4.3 that the subgroup B m,n ( S ) of B m + n ( S ) is isomorphic the semidirect product of B m (cid:0) S ( n ) (cid:1) ⋊ B n ( S ) . We also recall that thepresentations of the surface braid groups are detailed in Proposition 3.27. Since we have the relation σ i σ i +1 σ i = σ i +1 σ i σ i +1 for the braid generators of B n ( S ) , we deduce that γ ( σ i ) = γ ( σ i +1 ) forall i ∈ { , . . . , n } .Furthermore, it is a standard observation that the conjugation action by the braid generators { σ i } i ∈{ ,...,n − } of B n ( S ) on the generators X = { ξ j } j ∈{ ,...,n } of B m (cid:0) S ( n ) (cid:1) is the same as theaction of the braid group B n on the fundamental group π ( D n , p ) . More precisely, using theidentification ξ j = ( σ if j = 1 σ − ◦ σ − ◦ · · · ◦ σ − i − ◦ σ i ◦ σ i − ◦ · · · ◦ σ ◦ σ if j ∈ { , . . . , n } , it follows from the presentation of B m + n ( S ) that σ − i ξ j σ i = ξ i +1 if j = i ; ξ − i +1 ξ i ξ i +1 if j = i + 1; ξ j if j / ∈ { i, i + 1 } .Hence γ ( ξ i ) = γ ( ξ i +1 ) and we deduce from the presentations of B m (cid:0) S ( n ) (cid:1) and B n ( S ) that thenumbers of generators of B m,n ( S ) /Γ ( B m,n ( S )) is fixed when n varies.Moreover, a straightforward computation from their presentations shows that B n (Σ g, ) /Γ ( B n (Σ g, )) ∼ = Z g ⊕ Z / Z and B n ( N c, ) /Γ ( B n ( N c, )) ∼ = Z g ⊕ Z and are thus independent of n . A fortiori ker (cid:16) Λ S ,m,n (cid:17) and ker (cid:16) Λ S ,m,n +1 (cid:17) are isomorphic. Remark 4.9
For the orientable case, [BGG17, Proposition 3.3] shows that B m,n (Σ g, ) /Γ ( B m,n (Σ g, )) ∼ = Z g ⊕ ( Z / Z ) d m,n where d m,n = if n = m = 1 ; if m = 1 and n > or m > and n = 1 ; if m > and n > . 45 otation 4.10 We denote ker (cid:16) Λ S ,m,n (cid:17) by A ,m ( S ) and φ S ,m,n by φ S ,m for all natural numbers n . This notation is consistent for n > by Lemma 4.8. Although for n = 0 and n = 1 the kernelsare not isomorphic to the others, we use this slight abuse of notations for simplicity.Furthermore, Assumption 2.16 is satisfied using the morphism θ m (cid:0) S ( n ) (cid:1) and the quotient group A ,m ( S ) . Indeed, we have the following key property. Proposition 4.11
For all natural numbers m > , the action of B n ( S ) on the group A ,m ( S ) istrivial.Proof. We consider ψ ∈ B n ( S ) and x ∈ A ,m ( S ) . We fix ˜ x ∈ B m (cid:0) S ( n ) (cid:1) so that φ S ,m (˜ x ) = x . Recall that the action of ψ on x is induced by the conjugation as elements of B m,n ( S ) : (cid:0) θ m,n ( ψ ) ( x ) , B n ( S ) (cid:1) = (cid:16) B m ( S ( n ) ) , ψ − (cid:17) (cid:0) ˜ x, B n ( S ) (cid:1) (cid:16) B m ( S ( n ) ) , ψ (cid:17) . Therefore γ (cid:0) ψ − ˜ xψ (cid:1) = γ (˜ x ) . Then it follows from the commutativity of the diagram (4.1) that φ S ,m (˜ x ) = x = ψ − xψ . Contrary to classical braid groups (see Remark 4.4), another possibility to build representations forthe surface braid groups is to consider the third lower central quotients of the mixed braid groups,which are generally speaking different from the second ones in this case. As for the situation of§4.2.1, the following lemma shows that the quotient groups Λ S ,m,n do not depend on n if n > .Again such a property will be used to prove some polynomiality results in §8. Lemma 4.12
For all m > and n > , there is an isomorphism ker (cid:16) Λ S ,m,n (cid:17) ∼ = ker (cid:16) Λ S ,m,n +1 (cid:17) .Proof. Recall the group B m,n ( S ) is isomorphic the semidirect product of B m (cid:0) S ( n ) (cid:1) ⋊ B n ( S ) .Recall that we have the relation σ i σ i +1 σ i = σ i +1 σ i σ i +1 for the braid generators of B n ( S ) byProposition 3.27. Since γ ([ σ i , [ σ i +1 , σ i ]]) is trivial, we deduce that γ ( σ i ) = γ ( σ i +1 ) for all i ∈ { , . . . , n } in the metabelian quotients B m,n ( S ) /Γ ( B m,n ( S )) and B n ( S ) /Γ ( B n ( S )) .Furthermore, it follows from the presentation of B m + n ( S ) that the conjugation action by the braidgenerators { σ i } i ∈{ ,...,n − } of B n ( S ) on the generators { ξ j } j ∈{ ,...,n } of B m (cid:0) S ( n ) (cid:1) is defined by σ − i ξ j σ i = ξ i +1 if j = i ; ξ − i +1 ξ i ξ i +1 if j = i + 1; ξ j if j / ∈ { i, i + 1 } .We deduce from the relation γ ([ ξ i +1 , [ σ i , ξ i +1 ]]) = 1 B m,n ( S ) that γ ( ξ i ) = γ ( ξ i +1 ) . Then, usingthe fact that γ ( σ i ) = γ ( σ i +1 ) , we obtain γ ( ξ i +1 ) = γ (cid:0) σ − i ξ i σ i (cid:1) = γ (cid:0) σ − i +1 ξ i σ i +1 (cid:1) = γ (cid:0) ξ − i +1 ξ i ξ i +1 (cid:1) = γ ( ξ i ) . The numbers of generators of B m,n ( S ) /Γ ( B m,n ( S )) does not depend on n because of the pre-sentations of B m (cid:0) S ( n ) (cid:1) and B n ( S ) (see Proposition 3.27) and the semidirect product structure.Moreover, the metabelian quotient B n ( S ) /Γ ( B n ( S )) is also independent of n . Hence ker (cid:16) Λ Sm,n (cid:17) and ker (cid:16) Λ Sm,n +1 (cid:17) are isomorphic. Remark 4.13
For the orientable case, [BGG17, Corollary 3.9] proves that for m, n > and g > B m,n (Σ g, ) /Γ ( B m,n (Σ g, )) ∼ = (cid:0) Z × Z g (cid:1) ⋊ Z g . Notation 4.14
For simplicity, we denote ker (cid:16) Λ S ,m,n (cid:17) by A ,m ( S ) for all natural numbers n .46 .3 Mapping class groups The use of configuration spaces to construct homological representations for the surface braidgroups in §4.2 can be repeated for mapping class groups of surfaces. Indeed the quotients givenby the lower central series associated with some ordered and unordered configuration spaces giverise to interesting families of homological representations of these groups. We will see in §5.3 thatthe Magnus representations and those introduced by [Mor07] are particular cases of homologicalrepresentations. For the remainder of §4.3, we fix two natural numbers g > and c > andconsider a surface S which is either the orientable surface Σ g, or the non-orientable surface N c, ,as denoted in §3.4.2. A first idea to construct homological representations for the mapping class groups of surfacesmakes use of ordered configuration spaces. We fix a natural number m > . Recall from §3.4.3that F m ( S ) denotes the ordered configuration space of m points on the surface S . Considering thefollowing subspace of S × m D m ( S ) := (cid:8) ( x , . . . , x m ) ∈ S × m | x i = x j for some i = j (cid:9) , then F m ( S ) = S × m \ D m ( S ) . We fix m distinct points { z i } i ∈{ ,...,m } in the preferred boundary-component of S , and another point p in the same boundary-component, distinct from all z i . Wechoose the configuration c = ( z , . . . , z m ) as the basepoint of F m ( S ) .Each diffeomorphism ϕ of S that fixes the preferred boundary-component of S also fixes c and thediagonal action of ϕ on S × m preserves the subspace D m ( S ) . Hence the natural action of Diff ∂ ( S ) on the coordinates of the configuration space F m ( S ) defines a continuous group morphism θ F m ( S ) : Diff ∂ ( S ) −→ Homeo c ( F m ( S )) . We consider the canonical projection on the quotient by the l th term of the lower central series of π ( F m ( S ) , c ) , denoted by γ l . Assumption 2.13 is automatically satisfied since Γ l ( π ( F m ( S ) , c )) is a characteristic subgroup of π ( F m ( S ) , c ) . Then Definition 2.15 gives a representation, for allindex k > : L k (cid:0) F θ Fm ( S ) ,γ l (cid:1) : π Diff ∂ ( S ) −→ Aut Z ( H k (( F m ( S )) γ l ; Z )) . An interesting modification of this construction consists in removing the basepoint p from the con-figuration space: the version with Borel-Moore homology of this alternative is then endowed with anatural free generating set (see §6) and recovers Moriyama representations (see §5.3.2). We denotethe surface S \ { p } by S • . Namely, we consider now the configuration space F m ( S • ) = S × m \ ( D m ( S ) ∪ A m ( S, p )) where A m ( S, p ) is the space { ( x , . . . , x m ) ∈ S × m | x i = p for some i } .Again, any diffeomorphism ϕ of S that fixes the preferred boundary-component of S also fixes c and the diagonal action of ϕ on S × m preserves the subsets D m ( S ) and A m ( S, p ) . Hencethe natural action of Diff ∂ ( S ) on the coordinates of the configuration space F m ( S • ) definesa continuous group morphism θ F m ( S • ) : Diff ∂ ( S ) → Homeo c ( F m ( S • )) . Assumption 2.13 beingagain satisfied since Γ l ( π ( F m ( S • ) , c )) is a characteristic subgroup of π ( F m ( S • ) , c ) , Definition2.15 gives a representation, for all index k > : L k (cid:16) F θ Fm ( S • ) ,γ l (cid:17) : π Diff ∂ ( S ) −→ Aut Z ( H k ( F m ( S • ) γ l ; Z )) . Another natural idea to apply the general construction of §2 for mapping class groups of surfacesis to consider the lower central quotient groups of unordered configuration spaces.We fix a natural number m > . Recall from §3.4.3 that C m ( S ) denotes the unordered configurationspace of m points on the surface S . Using the space D m ( S ) introduced in §4.3.1, C m ( S ) can be47iewed as the space ( S × m \ D m ( S )) / S m . We fix m distinct points { z i } i ∈{ ,...,n } in the boundary S . We choose the configuration c = { z , . . . , z m } as the basepoint of C m ( S ) . Recall from §3.4.3that the braid group B m ( S ) is the fundamental group of the configuration space π ( C m ( S ) , c ) .The action by permutation of coordinates of the symmetric group S n on F m ( S ) commutes withthe action of any diffeomorphism ϕ of S on the coordinates of the elements of F m ( S ) . This inducesa canonical surjective continuous group morphism Homeo c ( F m ( S )) ։ Homeo c ( C m ( S )) . Thecomposite of the continuous group morphism θ F m ( S ) defined in §4.3.1 by this canonical surjectiongives the natural action of Diff ∂ ( S ) on the coordinates of the configuration space C m ( S ) defininga continuous group morphism: θ C m ( S ) : Diff ∂ ( S ) → Homeo c ( C m ( S )) . Again, we consider the canonical projection on the quotient by the l th term of the lower cen-tral series of π ( C m ( S ) , c ) , denoted by γ l . Assumption 2.13 is automatically satisfied since Γ l ( π ( C m ( S ) , c )) is a characteristic subgroup of π ( C m ( S ) , c ) . Therefore, Definition 2.15 givesa representation, for all index k > : L k (cid:0) F θ Cm ( S ) ,γ l (cid:1) : π Diff ∂ ( S ) −→ Aut Z ( H k (( C m ( S )) γ l ; Z )) . Additional properties for orientable surfaces:
The lower central series and quotients forthe surface braid groups for an orientable surface S = Σ g, have already been the subject of anintensive study in the literature. This allows us to give additional properties on the homologicalrepresentations in this situation.Taking m = 1 , the first homology group is the only one which produces a non-trivial homologicalrepresentation L ( F θ S ,γ l ) for any l . Since the lower central series of a free group on two or moregenerators does not stabilise, i.e. Γ l ( F g ) = Γ l +1 ( F g ) for all l > and g > , the study of thelower central quotient of the fundamental group of the surface Σ g, is an active research topic. Werefer the reader to [MKS04] for further details on this question. For l = 0 , the action correspondsto the natural action on the first homology group of the surface and its kernel is the Torelli group.For convenience we denote it by a g : Γ g, → Aut Z ( H (Σ g, , Z )) .Surface braid groups on m = 2 strands represent a more difficult situation. For instance, for thetorus with one boundary component Σ , , [BGG08, Section 4] proves that the lower central seriesdoes not stabilise: Γ l ( B (Σ , )) = Γ l +1 ( B (Σ , )) for all l > . Actually the question of whethersurface braid group B (Σ g, ) for any g > is residually nilpotent is still open.We now fix m > . [BGG08] and [BGG17] give a complete study of the lower central quotientgroups of B m (Σ g, ) . In particular they show that the lower central series stabilises, namely that Γ ( B m (Σ g, )) = Γ ( B m (Σ g, )) . Therefore it is relevant to define the constructed representations L k (cid:16) F θ Cm ( Σ g, ) ,γ l (cid:17) only for l . Moreover, they prove the following key results: Proposition 4.15 ([BGG08, Theorem 1] [BGG17, Corollary 3.12])
The abelianisation of B m (Σ g, ) is isomorphic to the product Z g × Z / Z . The third lower central quotient B m (Σ g, ) / Γ ( B m (Σ g, )) is isomorphic to the semidirect product ( Z × Z g ) ⋊ Z g . More precisely, recalling the presentation of Proposition 3.27, the first factor Z is central andis generated by σ := γ ( σ i ) for all i ∈ { , . . . , m − } , the second factor Z g is generated by { a i := γ ( a i ) } i ∈{ ,...,g } , and the third factor Z g is generated by { b i := γ ( b i ) } i ∈{ ,...,g } ; for all j ∈ { , . . . , g } , the generator b j acts trivially on a i for i ∈ { , . . . , g } \ { j } and a j b j = σ b j a j . The result on the third lower central quotient allows us to obtain an additional property forthe associated homological representation: we can find the best subgroup of the mapping classgroups Γ g, which acts on H m ( C m (Σ g, ) γ , Z ) as a B m (Σ g, ) / Γ ( B m (Σ g, )) -module, i.e. sothat Assumption 2.16 is satisfied. Beforehand, we have to refine the result of Proposition 4.15.48ccording to the unpublished work [BGG11], the ideas developed below were already explored byChristian Blanchet but have not yet been published.Let { α i , β i } i ∈{ ,...,g } be loops in Σ g, based at p that form a system of meridians and parallelsof the surface: α i and β i respectively encircle the meridian and the parallel of the i th handle.They give a free generating set { [ α i ] , [ β i ] } i ∈{ ,...,n } for the fundamental group π (Σ g, , p ) . Wealso assume that these loops are such that the product of commutators is a positively orientedloop around the boundary component. The abelianisation γ thus induces a symplectic basis { A i , B i } i ∈{ ,...,n } for the first homology group of the surface H g := H (Σ g, ; Z ) with respect tothe algebraic intersection form ω g : H g × H g → Z . Moreover, the operation ( k, c ) · ( k, c ) = ( k + k ′ + ω g ( c, c ′ ) , c + c ′ ) for all k, k ′ ∈ Z and c, c ′ ∈ H g defines a group structure on the set Z × H g : we denote by Z × ω g H g this central extension. Then: Lemma 4.16
The third lower central quotient B m (Σ g, ) / Γ ( B m (Σ g, )) is isomorphic to thecentral extension Z × ω g H g . Moreover the action of the mapping class group Γ g, on the quotient B m (Σ g, ) / Γ ( B m (Σ g, )) ։ H g is the natural action a g .Proof. The isomorphism is given by sending σ to the generator of Z in the central extension, a i to A i and b i to B i for all i ∈ { , . . . , g } . The relation a j b j = σ b j a j is preserved through thismorphism by the definition of the intersection form (up to the sign convention).As recalled in Remark 3.28, the generators { a i , b i } i ∈{ ,...,g } define a generating set for the funda-mental group of Σ g, : the action of Γ g, induced by θ C m (Σ g, ) on the image of these generators inthe quotient H g is thus exactly the symplectic representation of the mapping class group.Hence Lemma 4.16 shows that we must restrict to a subgroup of the Torelli group I g, to obtaina trivial action on the third lower central quotient. Since Aut Z ( Z ) ∼ = Z / Z , there exists a map k : Γ g, → Hom ( H g , Z ) so that the homological representation is defined by L m (cid:16) F θ Cm ( Σ g, ) ,γ (cid:17) = (cid:20) ± Id Z k a g (cid:21) . The following result shows that this map k is related to the well-known Chillingworth homomor-phism
Chill introduced in [Chi72] which describes the action of the Torelli group I g, on the windingnumbers of the curves of Σ g, . Lemma 4.17
The map k is a crossed homomorphism and its kernel coincides with the kernel ofChillingworth homomorphism.Proof. Since L m (cid:16) F θ Cm ( Σ g, ) ,γ (cid:17) is a morphism, we deduce that that k ( ϕ ◦ ψ ) = k ( ψ )+ k ( ϕ ) a g ( ψ ) for all ϕ, ψ ∈ Γ g, : this proves that k is a crossed homomorphism. Moreover [Mor89] proves that H ( Γ g, , H g ) ∼ = Z . Hence k = λ · Chill + c where λ ∈ Z and c is a principal crossed homomorphism:restricting to the Torelli group, we deduce that ker ( k ) = ker ( Chill ) .Hence the homological representation L m (cid:16) F θ Cm ( Σ g, ) ,γ (cid:17) acts on the homology H m (( C m (Σ g, )) γ , Z ) as a (cid:18) Z × ω g H g (cid:19) -module if we restrict to any subgroup of the (index- or index- ) subgroup of thekernel of the Chillingworth homomorphism so that the sign of the ± Id Z is positive. For in-stance, [Joh83] proves that the Johnson subgroup K g, , which is the kernel of the natural map Γ g, → Aut Z ( π (Σ g, , p ) /Γ ( π (Σ g, , p ))) , is a subgroup of ker ( Chill ) . Remark 4.18
If we assume that g > , then it follows from [BGP14] that every proper subgroupof the mapping class group Γ g, has index at least : a fortiori there are no non-trivial morphisms Γ g, → Z / Z in this case. Thus the subgroup of the kernel of the Chillingworth homomorphismmentioned above is of index (i.e., equal to it). So it suffices to restrict to any subgroup the kernel of49he Chillingworth homomorphism in order that the homological representation L m (cid:16) F θ Cm ( Σ g, ) ,γ (cid:17) acts on the homology H m (( C m (Σ g, )) γ , Z ) as a (cid:18) Z × ω g H g (cid:19) -module if g > . In this subsection, we reinterpret some of the constructions of sections 4.1–4.3 using the generalprocedure summarised in §2.5. In each case, this amounts to defining a continuous functor C t −→ Cov Q or C t −→ g Cov Q , where C t is an appropriate topological category (obtained by applying the topological version ofQuillen’s bracket construction, as described in §3.4.5), and Q is a group. In each of our examples for§§4.1–4.3, objects of C t are surfaces and morphism spaces are spaces of embeddings. The generalpattern of the construction is as follows: we send each object (surface) to a configuration spaceon that surface, equipped with a choice of quotient of its fundamental group onto Q . Embeddingsbetween surfaces clearly induce maps of configuration spaces, and we then check that either (a) theinduced homomorphisms of fundamental groups commute with the chosen quotients to Q , in whichcase we obtain a functor C t → Cov Q , or (b) the weaker property that the induced homomorphismsof fundamental groups at least preserve the kernels of these quotients, in which case we obtain afunctor C t → g Cov Q . Let us fix an integer m > and set Q = A m = Z ⊕ B ab m , which is Z for m = 1 and Z for m > .Let U β t be the topological category defined in Example 3.37 and let U β t > be its full subcategoryon all objects except the zero-punctured disc D = D . We define a continuous functor F : U β t > −→ Cov A m (4.2)as follows. For an object (i.e., a punctured disc) D n of U β t > we define F ( D n ) to be the unorderedconfiguration space C m ( D n ) , based at a fixed configuration c contained in ∂ D , and we choosethe quotient π ( C m ( D n ) , c ) −→→ A m = Z ⊕ B ab m to be the homomorphism φ m,n constructed in §4.1. Morphisms of U β t > are certain embeddings ϕ : D n → D n ′ (see Example 3.37 for a precise description), and induce maps of configuration spaces C m ( ϕ ) : C m ( D n ) → C m ( D n ′ ) , so we may set F ( ϕ ) = C m ( ϕ ) . This clearly defines a continuousfunctor if one ignores the condition that C m ( ϕ ) ∗ must commute with the quotients φ m,n and φ m,n ′ ,so it remains to check this condition. Now, all morphisms of U β t > may be written as a compositionof a “standard embedding” D n → D n ′ followed by an automorphism of D n ′ . This is because, asexplained in Example 3.37, the morphism space U β t ( D n , D n ′ ) consists of embeddings that maybe extended to a diffeomorphism of D n ′ . Hence it suffices to check the commutativity conditionfor ϕ either a standard embedding or an automorphism. For automorphisms, this is exactly thecontent of Lemma 4.1, and for standard embeddings, this is what was checked explicitly just aboveDefinition 4.2. Thus we have a well-defined continuous functor F .This gives us the first input of Definition 2.21, with C t = U β t > . Taking the ground ring k to be Z and defining R = M = Z [ A m ] (i.e., not twisting the coefficients), we therefore obtain a functor L i ( F ) : π (cid:0) U β t > (cid:1) = U β > −→ Z [ A m ] − Mod , for any i > . Since U β is equivalent to U β > with an initial object adjoined, we may extend thisto a functor L i ( F ) : U β −→ Z [ A m ] − Mod by sending the initial object of U β (the zero-punctured disc) to the initial object of Z [ A m ] − Mod (the trivial Z [ A m ] -module). This is exactly the functor LB m of Proposition 4.3.We summarise this discussion as: 50 roposition 4.19 The continuous functor (4.2) determines, through the construction of §2.5, thefunctor LB m : U β −→ Z [ A m ] − Mod of Proposition 4.3, and hence in particular a family of representations of the classical braid groups.
Now let us fix a surface S = Σ g, or N c, , an integer m > and l ∈ { , } . Recall that the group ker (cid:16) Λ Sl,m,n (cid:17) is defined in diagram (4.1); it is the subgroup ker (cid:16) Λ Sl,m,n (cid:17) = B m ( S ( n ) ) ∩ Γ l ( B m,n ( S )) of the surface braid group B m ( S ( n ) ) , where S ( n ) = D n ♮S . According to Lemmas 4.8 and 4.12, itis independent of n for n > , so we may define A l,m ( S ) to be this group for any n > .We set Q = A l,m ( S ) and let h β t , B ( S ) t i be the topological category defined in Example 3.39,where B ( S ) = B g, +2 if S = Σ g, and B ( S ) = B c, − if S = N c, . Let h β t , B ( S ) t i > denote itssubcategory on all objects of the form S ( n ) = D n ♮S for n > . We define continuous functors G : h β t , B ( S ) t i > −→ Cov A ,m ( S ) G : h β t , B ( S ) t i > −→ g Cov A ,m ( S ) (4.3)as follows. We send an object S ( n ) to the unordered configuration space C m ( S ( n ) ) , based at aconfiguration c contained in the parametrised interval I ⊂ ∂S , and we choose the quotient π ( C m ( S ( n ) ) , c ) = B m ( S ( n ) ) −→→ A l,m ( S ) to be the quotient map φ Sl,m,n constructed in §4.2 (see in particular diagram (4.1)).Morphisms of h β t , B ( S ) t i > are certain embeddings ϕ : S ( n ) → S ( n ′ ) for n n ′ (see Example3.39 for a precise description). Hence they induce maps of configuration spaces C m ( ϕ ) : C m ( S ( n ) ) → C m ( S ( n ′ ) ) , so we may set G l ( ϕ ) = C m ( ϕ ) . This clearly defines a continuous functor, if we ignorethe condition that C m ( ϕ ) ∗ must commute with the quotients φ Sl,m,n and φ Sl,m,n ′ (in the case l = 2 )or simply preserve the kernels of these quotients (in the case l = 3 ), so it remains to check theseconditions. As in the case of U β t , all morphisms of h β t , B ( S ) t i may be written as a compositionof the “standard inclusion” S ( n ) = D n ♮S → D n ′ ♮S = S ( n ′ ) followed by an automorphism of D n ′ ,due to the fact that the morphism space of h β t , B ( S ) t i ( S ( n ) , S ( n ′ ) ) consists of embeddings thatmay be extended to diffeomorphisms of S ( n ′ ) , as explained in Example 3.39. Hence it suffices tocheck the conditions on π for ϕ either a standard inclusion or an automorphism. For l = 2 , thefact that the induced map C m ( ϕ ) ∗ on π commutes with the quotients φ S ,m,n and φ S ,m,n ′ followsfrom Lemma 4.8 when ϕ is a standard inclusion and from the combination of Propositions 4.6 and4.11 when ϕ is an automorphism. For l = 3 , the fact that C m ( ϕ ) ∗ sends the kernel of φ S ,m,n to thekernel of φ S ,m,n ′ follows from Lemma 4.12 when ϕ is a standard inclusion, and from Proposition4.6 when ϕ is an automorphism. Thus we have well-defined continuous functors G and G .This gives us the first input of Definition 2.21 (for l = 2 ) or Definition 2.22 (for l = 3 ), with C t = h β t , B ( S ) t i > . Taking the ground ring k to be Z and defining R = M = Z [ A l,m ( S )] (i.e.,not twisting the coefficients), we therefore obtain functors L i ( G ) : π ( h β t , B ( S ) t i > ) = h β , B ( S ) i > −→ Z [ A ,m ( S )] − Mod L i ( G ) : π ( h β t , B ( S ) t i > ) = h β , B ( S ) i > −→ Z [ A ,m ( S )] − Mod tw , (4.4)for any i > . This extends functorially the families of homological representations constructed in§4.2, for any m > and for l ∈ { , } . We summarise this discussion as: In fact, when ϕ is a standard inclusion, Lemma 4.12 actually shows that C m ( ϕ ) ∗ commutes with the quotients φ S ,m,n and φ S ,m,n ′ . roposition 4.20 For any m > , the continuous functors (4.3) determine, via the constructionof §2.5, functors of the form (4.4) that extend the families of representations defined in §4.2 for thesurface braid groups B n ( S ) for S any compact, connected surface with one boundary-component. Remark 4.21
We do not have analogous functorial extensions, in the setting of §2.5, of therepresentations constructed in §4.3 of mapping class groups of surfaces S . This is because thequotient groups Q in those constructions are typically not independent of the surface S . Forexample, in §4.3.2 we constructed representations of π Diff ∂ ( S ) using Q = B m ( S ) /Γ l ( B m ( S )) ,but this depends non-trivially on S for any m > and l ∈ { , } by Proposition 4.15. By contrast,for example, the quotient groups Q that we used in the case of the classical braid groups B n areall isomorphic to Z (for m = 1 ) or Z (for m > ). This allowed us, for fixed m > , to constructa functor out of U β using the category Cov Z (for m = 1 ) or Cov Z (for m > ). We first describe certain spaces of unlinks, which will be used in our construction of functorialhomological representations of (extended and non-extended) loop braid groups. First, recall fromabove that there are (at least) two interpretations of loop braid groups LB n , as follows: • LB n ∼ = π (Diff +2 D ( D , L )) ,where L is an n -component unlink in the interior of the unit -disc D that diffeomorphisms areassumed to fix as a subset, the subscript “ D ” indicates that diffeomorphisms are assumed to fixpointwise a neighbourhood of a chosen pair of disjoint -discs in ∂ D and the superscript “ + ”indicates that diffeomorphisms are assumed to preserve the orientation of L (see Lemma 3.43).Secondly: • LB n ∼ = π ( U + n ) , where U + n denotes the space of oriented n -component unlinks in D , topologised as a subspace of Emb( n S , D ) / Diff + ( n S ) , where n S is the disjoint union of n copies of the circle (see §3.4.4). Wesimilarly have two interpretations of the extended loop braid groups LB ext n , as follows: • LB ext n ∼ = π (Diff D ( D , L )) , • LB ext n ∼ = π ( U n ) ,where, in the first interpretation, we have dropped the condition that diffeomorphisms preservethe orientation of L , and the space U n denotes the space of unoriented n -component unlinks in D , topologised as a subspace of Emb( n S , D ) / Diff( n S ) .Recall (cf. §3.4.4) that LB n is generated by two finite families of elements: certain elements τ i thateach correspond to a loop in U + n that interchanges two unknots without either of them passingthrough the other, and certain elements σ i that each correspond to a loop in U + n that interchangestwo unknots, while one passes through the other. The extended version LB ext n is generated bythese elements together with certain elements ρ i that each correspond to a loop in U n that rotatesa single circle by 180 degrees. See Figure 1 of [BH13] for a picture (note that our notation differsfrom theirs by a cyclic permutation of the letters τ σ ρ τ ). Using the presentations of LB n and LB ext n calculated in §3 of [BH13] (Proposition 3.3 for U R n ∼ = LB n and Proposition 3.7for R n ∼ = LB ext n ), we see that: • The abelianisation of LB n is isomorphic to Z ⊕ Z / Z for n > and trivial for n = 1 , wherethe Z summand is generated by an element σ , which is the image of all generators of theform σ i , and the Z / Z summand is generated by an element τ , which is the image of allgenerators of the form τ i . • The abelianisation of LB ext n is isomorphic to ( Z / Z ) for n > and Z / Z for n = 1 , where thethree summands are generated by elements σ , τ and ρ , which are the images of all generatorsof the form σ i , τ i and ρ i respectively, and the summand generated by ρ is the one presentalso in the n = 1 case. See Remark 4.23 below for a discussion of issues related to choosing basepoints. efinition 4.22 For an n -component unlink L in the interior of D and m > , we define X m ( L ) = { oriented m -component links L ′ in D , disjoint from L , so that L ⊔ L ′ is an unlink } ,Y m ( L ) = { unoriented m -component links L ′ in D , disjoint from L , so that L ⊔ L ′ is an unlink } , topologised as subspaces of Emb( m S , D ) / Diff + ( m S ) and Emb( m S , D ) / Diff( m S ) respectively.We now construct certain quotients of π ( X m ( L )) and π ( Y m ( L )) . The alpha quotient.
First, note that we have natural inclusions i : X m ( L ) ֒ −→ U + m and j : X m ( L ) ֒ −→ U + m + n , given respectively by forgetting the unlink L or adjoining it to the given m -component unlink.Here, n denotes the number of components of L . Note that j depends a choice of orientation of L (but i does not, of course). Taking induced maps on π and abelianising, we obtain a map α m ( L ) : π ( X m ( L )) −→ ( Z ⊕ Z / Z for m = 1 , ( Z ⊕ Z / Z ) ⊕ ( Z ⊕ Z / Z ) for m > (4.5)defined by α m ( L ) = ((ab ◦ i ∗ ) × (ab ◦ j ∗ )) ◦ ∆ . This is not surjective, but it is an easy exercise tocalculate its image. When m = 1 , its image is spanned by the element στ , which has infinite order.When m > , its image is spanned by the elements (1 , στ ) , ( στ, στ ) and ( τ, τ ) , which generatea subgroup isomorphic to Z ⊕ Z ⊕ Z / Z . Writing ˆ α m ( L ) for the map α m ( L ) after restricting itscodomain to its image, we have therefore constructed a quotient map ˆ α m ( L ) : π ( X m ( L )) −→→ ( Z for m = 1 , Z ⊕ Z ⊕ Z / Z for m > , (4.6)depending on a choice of orientation of the unlink L . Remark 4.23
There is a subtlety related to basepoints that arises in the above construction. Foreach r > let us choose an r -component unlink c r contained in ∂ D . We also fix an orientation of D , which therefore induces an orientation of ∂ D and hence an orientation of each c r , so we maythink of c r as an element of U + r , and also of X r ( L ) for any unlink L in the interior of D . To makeprecise the identification of LB n with π ( U + n ) above, we choose an isomorphism LB n ∼ = π ( U + n , c n ) .Now, the inclusion i sends c m to itself, so it induces a homomorphism i ∗ : π ( X m ( L ) , c m ) −→ π ( U + m , c m ) ∼ = LB m using the chosen identification with LB m . We may then compose this with the abelianisation mapof LB m to obtain the first component of the map α m ( L ) . On the other hand, the inclusion j sends c m to c m ⊔ L = c m + n . We therefore choose some path c m ⊔ L c m + n in U + m + n and use this choiceto identify the target of j ∗ with LB m + n : j ∗ : π ( X m ( L ) , c m ) −→ π ( U + m + n , c m ⊔ L ) ∼ = π ( U + m + n , c m + n ) ∼ = LB m + n . Composing this with the abelianisation map LB m + n , we obtain a map ab × j ∗ : π ( X m ( L ) , c m ) −→ Z ⊕ Z / Z , which (a priori) depends on the choice of path c m ⊔ L c m + n . Modifying this choice correspondsto inserting a conjugation automorphism of π ( U + m + n , c m + n ) into the composition. However, sincethe target of the homomorphism is abelian, this does not actually change anything, and so themap ab × j ∗ is in fact well-defined, independent of the choice of path c m ⊔ L c m + n .A similar issue arises in the next construction, which is resolved in the same way.53 + L − Figure 4.1
The sign convention for the geometric description of (one summand of) the quotient ˆ α m ( L ). The beta quotient.
In the unoriented case, we may perform a similar construction. We havenatural inclusions i : Y m ( L ) ֒ −→ U m and j : Y m ( L ) ֒ −→ U m + n , given respectively by forgetting the unlink L or adjoining it to the given m -component unlink.This time both i and j are independent of any choice of orientation of L . Taking induced maps on π and abelianising, we obtain a map β m ( L ) : π ( Y m ( L )) −→ ( ( Z / Z ) ⊕ ( Z / Z ) for m = 1 , ( Z / Z ) ⊕ ( Z / Z ) for m > (4.7)defined by β m ( L ) = ((ab ◦ i ∗ ) × (ab ◦ j ∗ )) ◦ ∆ . This is again not surjective, but it is an easyexercise to calculate its image. When m = 1 , its image is spanned by the two elements ( ρ, ρ ) and (1 , στ ) , which generate a subgroup isomorphic to ( Z / Z ) . When m > , its image is spanned bythese elements together with ( τ, τ ) and ( στ, στ ) , which generate a subgroup isomorphic to ( Z / Z ) .Writing ˆ β m ( L ) for the map β m ( L ) after restricting its codomain to its image, we have thereforeconstructed a quotient map ˆ β m ( L ) : π ( Y m ( L )) −→→ ( ( Z / Z ) for m = 1 , ( Z / Z ) for m > . (4.8)Note that this quotient does not depend on a choice of orientation of the unlink L . The gamma quotient.
Finally, it will be useful to construct a further quotient of ˆ α m ( L ) that isindependent of an orientation of the unlink L . Recall that the image of ˆ α m ( L ) is generated by theelement στ of LB ab > when m = 1 and by the elements (1 , στ ) , ( στ, στ ) and ( τ, τ ) of LB ab > ⊕ LB ab > when m > . Geometrically, the summand generated by στ when m = 1 and by (1 , στ ) when m > counts the number of times that a component of the m -component unlink passes throughone of the components of the fixed n -component unlink L . This is counted with sign, as depictedin Figure 4.1: note that the orientation of L is important to determine this sign, whereas theorientation of the component that is passing through it does not matter. Thus, reversing theorientation of L changes the sign of this count. On the other hand, the other two summands (when m > ), generated by ( στ, στ ) and ( τ, τ ) , count certain motions of the m -component unlink thatdo not involve L , so reversing the orientation of L does not affect these counts. If we write − L forthe unlink L with the opposite orientation, this discussion implies that the two quotients ˆ α m ( L ) and ˆ α m ( − L ) differ by the automorphism of their target given by x
7→ − x (in the case m = 1 )respectively ( x, y, z ) ( − x, y, z ) (in the case m > ). Thus, to obtain a further quotient of ˆ α m ( L ) that is independent of the choice of orientation of L , we simply need to quotient one Z summandof its image to a Z / Z summand. In summary, we have constructed a quotient map γ m ( L ) : π ( X m ( L )) −→→ ( Z / Z for m = 1 , Z / Z ⊕ Z ⊕ Z / Z for m > , (4.9)which is independent of any choice of orientation of the unlink L .We now apply these quotients and the construction of §2.5 to produce functorial homologicalrepresentations of the (extended and non-extended) loop braid groups.54 he alpha representations of the non-extended loop braid groups. Fix an integer m > and set Q to be the group Z if m = 1 and Z ⊕ Z ⊕ Z / Z if m > . Let U L β t be the topologicalcategory described in Remark 3.42: its objects are oriented unlinks in the interior of D and itsmorphisms from L to L ′ are embeddings ϕ : D → D fixing pointwise a disc in ∂ D , sending L into L ′ preserving their orientations, and which admit an extension to a diffeomorphism D ♮ D ∼ = D taking L ⊔ L ′′ onto L ′ , where L ′′ is some oriented unlink in the second copy of D in the boundaryconnected sum. Note that this last condition implies that the preimage of L ′ under ϕ must beequal to L (and not larger). See Remark 3.42 for the precise details of these morphism spaces.Let U L β t > be the full subcategory on all objects except the empty unlink. We define a continuousfunctor F αm : U L β t > −→ Cov Q (4.10)as follows. For an object (i.e., an oriented unlink) L of U L β t > we define F αm ( L ) to be the space X m ( L ) equipped with the quotient ˆ α m ( L ) of its fundamental group. Given a morphism ϕ : D → D from L to L ′ , we claim first of all that it induces a well-defined map X m ( ϕ ) : X m ( L ) → X m ( L ′ ) ,i.e., that it sends a link L ′′ in D r L such that L ⊔ L ′′ is an unlink to a link ϕ ( L ′′ ) in D r L ′ such that L ′ ⊔ ϕ ( L ′′ ) is an unlink. The fact that ϕ ( L ′′ ) is disjoint from L ′ follows from the factthat ϕ − ( L ′ ) = L , as noted above. To see that L ′ ⊔ ϕ ( L ′′ ) is an unlink, note that L ⊔ L ′′ bounds adisjoint union of -discs in D , since it is an unlink, so ϕ ( L ) ⊔ ϕ ( L ′′ ) also bounds a disjoint union of -discs, so it is an unlink, and hence so is its sub-link L ′ ⊔ ϕ ( L ′′ ) . Hence setting F αm ( ϕ ) = X m ( ϕ ) defines a continuous functor, if we ignore the condition that the induced map X m ( ϕ ) ∗ on π mustcommute with the quotients ˆ α m ( L ) and ˆ α m ( L ′ ) , so it remains to check this condition.All morphisms of U L β t > may be written as a composition of a “standard inclusion” ( D , L ) → ( D , L ′ ) followed by an automorphism of ( D , L ′ ) , due to the fact, recalled above, that morphismsare certain embeddings that admit an extension to a diffeomorphism. Hence it suffices to check thecommutativity condition for X m ( ϕ ) ∗ when ϕ is either a standard inclusion or an automorphism.Now, the commutativity condition says geometrically that the map X m ( ϕ ) ∗ : π ( X m ( L )) −→ π ( X m ( L ′ )) must preserve certain invariants of loops of m -component unlinks. If m = 1 , there is just oneinvariant, which is the number of times (with sign determined by Figure 4.1) that a component ofthe m -component unlink passes through a component of the fixed link (namely L or L ′ ). If m > ,there are two further invariants, which count (a) the number of times that one component of the m -component unlink passes through another of its components (with sign), and (b) the number oftimes (mod ) that two of its components are interchanged (without passing through each other).It is clear that standard inclusions preserve all of these invariants of loops. We therefore just haveto check that X m ( ϕ ) ∗ preserves these invariants when ϕ is an automorphism of L = L ′ in U L β t > ,which corresponds (on π ) to an element of LB n , where n is the number of components of L . Firstof all, we may assume that this diffeomorphism of D is supported in a small disc neighbourhoodof L , and so it clearly does not affect the invariants (a) and (b), since these involve loops of m -component unlinks that do not interact with L (they may be isotoped to be disjoint from thesmall disc neighbourhood of L ). So we just need to show that [ ϕ ] ∈ LB n preserves the invariantthat counts the number of times that a component of the m -component unlink passes through acomponent of L . It suffices to check this for [ ϕ ] = τ i and [ ϕ ] = σ i generators of LB n . This mayeasily be checked on a case-by-case basis, drawing a picture of the effects that τ i and σ i ∈ LB n have on a simple loop of m -component unlinks in D r L , where m − of the components are fixed,far away from L , and the remaining component passes once through one of the components of L .Thus we have a well-defined continuous functor F αm as above. This gives us the first input ofDefinition 2.21, with C t = U L β t > . Taking the ground ring k to be Z and defining R = M = Z [ Q ] (i.e., not twisting the coefficients), we therefore obtain a functor L i ( F αm ) : π ( U L β t > ) = U L β > −→ Z [ Q ] − Mod , for any i > . Since U L β is equivalent to U L β > with an initial object adjoined, we may extendthis to a functor L i ( F αm ) : U L β −→ Z [ Q ] − Mod
55y sending the initial object of U L β (the empty unlink) to the initial object of Z [ Q ] − Mod (thetrivial module). We summarise this construction as:
Theorem 4.24
For any m > and i > , the continuous functor (4.10) determines, through theconstruction of §2.5, a functor L i ( F αm ) : U L β −→ Z [ Q ] − Mod , (4.11) where Q = Z when m = 1 and Q = Z ⊕ Z ⊕ Z / Z when m > . In particular, this gives coherentfamilies of representations of the loop braid groups { LB n } n > defined over the Laurent polynomialrings Z [ Z ] = Z [ x ± ] when m = 1 and defined over Z [ Z ⊕ Z ⊕ Z / Z ] = Z [ x ± , y ± , z ± ] / ( z ) when m > . The beta representations of the extended loop braid groups.
Fix an integer m > andset Q to be the group ( Z / Z ) if m = 1 and ( Z / Z ) if m > . Let U ( L β ext ) t be the categorydescribed in Remark 3.42: its objects are unoriented unlinks in the interior of D and its morphismsfrom L to L ′ are embeddings ϕ : D → D , defined exactly as described for U L β t above, but withoutthe orientation-preserving condition. Let U ( L β ext ) t > be its subcategory on all objects except theempty unlink. We define a continuous functor F βm : U ( L β ext ) t > −→ Cov Q (4.12)as follows. For an object (i.e., a non-empty unlink) L of U ( L β ext ) t > , we define F βm ( L ) to be thespace Y m ( L ) equipped with the quotient ˆ β m ( L ) of its fundamental group. A morphism ϕ : D → D of U ( L β ext ) t > from L to L ′ induces a map Y m ( ϕ ) : Y m ( L ) → Y m ( L ′ ) , which one may see is well-defined by exactly the same argument as above (for the functor F αm ). Hence setting F βm ( ϕ ) = Y m ( ϕ ) defines a continuous functor, if we ignore the condition that the induced map Y m ( ϕ ) ∗ on π mustcommute with the quotients ˆ β m ( L ) and ˆ β m ( L ′ ) , so it remains to check this condition, which saysgeometrically that the map Y m ( ϕ ) ∗ : π ( Y m ( L )) −→ π ( Y m ( L ′ )) must preserve certain invariants of loops of m -component unlinks. As before, it is enough to checkthis condition when ϕ is either a standard inclusion (in which case it is clear) or an automorphismof L = L ′ in U ( L β ext ) t > , which corresponds (on π ) to an element of LB ext n , where n is the numberof components of L . As in the previous setting, all but one of these invariants count certain motionsof m -component unlinks that do not interact with L (in the sense that they are supported awayfrom a small disc neighbourhood of L ), so they are clearly preserved, since we may ensure by anisotopy that ϕ is supported only on a small disc neighbourhood of L . The last invariant that wemust check is preserved by [ ϕ ] ∈ LB ext n , is the number (mod ) of times that a component ofthe m -component unlink passes through a component of L . It suffices to check this for [ ϕ ] = τ i , [ ϕ ] = σ i and [ ϕ ] = ρ i generators of LB ext n . As in the previous setting, this may easily be checkedon a case-by-case basis. In fact, the generators τ i and σ i preserve this invariant as an integer (notjust mod ), whereas the generator ρ i preserves it only mod , since it reverses the orientation ofone component of L .Thus we have a well-defined continuous functor F βm as above. This gives us the first input ofDefinition 2.21, with C t = U ( L β ext ) t > . Taking the ground ring k to be Z and defining R = M = Z [ Q ] (i.e., not twisting the coefficients), we therefore obtain a functor L i ( F βm ) : π ( U ( L β ext ) t > ) = U ( L β ext ) > −→ Z [ Q ] − Mod , Although, of course, this integer depends on an arbitrary choice of orientation of L (since L is not equippedwith one), and it is only independent of this choice mod 2. i > . Since U ( L β ext ) is equivalent to U ( L β ext ) > with an initial object adjoined, we mayextend this to a functor L i ( F βm ) : U ( L β ext ) −→ Z [ Q ] − Mod by sending the initial object of U ( L β ext ) (the empty unlink) to the initial object of Z [ Q ] − Mod (thetrivial module). We summarise this construction as:
Theorem 4.25
For any m > and i > , the continuous functor (4.12) determines, through theconstruction of §2.5, a functor L i ( F βm ) : U ( L β ext ) −→ Z [ Q ] − Mod , (4.13) where Q = ( Z / Z ) when m = 1 and Q = ( Z / Z ) when m > . In particular, this gives coherentfamilies of representations of the extended loop braid groups { LB ext n } n > defined over the Laurentpolynomial rings Z [( Z / Z ) ] = Z [ x ± , y ± ] / ( x , y ) when m = 1 and defined over Z [( Z / Z ) ] = Z [ x ± , y ± , z ± , w ± ] / ( x , y , z , w ) when m > . The gamma representations of the extended loop braid groups.
Clearly the beta repre-sentations of the extended loop braid groups constructed in Theorem 4.25 restrict to representa-tions of the non-extended loop braid groups, since these are subgroups of the extended loop braidgroups (and indeed U L β is a subcategory of U ( L β ext ) ). Conversely, one may wonder whether the alpha representations of the non-extended loop braid groups constructed in Theorem 4.24 may beextended to representations of the extended loop braid groups. Functorially, this is the question ofwhether the functors L i ( F αm ) defined on U L β may be extended to U ( L β ext ) . In fact, they may not,but our next construction shows that, after changing the ring over which we define this functor —specifically, dividing the Laurent polynomial rings of Theorem 4.24 by the ideal ( x ) — they maybe so extended, using the further quotients γ m ( L ) of the quotients ˆ α m ( L ) constructed above.Fix an integer m > and set Q to be the group Z / Z if m = 1 and Z ⊕ ( Z / Z ) if m > . Wedefine a continuous functor F γm : U ( L β ext ) t > −→ Cov Q (4.14)as follows. We send the object (i.e., unoriented unlink) L of U ( L β ext ) t > to the space X m ( L ) equipped with the quotient γ m ( L ) of its fundamental group. Note that this is valid since weensured that γ m ( L ) does not depend on a choice of orientation of L . A morphism ϕ : D → D of U ( L β ext ) t > from L to L ′ induces a well-defined map X m ( ϕ ) : X m ( L ) → X m ( L ′ ) . This was shownabove (in the construction of F αm ) when ϕ satisfied the additional assumption that its restrictionto a map L → L ′ was orientation-preserving. However, this fact was not used in that argument,so it applies equally well to this more general setting. Hence setting F γm ( ϕ ) = X m ( ϕ ) defines acontinuous functor, if we ignore the condition that the induced map X m ( ϕ ) ∗ : π ( X m ( L )) −→ π ( X m ( L ′ )) must commute with the quotients γ m ( L ) and γ m ( L ′ ) , so it remains to check this condition. If ϕ isa morphism of U L β t > ⊂ U ( L β ext ) t > , then we have already checked, during the construction of F αm , that X m ( ϕ ) ∗ commutes with thequotients ˆ α m ( L ) and ˆ α m ( L ′ ) . Since γ m ( L ) factors through ˆ α m ( L ) , this means that X m ( ϕ ) ∗ alsocommutes with the quotients γ m ( L ) and γ m ( L ′ ) . Any morphism of U ( L β ext ) t > may be written asa composition of a morphism of U L β t > together with an automorphism of L = L ′ in U ( L β ext ) t > that corresponds, on π , to a generator of LB ext n (where n is the number of components of L )of the form ρ i , which rotates one component of L by 180 degrees. It therefore remains to checkthat X m ( ϕ ) ∗ commutes with the quotients γ m ( L ) and γ m ( L ′ ) when [ ϕ ] = ρ i . This corresponds to57hecking that X m ( ρ i ) ∗ preserves (mod ) the number (counted with sign) of times that a componentof the m -component unlink passes through a component of L . Let us consider a simple loop θ of m -component unlinks in D r L , where m − of the components are fixed, far away from L , and theremaining component passes once through the j -th component of L . Let us say that an orientationof L has been chosen such that the number of passes of θ through L is +1 . Then, for i = j ,the number of passes of X m ( ρ i ) ∗ ( θ ) through L is also +1 , and for i = j , the number of passes of X m ( ρ i ) ∗ ( θ ) through L is − . But − ≡ +1 (mod ).Thus we have a well-defined continuous functor F γm as above. This gives us the first input ofDefinition 2.21, with C t = U ( L β ext ) t > . Taking the ground ring k to be Z and defining R = M = Z [ Q ] (i.e., not twisting the coefficients), we therefore obtain a functor L i ( F γm ) : π ( U ( L β ext ) t > ) = U ( L β ext ) > −→ Z [ Q ] − Mod , for any i > . Since U ( L β ext ) is equivalent to U ( L β ext ) > with an initial object adjoined, we mayextend this to a functor L i ( F γm ) : U ( L β ext ) −→ Z [ Q ] − Mod by sending the initial object of U ( L β ext ) (the empty unlink) to the initial object of Z [ Q ] − Mod (thetrivial module). We summarise this construction as:
Theorem 4.26
For any m > and i > , the continuous functor (4.14) determines, through theconstruction of §2.5, a functor L i ( F γm ) : U ( L β ext ) −→ Z [ Q ] − Mod , (4.15) where Q = Z / Z when m = 1 and Q = Z ⊕ ( Z / Z ) when m > . In particular, this gives coherentfamilies of representations of the extended loop braid groups { LB ext n } n > defined over the Laurentpolynomial rings Z [ Z / Z ] = Z [ x ± ] / ( x ) when m = 1 and defined over Z [ Z / Z ⊕ Z ⊕ Z / Z ] = Z [ x ± , y ± , z ± ] / ( x , z ) when m > . These are extensions of the representations of { LB n } n > constructed in Theorem4.24, after dividing the Laurent polynomial rings by the ideal ( x ) . In this section, we present the various already existing families of representations that can berecovered as particular homological representations. Indeed for several families of groups suchmapping class groups or classical and surface braid groups, many families of representations happento be more or less explicitly applications of the general procedure encoded by the homologicalfunctors described in §2 for several families of groups such mapping class groups or classical andsurfaces braid groups. Furthermore, we exhibit in §5.4 a connection between the constructionof linear representations introduced in [Lon94] and generalised in [Sou18], called the
Long-Moody construction, and a one of the homological functors of §2.
In [Big04a], Bigelow introduces a general method to construct a representation of the braid group B n from a representation of the braid group B m for two integers m and n . It builds the so-calledfamilies of Lawrence-Bigelow representations , first introduced by Lawrence [Law90] in a different Note that the unlink L does not come with an orientation, so this choice is purely arbitrary; however, weshowed earlier (in the construction of the gamma quotients) that the resulting count (with sign) of passes through L is independent of this choice mod 2. Lawrence-Krammer-Bigelow representations that Bigelow [Big01] andKrammer [Kra02] independently proved to be faithful. We review here this construction and thenshow that the representations constructed in §4.1 are the Lawrence-Bigelow representations.
Bigelow’s construction:
Introducing Bigelow’s construction requires the following tools. Werecall that D n denotes the unit -disc where n distinct smaller -discs are removed from the interiorand that C m ( D n ) is the configuration space of m unordered points in D n . We fix m distinct points { z i } i ∈{ ,...,m } in the boundary of D n and the configuration c = ( z , . . . , z m ) as the basepoint of C m ( D n ) and assume that D is the unit disc in the complex plane centered at .Let f : C m ( D n ) → C ∗ be the map defined by ( x , . . . , x m ) Y i ∈{ ,...,m } Y j ∈{ ,...,n } ( x i − p j ) and recall that W : π ( C m ( D n ) , c ) → Z is the map induced by sending a loop γ to the windingnumber of f ◦ γ around . Finally, recall that the inclusion D n ֒ → D defined by gluing discs onall the interior boundary components induces an inclusion map i : C m ( D n ) ֒ → C m (cid:0) D (cid:1) , whichinduces surjective homomorphism in homotopy i ∗ : π ( C m ( D n ) , c ) → B m .Bigelow’s construction starts with a representation ρ : B m = C m ( D ) → GL K ( V ) where K is anintegral domain and we fix q a unit in K . The first key point of Bigelow’s construction is to considerthe following deformation of ρ . Definition 5.1
Let ρ ′ q,n : π ( C m ( D n ) , c ) → GL K ( V ) be the representation defined by ρ ′ q,n ( g ) ( v ) = q w ( g ) ρ ( i ∗ ( g )) ( v ) for all g ∈ π ( C m ( D n ) , c ) and all v ∈ V .We denote by ^ C m ( D n ) the universal covering space of C m ( D n ) . We consider the quotient space L ρ := (cid:16) ^ C m ( D n ) × V (cid:17) /π ( C m ( D n ) , c ) , where the diagonal action of π ( C m ( D n ) , c ) on ^ C m ( D n ) is given by deck transformations and theone on V is given by ρ ′ q,n . It is the canonical bundle (also known as Borel construction) associatedwith the action ρ ′ q,n of C m ( D n ) on V .Then, Bigelow’s construction consists in the natural action of the braid group on the (ordinary)homology group H ∗ ( C m ( D n ) , L ρ ) or the Borel-Moore homology group H BM ∗ ( C m ( D n ) , L ρ ) of theconfiguration space C m ( D n ) with local coefficients L ρ : Theorem 5.2 [Big04a, Section 2] From a representation ρ : B m = C m ( D ) → GL K ( V ) , the braidgroup B n acts on H ∗ ( C m ( D n ) , L ρ ) and H BM ∗ ( C m ( D n ) , L ρ ) , thus defining Big k,n ( ρ ) : B n → H k ( C m ( D n ) , L ρ ) and Big
BMk,n ( ρ ) : B n → H BMk ( C m ( D n ) , L ρ ) for all natural numbers k and n .They are called Bigelow’s representation and Borel-Moore Bigelow’s representation.Proof. Any f ∈ Diff ( D n ) that acts as the identity on the boundary defines an element of Diff c ( C m ( D n )) and then induces an automorphism f ∗ of π ( C m ( D n ) , c ) . Since f fixes the m points in C m ( D n ) ,we deduce that f ∗ ( g ) = g for all g ∈ π ( C m ( D n ) , c ) . Then, there exists a unique lift ˜ f ∗ : L ρ → L ρ of f ∗ acting as the identity on the fiber over c , which depends only on the homotopy class of f :this induces the action of B n on the homology groups. Lawrence-Bigelow representations:
Using the previous notations, we assign V = K = Z [ A m ] where A = (cid:10) q ± (cid:11) and A m = (cid:10) q ± , t ± (cid:11) if m > . Let X m : B m → K ∗ be the morphism defined bysending each Artin generator σ i to the multiplication by t if m > and the trivial representationif m = 1 . The Lawrence-Bigelow representations of braid groups are the applications of Bigelow’sconstruction to these representations: 59 efinition 5.3 For all natural numbers n , Big m,n ( X m ) and Big
BMm,n ( X m ) are respectively called Lawrence-Bigelow representations and
Borel-Moore Lawrence-Bigelow representations . Remark 5.4
For m = 1 and each natural number n , Big ,n ( X ) and Big BM ,n ( X ) both arethe well-known reduced Burau representation (see [KT08, Section 3.3] for more details about theassociated family of representations). Also, for m = 2 and each natural number n , Big ,n ( X ) and Big BM ,n ( X ) both are the Lawrence-Krammer-Bigelow representation [Big03; Big01] introduced toprove that braid groups are linear. In addition, using [PP02, Theorem 1.2], the tensor product withthe field of fractions Big ,n ( X ) ⊗ Z Q ( q, t ) is isomorphic to the Lawrence-Krammer representation[Kra02] for all natural numbers n . Recovering:
The representations which form the functors constructed in §4.1 are actually equiv-alent to the Lawrence-Bigelow representations:
Proposition 5.5
As representations of the braid group B n , L m (cid:0) F θ m ,φ m,n (cid:1) and L BMm ( F θ,φ ) n arerespectively equivalent to Big m,n ( X m ) and Big
BMm,n ( X m ) .Proof. Recall that the local coefficient system L X m is the canonical fiber bundle (cid:16) ^ C m ( D n ) × Z [ A m ] (cid:17) /π ( C m ( D n ) , c ) ξ −→ C m ( D n ) so that ξ − ( c ) ∼ = Z [ A m ] for all c ∈ C m ( D n ) . Denoting C • ( X ) the singular chain complex of thetopological space X , there is an isomorphism of abelian groups: H m ( C m ( D n ) , L X m ) ∼ = H m (cid:18) C • (cid:16) ^ C m ( D n ) (cid:17) ⊗ Z [ π ( C m ( D n ) ,c )] ξ − ( c ) (cid:19) . Recall that φ ,n is the composite γ ◦ Σ (where γ : F n → Z n is the abelianisation map and Σ isthe sum map) and that, for m > , φ m,n : π ( C m ( D n ) , c ) → Z is defined by γ ( T ( γ ) , W ( γ )) ,where T ( γ ) counts the total number of half-twists for a curve representing γ . In any case, themorphism φ m,n induces a representation π ( C m ( D n ) , c ) φ m,n / / A m (cid:31) (cid:127) / / Aut Z [ A m ] ( Z [ A m ]) where the second arrow is the morphism induced by the multiplication by the elements of A m :this is exactly the morphism ( X m ) ′ q of Definition 5.1 induced by X m . Using Shapiro’s lemma, wededuce that as abelian groups: H m ( C m ( D n ) , L X m ) ∼ = H m (cid:16) C m ( D n ) φ m,n , Z (cid:17) and the action of Big m,n ( X m ) for an element σ of B n on H ∗ (cid:16) C m ( D n ) φ m,n , Z (cid:17) through this isomor-phism is given by the unique lift of σ for the covering space C m ( D n ) φ m,n . Hence, L m (cid:0) F θ m ,φ m,n (cid:1) ∼ = Big m,n ( X m ) as representations of B n .The proof that L BMm ( F θ,φ ) n ∼ = Big
BMm,n ( X m ) follows repeating mutatis mutandis the previous one,using Borel-Moore homology instead of standard homology.Proposition 5.5 justifies the notation LB m for the functor defined by the representations L m (cid:0) F θ m ,φ m,n (cid:1) :this functor actually encodes the Lawrence-Bigelow representations. Remark 5.6
In [Sou19, Section 1.2], it is proved that the family of reduced Burau and Lawrence-Krammer representations define functors over the category h β , β i . More precisely, they respectivelyform functors Bur : h β , β i → C (cid:2) t ± (cid:3) -Mod and LK : h β , β i → C (cid:2) t ± , q ± (cid:3) -Mod. By Remark 5.4,these functors are actually respectively equivalent to the functors LB and LB tensored (andtheir respective alternative using Borel-Moore homology) by C .60 .2 For surface braid groups The application of the general construction of homological representations to surface braid groupsin §4.2.2 provides several families of representations of these groups. Actually one of them isalready defined by An and Ko in [AK10], where they describe an extension of homological repre-sentations from the classical braid groups to the surface braid groups. Some of the technical keytools of their construction are reinterpreted by Bellingeri, Godelle and Guaschi in [BGG17] usingmetabelian quotients of the surface braid groups. We review here the An-Ko representations withthis approach.We use the framework and notations of §4.2.2 where we introduced the representation L m (cid:18) F θ m ,φ Σ g, ,m,n (cid:19) : B n (Σ g, ) −→ Aut Z H k C m (cid:16) Σ ( n ) g, (cid:17) φ Σ g, ,m,n , Z !! . Furthermore, any morphism ψ Q m,n : B m,n (Σ g, ) /Γ ( B m,n (Σ g, )) → Q m,n induces a B n (Σ g, ) - A ,m,n -bimodule structure. We denote by ψ Q n the action of B n (Σ g, ) on Z [ Q m,n ] induced by the mul-tiplication. Since the space of the representation is naturally endowed with a Z [ A ,m,n ] -modulestructure, then the tensor product of ψ Q n and L m (cid:18) F θ m ,φ Σ g, ,m,n (cid:19) defines a morphism B n (Σ g, ) / / Aut Z [ Q m,n ] Z [ Q m,n ] ⊗ Z [ A ,m,n ] H k C m (cid:16) Σ ( n ) g, (cid:17) φ Σ g, ,m,n , Z !! In [AK10], the authors introduce a particular H Σ in an had hoc way. [BGG17, Section 4] actuallyprove that H Σ is a quotient of the third lower central quotient group B m,n (Σ g, ) /Γ ( B m,n (Σ g, )) .Then the An-Ko representations are ψ H Σ n ⊗ Z [ A ,m,n ] L m (cid:18) F θ m ,φ Σ g, ,m,n (cid:19) . Remark 5.7
The groups A ,m,n and H Σ are defined abstractly in [AK10] in terms of grouppresentation to satisfy certain technical homological constraints, without any connection to thethird lower central quotient. The method applied in §4.2.2 underlines the mainspring of thesegroups. Also the use of the third lower central quotient is a valuable tool to define the homologicalrepresentations: as it is done in Proposition 4.6, it allows to straightforwardly prove that the keyAssumption 2.13 is satisfied. Hence it gives an alternative to the ad hoc technical [AK10, Lemma3.1]. Some of the homological representations for mapping class groups of §4.3 have already been in-troduced in a different way and studied. We review here these cases. Recall that, for g a naturalnumber, we consider the mapping class group Γ g, of the compact surface Σ g, and that p is abasepoint on the boundary of Σ g, . The Magnus representations of mapping class groups compact of connected oriented smooth sur-faces and have been a fundamental tool in combinatorial group theory for many years. They wereoriginally defined using the Fox free differential calculus. We refer the reader to [Bir74; Sak12] forfurther details on this definition. However, Suzuki [Suz05] introduced an equivalent topologicaldefinition that we present in this section: this interpretation shows that Magnus representationsare a particular case of homological representations introduced in §4.3.2.Recall that γ denotes the abelianisation of π (Σ g, , p ) . Let ξ γ : Σ γ g, → Σ g, regular coveringspace associated with the abelianisation and we fix a lift p γ ∈ ξ − γ ( p ) in Σ γ g, . Since the commu-tator subgroup [ π (Σ g, , p ) , π (Σ g, , p )] is a characteristic subgroup of the fundamental group,61ssumption 2.13 is satisfied. Therefore, it follows from Proposition 2.14 that there is a well-definedaction of the mapping class group on the reduced homology groups on the covering Σ γ g, : this isthe Magnus representation of the mapping class group
Mag Γ ( g ) : Γ g, → Aut (cid:0) H (cid:0) Σ γ , ξ − γ ( p ) ; Z (cid:1)(cid:1) . Note that Assumption 2.16 is satisfied if and only if we restrict
Mag Γ to the Torelli group of thesurface. This restriction defines the Magnus representation of the Torelli group
Mag I ( g ) : I g, → Aut Z [ H (Σ g, ; Z )] (cid:0) H (cid:0) Σ γ , ξ − γ ( p ) ; Z (cid:1)(cid:1) . In [Mor07], Moriyama studies the natural action of the mapping class group Γ g, of the surface Σ g, on some relative homology groups of the configuration spaces of n -points on a surface Σ g, .The main result of his work is that the kernel of the action of Γ g, coincides with the kernel of thenatural action on the n th lower central quotient group of the fundamental group of Σ g, . We provehere that this family of representations is a particular case of the general construction introducedin §2.For any diffeomorphim ϕ of Σ g, which fixes pointwise the boundary component, the diagonalaction of ϕ on Σ × mg, preserves the subsets D m (Σ g, ) and A m (Σ g, , p ) (introduced in §4.3.1) of Σ × mg, . Then the relative homology group H m (cid:0) Σ × mg, , ( D m (Σ g, ) ∪ A m (Σ g, , p )) ; Z (cid:1) is equipped with a Γ g, -module structure induced from the diagonal action. We call this structurethe m th Moriyama representation of Γ g, and denote it by Mor m ( g ) . Let M + ,gen be the fullsubcategory of M +2 of the orientable surface with no marked points. The monoidal structure (cid:0) M +2 , ♮, (cid:1) restricts to a braided monoidal structure both on the subgroupoids M +2 . Hence therepresentations considered in [Mor07] define a functor Mor m : M + ,gen → Z - Mod .Recall that F m (cid:0) Σ ′ g, (cid:1) denotes the ordered configuration space of m points on the surface Σ g, \{ p } .Note that F m (cid:0) Σ ′ g, (cid:1) is homeomorphic to the complement of the set ( D m (Σ g, ) ∪ A m (Σ g, , p )) in Σ × mg, . Hence, as the Γ g, -module structure is induced from the diagonal action in both situations,it follows from the definition of Borel-Moore homology that we have the following Γ g, -modulesisomorphism H BMm (cid:0) F m (cid:0) Σ ′ g, (cid:1) , Z (cid:1) ∼ = Mor m ( g ) for all natural numbers m and g . Hence the representation L m ( F θ,γ ) n introduced in 4.3.1 as-sociated with the universal covering induced by γ = id π ( F m ( Σ ′ g, ) ,c ) is equivalent to the m thMoriyama representation. In [Lon94], Long and Moody consider in a very general recipee for constructing homological rep-resentations for braid groups. This method and its variants have been studied with a functorialpoint of view in [Sou19] and then generalised in [Sou18] for general families of groups. We firstreview here these construction and then gives their connections to the homological functors of §2.Let ( G , ♮, be a strict monoidal groupoid and ( M , ♮ ) be a left-module over G . We consider afunctor A : hG , Mi → Gr . We also assume that Obj ( G ) and Obj ( M ) are both isomorphic to thenatural numbers N and that there exist two objects ∈ Obj ( G ) and ˆ0 ∈ Obj ( G ) so that any object X of M is isomorphic to the monoidal product n := 1 ♮n ♮ . We denote by G n the isomorphismgroup Iso hG , Mi ( n ) .For R a commutative ring, let R -Alg be the category of unital R -algebras. For all groups G , thegroup rings R [ G ] and the augmentation ideals I R [ G ] respectively assemble to define the group62lgebra functor R [ − ] : Gr → R -Alg and the augmentation ideal functor I R [ − ] : Gr → R -Alg. Werespectively denote by R [ A ] and I A the composite functors R [ − ] ◦ A and I R [ − ] ◦ A .Then R [ A ] is a monoid object in Fct ( hG , Mi , R -Mod ) . Hence pointwise tensor product of functorsautomatically induce the tensor product functor over R [ A ] (see [Sou18, Definition 2.1]) − ⊗ R [ A ] − : Mod- R [ A ] × R [ A ] -Mod → Fct ( hG , Mi , R -Mod ) where R [ A ] -Mod and Mod- R [ A ] respectively denote the categories of left and right modules R [ A ] .Moreover I A is a right R [ A ] -module and therefore defines a functor I A ⊗ R [ A ] − : R [ A ] -Mod → Fct ( hG , Mi , R -Mod ) .Recall from [MM94, Chapter 1, Section 5] that the Grothendieck construction over A denotedby R A , is the category of pairs ( · c , c ) , where c ∈ Obj ( M ) and · c is a group a group (viewedas a category with one object); a morphism in Hom R A (( · c , c ) , ( · c ′ , c ′ )) is a pair ( α, f ) , where f ∈ Hom hG , Mi ( c, c ′ ) and α ∈ · c ′ . For ( α, f ) : ( · c , c ) → ( · c ′ , c ′ ) and ( β, g ) : ( · c ′ , c ′ ) → ( · c ′′ , c ′′ ) ,the composition is defined by ( β, g ) ◦ ( α, f ) = ( β ◦ F ( g ) ( α ) , g ◦ f ) . Note that this definition isdual to the one used in §2.2 for a contravariant functor. There is a canonical projection functor R A → hG , Mi , given by sending an object ( · c , c ) to c . A section s A : hG , Mi = R ֒ → R A to thisprojection functor is induced by the trivial natural transformation → A (where is the trivialfunctor hG , Mi →
Gr). We recall from [Sou18, Proposition 2.4] that the precomposition by s A defines a natural equivalence Fct (cid:0)R A , R -Mod (cid:1) ∼ = R [ A ] -Mod.Finally the key to define a functor which describes a Long-Moody construction is to consider afunctor ς : R A → hG , Mi , so that the following diagram is commutative: hG , Mi (cid:31) (cid:127) s A / / ♮ − % % ❏❏❏❏❏❏❏❏❏❏ R A ς (cid:15) (cid:15) hG , Mi , where ♮ − : hG , Mi → hG , Mi denotes the functor defined by (1 ♮ − ) ( X ) = 1 ♮X for all X ∈ Obj ( M ) and (1 ♮ − ) ( φ ) = id ♮φ for all morphisms φ of hG , Mi . Definition 5.8 [Sou18, Section 2] The Long-Moody functor LM A ,ς associated with the functors A and ς is the Fct ( hG , Mi , R -Mod ) s ∗A ◦ ς ∗ / / R [ A ] -Mod I A ⊗ R [ A ] − / / Fct ( hG , Mi , R -Mod ) , where s ∗A and ς ∗ respectively denote the precomposition functors s A and ς .The appropriate data A and ς naturally arise for many families of groups in connection with topol-ogy. We refer the reader to [Sou18, Section 3] for the introduction of non-trivial and natural suchfunctors for the families of braid groups { B n } n ∈ N , surface braid groups { B n (Σ g, ) } n ∈ N , mappingclass groups { Γ n, } n ∈ N and (cid:8) Γ ng, (cid:9) n ∈ N . Recovering:
We assume that there exists a topological category hG , Mi t so that π hG , Mi t = hG , Mi (which is the case for all the aforementioned examples). To recover the Long-Moodyconstructions encoded by the functors, we restrict to the full subcategory hG , Mi on the objectsisomorphic to n for some natural number n. We denote it by · n since its skeleton is the group G n viewed as a category, and by A n the restriction of A to · n . We pick a topological lift of · n in hG , Mi t , denoted by X n . Recall from §2 the category g Cov Q of based path-connected spaces X with a surjection π ( X ) ։ Q . Hence g Cov π ( X n ) encodes the universal cover of π ( X n ) . We denoteby U : g Cov π ( X n ) → Top ∗ the functor which forgets the surjections. We require the following mildproperty: 63 ssumption 5.9 There exists a functor b A n : X n → g Cov Q which image is a topological space witha boundary component and so that the following diagram is commutative X n b A n / / π (cid:15) (cid:15) g Cov π ( X n ) U / / Top ∗ π (cid:15) (cid:15) · n A n / / Gr . In all the examples of [Sou18], the various functors A for braid groups and mapping class groupsare induced by a geomtrical construction and a fortiori Assumption 5.9 is satisfied. We abuse thenotation LM A ,ς to denote the restriction of a Long-Moody functor to the category Fct (cid:0) · n , R -Mod (cid:1) .Then: Proposition 5.10
Let M be a R -module and ρ : G n +1 → GL R ( M ) be a representation. Thenthere is an equivalence of representations LM A ,ς ( ρ ) ∼ = L r (cid:16) b A n ; M (cid:17) .Proof. For the topological space, we denote by C • ( X, ∗ ) the singular chain complex of X relativeto a point p on the boundary and by ^ b A n ( X n ) the universal cover of b A n ( X n ) . Note that thefunctor ς induces the (cid:16) R h π (cid:16) e A n ( X n ) (cid:17)i , R (cid:17) -bimodule structure of M . There is an isomorphismof abelian groups: H C • (cid:18) ^ b A n ( X n ) , e p (cid:19) ⊗ π (cid:0) e A n ( X n ) (cid:1) M ∼ = I A n ⊗ π (cid:0) e A n ( X n ) (cid:1) M. The result then follows directly from the fact that the actions of both representations on the lefthand sides are induced by A n and the ones on M are defined by ρ . Most of the homological representations described in the previous sections are defined via actionson the homology of some configuration spaces of points of a topological space. Instead of usingordinary homology, we may instead use Borel-Moore homology of these configuration spaces. Thesealternative homology groups have useful properties (see §6.1) which allow one to compute freegenerating sets of the representations under consideration. We formulate this property in §6.1 anddiscuss applications in §6.2.
The following lemma gives a criterion for an inclusion of spaces to induce isomorphisms on the(possibly twisted) Borel-Moore homology of their unordered configuration spaces. It abstracts theessential ideas of Lemma 3.1 of [Big04b] and Lemma 3.3 of [AK10].
Lemma 6.1
Let M be a compact metric space, and let ∅ = P ⊆ Γ ⊆ M be subspaces, where P is finite and Γ is closed. Assume also that M and Γ are locally compact.Then the following conditions ensure that, for all k ∈ N , the inclusion C k (Γ r P ) ֒ −→ C k ( M r P ) induces isomorphisms on Borel-Moore homology with any twisted coefficients. There is a retraction π : M → Γ . • There is a continuous map h : [0 , ∞ ) → Emb ne π ( M, M ) , where Emb ne π ( M, M ) is the space,with the compact-open topology, of self-embeddings of M that commute with the projection π and are non-expanding , i.e. do not increase distances between points, such that: ◦ h = id , ◦ h t fixes Γ pointwise, ◦ for all s and t , we have h s ( h t ( M )) ⊆ h t ( M ) , ◦ for all t , the subset h t ( M ) contains a neighbourhood of P , ◦ for all sufficiently small ǫ > there is a value of t such that the fibres of the projection π t = π | h t ( M ) : h t ( M ) −→ Γ have diameter smaller than ǫ . Moreover, its restriction to π ◦ t = π | h t ( M ) ∩ π − (Γ r P ) : h t ( M ) ∩ π − (Γ r P ) −→ Γ r P is non-expanding and admits a homotopy incl ◦ π ◦ t ≃ id h t ( M ) ∩ π − (Γ r P ) through non-expandingmaps that preserve the fibres of π ◦ t .Sketch of proof. Consider the following commutative diagram of inclusions: ¯ C ˆ C t C t C ¯ C r A ǫ,t ˆ C t r A ǫ,t C t r A ǫ,t C r A ǫ,t (B) (C) (A) (6.1)where: • C = C k ( M r P ) , • ¯ C = C k (Γ r P ) , • C t = C k ( h t ( M ) r P ) , • ˆ C t ⊆ C t is the subset of those configurations { x , . . . , x k } such that π ( x i ) P for all i and π ( x i ) = π ( x j ) for all i = j , • A ǫ,t ⊆ C is the subset of those { x , . . . , x k } ∈ C such that d ( x i , x j ) > ǫ for all i = j and, foreach i , we have either x i h t ( M ) or d ( π ( x i ) , P ) > ǫ .It suffices to show that the inclusion of pairs ( ¯ C, ¯ C r A ǫ,t ) ֒ −→ ( C, C r A ǫ,t ) induces isomorphisms on twisted relative homology, for all sufficiently small ǫ > and sufficientlylarge t (where the lower bound on “sufficiently large t ” is permitted to depend on ǫ ), since theconditions imply that the map on twisted Borel-Moore homology induced by ¯ C ֒ → C is the inverselimit of these maps. One sees this as follows:(A) The horizontal inclusions in square (A) of (6.1) are homotopy equivalences, for all t and ǫ .This uses the homotopy given by h .(B) For given ǫ , we may choose t sufficiently large such that the horizontal inclusions in square(B) of (6.1) are also homotopy equivalences. This uses the homotopy from incl ◦ π ◦ t to theidentity assumed in the last property of the hypotheses.(C) For given ǫ , we may choose t sufficiently large such that square (C) of (6.1) is excisive . Thenon-trivial thing to check for this is that C t ∩ A ǫ,t ⊆ ˆ C t , which is ensured by our assumptionabout the diameter of the fibres of π t . Remark 6.2 If M is a connected, compact surface with one boundary-component and P is a finitesubset of its interior, we may take Γ to be an embedded, connected graph in the interior of M , with P = { p , . . . , p n } as its vertices, with n − edges between p i and p i +1 for i ∈ { , . . . , n − } and with − χ ( M ) edges from p to itself that generate H ( M ) . We thus see that the twisted Borel-Moorehomology of unordered configuration spaces on the punctured surface M r P is isomorphic to the65orel-Moore homology of unordered configuration spaces on a disjoint union of n − χ ( M ) openintervals, which is free and concentrated in degree k , if we are considering the configuration spaceof k unordered points. Its dimension is: (cid:18) n + k − χ ( M ) − k (cid:19) . Thus we recover Lemma 3.1 of [Big04b] and Lemma 3.3 of [AK10]. One may also deduce similarresults if some of the points of P lie on the boundary of the surface M , again taking Γ to be anembedded, connected graph. See also §6.2, where some of these cases are discussed in more detail,considering also the induced actions of the braid group resp. mapping class group. One can alsoconsider higher-dimensional applications of Lemma 6.1. Let W = ( S n × S n ) r int( D n ) and take M = W g = W ♮ · · · ♮W the g -fold boundary connected sum, for g > , and P a finite subset of itsinterior. We may then take Γ to be an embedded CW-complex whose -cells are P = { p , . . . , p n } and which has n − one-cells joining p i to p i +1 for i ∈ { , . . . , n − } and g distinct n -cells, eachattached trivially to p . We then deduce that the twisted Borel-Moore homology of unorderedconfiguration spaces on the punctured manifold W g r P is isomorphic to the twisted Borel-Moorehomology of unordered configuration spaces on a disjoint union of n − open intervals togetherwith g open n -discs. The key result of Lemma 6.1 allows one to prove further properties for the Borel-Moore version ofthe Lawrence-Bigelow representations and the Moriyama representations. More precisely, we gainan explicit description of the spaces of representations and prove that these families of representa-tions form functors over appropriate source categories.
The isomorphism of Lemma 6.1 introduces a convenient free generating set for Borel-Moore homol-ogy of configuration spaces. In particular, this gives a useful description and additional propertiesof the Lawrence-Bigelow functors (see §4.1). This new description is a key point to prove thepolynomiality results of §8.1.We fix two natural numbers n > and m > . Let P m ( n ) be the set of partitions of m into n numbers: P m ( n ) = ( ω , . . . , ω n ) | ω i ∈ N and X i n ω i = m if n > and P m (0) = ∅ . Considering the m th Lawrence-Bigelow functor using Borel-Moore homology LB BMm , we denoteby ( p i , p i +1 ) the open interval joining the punctures p i and p i +1 for all i ∈ { , . . . , n − } . Thedisjoint union I n := a i n − ( p i , p i +1 ) if n > and I = I = ∅ is the convenient subset of the punctured disc D n to apply Lemma 6.1. We represent this subsetby the following picture: p p p p p n − p n · · · .Then the configuration space C m ( I n ) is homeomorphic to a disjoint union of (cid:18) m + n − m (cid:19) = Card ( P m ( n − There is a small typo in the statement of Lemma 3.1 of [Big04b]: the top line of the binomial coefficient shouldbe n + m − n + m − m -balls which are parameterised by ( n − -tuples ( ω , . . . , ω n − ) of natural numbers so thatthe i -th interval ( p i , p i +1 ) contains ω i points from the configuration and ω + ω + · · · + ω n − = m .We deduce that we have an isomorphism of Z [ A m ] -modules LB BMm ( n ) ∼ = M ω ∈P m ( n − Z [ A m ] ω (6.2)if n > and LB BMm (0) = 0 .Moreover, the restriction to these subsets allows to fully understand the action of the morphismsof h β , β i on the configuration spaces given by the Lawrence-Bigelow functors. Recall that themorphisms of the category h β , β i are composite of automorphisms and some injections: it is thusenough to consider the action of the generators of the braid groups and the injection [1 , id n ] tofully describe the Lawrence-Bigelow functors on morphisms.First, for each the Artin generator σ i with i ∈ { , . . . , n − } , the action of σ i on each interval ( p i , p i +1 ) is defined by the classical action of the braid group (seen as a mapping class group) onthe fundamental group of the n -punctured disc. Hence this action is described for the whole subset I n by the following picture. σ i p p i − p i p i +1 p i +2 p n · · · · · · p p i − p i p i +1 p i +2 p n · · · · · · Let θ m,n ( σ i ) be the induced homeomorphism of the configuration space C m ( I n ) . Then LB BMm ( σ i ) is the morphism on Borel-Moore homology induced by the unique lift of this homeomorphism.Therefore the action of LB BMm ( σ i ) on each Borel-Moore homology class can be represented by theprevious picture of the action of σ i on the set I n where each interval is labelled with the numberof configurations points which sit inside.On the other hand, the morphism LB BMm ([1 , id n +1 ]) is the induced morphism on Borel-Moorehomology of the unique lift e φm,n of the configuration space map e m,n : C m ( D n ) → C m ( D n ) defined by adding a puncture on the left to the n -punctured surface D n . By Lemma 6.1, this isequivalent to the morphism on Borel-Moore homology induced by the embedding of I n into I n as the n − last intervals, i.e. sending the interval ( p i , p i +1 ) of I n to the interval ( p i +1 , p i +2 ) of I n +1 for each i ∈ { , . . . , n − } . [1 , id n +1 ] p p p p n − p n · · · p p p p p n p n +1 · · · At the level of the sets of partitions, the map e m,n induces the injective map P m ( e m,n ) : P m ( n − ֒ →P m ( n ) sending ( ω , . . . , ω n − ) to (0 , ω , . . . , ω n − ) for n > , and the trivial set maps P m ( e m, ) : ∅ → ∅ and P m ( e m, ) : ∅ → P m (1) = { ( m ) } . These induce injective morphisms (which is trivialfor n = 0 ) M ω ∈P m ( n − Z [ A m ] ω ֒ → M ω ∈P m ( n ) Z [ A m ] ω that we denote by ι P m ( n ) \P m ( n − ⊕ id P m ( n − for simplicity. Hence LB BMm ([1 , id n ]) is equivalentto this injection and its action on each Borel-Moore homology class can be represented by theabove picture of the embedding of I n in I n +1 where each interval is labelled with the number ofconfigurations points which sit inside ( being automatically the one of the interval ( p , p ) of thepicture on the right-hand side). Recall from §5.3.2 the representation
Mor m ( g ) with fixed g > of the mapping class group Γ g, introduced by Moriyama [Mor07] is isomorphic to an homological representation of §4.3.1 which67epresentation space is H BMm (cid:0) F m (cid:0) Σ ′ g, (cid:1) , Z (cid:1) , and defines a functor Mor m : M + ,gen → Z - Mod with m > a natural number. Lemma 6.1 then implies additional properties on the space of thisrepresentation. For n > a natural number, let Q m ( n ) be the set of arrangements of m into n numbers: Q m ( n ) = S m × P m (2 n ) . Note that m ! · (cid:18) m + n − m (cid:19) = Card ( Q m ( n )) . Lemma 6.3
For all natural numbers m and k , there is an abelian group isomorphism H BMk (cid:0) F m (cid:0) Σ ′ g, (cid:1) , Z (cid:1) ∼ = if k = m ; L ω ∈Q m ( g ) Z ω = Z ⊕ (2 g + m − g − if k = m .Proof. First of all, we recall that for X a topological space homeomorphic to the complement ofa closed subcomplex S in a finite CW -complex Y , the Borel-Moore homology group H BM ∗ ( X, L ) with local coefficient L is equivalent to the relative homology H BM ∗ ( Y, S ; L ) . The quotient map Σ × mg, → Σ × mg, / S m defines a regular cover of Σ × mg, / S m path connected space. Hence it inducesthe locally trivial fibration S m → Σ × mg, → Σ × mg, / S m , together with the compatible locally trivialfibration of subspaces S m → D m (Σ g, ) ∪ A m (Σ g, , p ) → D m (Σ g, ) ∪ A m (Σ g, , p ) / S m . Thenthe associated Serre spectral sequence for relative homology (see for instance [DK01, Theorem9.33]) has only one non-trivial row. Hence, for all natural numbers p > , we obtain the followingisomorphism: H BMp (cid:0) F m (cid:0) Σ ′ g, (cid:1) , Z (cid:1) ∼ = H BMp (cid:0) C m (cid:0) Σ ′ g, (cid:1) ; Z [ S m ] (cid:1) . Let W g be the wedge of g copies of the oriented circle S with the base point p which definea free generating set of the fundamental group of Σ g, . Then applying Lemma 6.1 to the subset W g \ { p } of Σ ′ g, (which is homeomorphic to the disjoint union of g open intervals), we have: H m (cid:0) C m (cid:0) Σ ′ g, (cid:1) , Z (cid:1) ∼ = ⊕ ω ∈P m ( n ) Z [ S m ] ω ∼ = M ω ∈Q m ( g ) Z ω . This result recovers [Mor07, Proposition 3.3, 4.2 and 4.3] using the Γ g, -modules isomorphism H BMm (cid:0) F m (cid:0) Σ ′ g, (cid:1) , Z (cid:1) ∼ = Mor m ( g ) , the techniques used in [Mor07] being different from the onespresented here. This isomorphism is crucial to prove the polynomiality results of §8.2. Remark 6.4
For each natural number m and g , the homology groups H BMm (cid:0) F m (cid:0) Σ ′ g, (cid:1) / S m , Z (cid:1) is isomorphic to Sym n ( H (Σ g, , Z )) where Sym n denotes the n th symmetric tensor power.Finally, Lemma 6.3 allows us to prove that the functor Mor m lifts to the category (cid:10) M + ,gen , M + ,gen (cid:11) : Lemma 6.5
The functor
Mor m extends to Mor m : (cid:10) M + ,gen , M + ,gen (cid:11) → Z - Mod by assigning forall Σ g, , Σ g ′ , ∈ Obj (cid:0) M + ,gen (cid:1) : Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) = ι Z ⊕ ( Q m ( g ′ + g ) \Q m ( g ) ) ⊕ id Z ⊕Q m ( g ) . Proof.
Relation (3.1) of Lemma 3.5 is trivially satisfied from the definition of
Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) .We consider ϕ ∈ Γ g, and ϕ ′ ∈ Γ g ′ , . It follows from Lemma 6.3 that Mor m ( ϕ ) is an automor-phism of Z ⊕Q m ( g ) in Z ⊕ ( Q m ( g ′ + g ) \Q m ( g ) ) ⊕ Z ⊕Q m ( g ) ∼ = Z ⊕Q m ( g ′ + g ) and that Mor m (cid:16) id Σ g ′ , ♮ϕ (cid:17) = id Z ⊕ ( Q m ( g ′ + g ) \Q m ( g ) ) ⊕ Mor m ( ϕ ) . Hence Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) ◦ Mor m ( ϕ ) = Mor m (cid:16) id Σ g ′ , ♮ϕ (cid:17) ◦ Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) . Mor m (cid:0) ϕ ′ ♮id Σ g, (cid:1) is an automorphism of Z ⊕Q m ( g ′ ) ֒ → Z ⊕ ( Q m ( g ′ + g ) \Q m ( g ) ) in Z ⊕ ( Q m ( g ′ + g ) \Q m ( g ) ) ⊕ Z ⊕Q m ( g ) = Z ⊕Q m ( g ′ + g ) . In particular, it followsfrom the definition of ι Z ⊕ ( Q m ( g ′ + g ) \Q m ( g ) ) that Mor m (cid:0) ϕ ′ ♮id Σ g, (cid:1) ◦ Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) = Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) . Hence, we deduce that
Mor m ( ϕ ′ ♮ϕ ) ◦ Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) = Mor m (cid:16)h Σ g ′ , , id Σ g ′ , ♮ Σ g, i(cid:17) ◦ Mor m ( ϕ ) . Hence Relation (3.2) of Lemma 3.5 is satisfied, which implies the desired result.
In this section, we review the notions of (very) strong and weak polynomial functors with respectto the framework of the present paper. In [DV19, Section 1], Djament and Vespa introduce thesenotions in the context of a functor category
Fct ( M , A ) , where M is a for symmetric monoidal(small) categories where the unit is an initial object and A is a Grothendieck category. They define strong polynomial functors to extend the classical concept of polynomial functors, which were firstdefined using cross effects by Eilenberg and Mac Lane in [EM54] for functors on module categories.In particular, one reason for interesting in strong polynomial functors is their homological stabilityproperties studied in [RW17]. Furthermore, the notion of weak polynomial functor is first intro-duced in [DV19, Section 1] and happens to be more appropriate to study the stable behavior forobjects of the category Fct ( M , A ) (see [DV19, Section 5] and [Dja17]) and give a new tool forclassifying the families of representations of families of groups (see §7.2 and §8.3.2). Then, thenotions of strong and weak polynomial functors are extended in [Sou18, Section 4] to the largersetting where M is a full subcategory of a pre-braided monoidal category where the unit is aninitial object. We also refer to [Pal17] for a comparison of the various instances of the notions oftwisted coefficient system and polynomial functor.This section thus recollects the definitions and properties of [Sou19, Section 3] and [Sou18, Section4] to the present slightly larger framework, the various proofs being mutatis mutandis generalisa-tions of these previous works. For the remainder of §7, we fix a left-module ( M , ♮ ) over strict monoidal groupoid ( G , ♮, , where we assume that M is a groupoid, ( G , ♮, has no zero divisors and Aut G (0) = { id } . We also fix a Grothendieck category A . We recall that therefore thefunctor category
Fct ( M , A ) is a Grothendieck category. Let X be an object of G . Let τ X : Fct ( hG , Mi , A ) → Fct ( hG , Mi , A ) be the functor defined by τ X ( F ) = F ( n♮ − ) . It is called the translation functor. Let i X : Id → τ X be the natural transforma-tion of Fct ( M , A ) induced by precomposition with the morphisms { [ X, id
X♮A ] : 0 ♮Y → X♮A } A ∈ Ob( M ) .We define δ X = coker ( i X ) , called the difference functor, and κ X = ker ( i X ) , called the evanes-cence functor. We recall the following elementary properties of the translation, evanescence anddifference functors: Proposition 7.1
The translation functor τ X is exact and induces the following exact sequence ofendofunctors of Fct ( hG , Mi , A ) : −→ κ X Ω X −→ Id i X −→ τ X ∆ X −→ δ X −→ . (7.1) Moreover, for a short exact sequence −→ F −→ G −→ H −→ in the category Fct ( hG , Mi , A ) ,there is a natural exact sequence in the category Fct ( hG , Mi , A ) : −→ κ X ( F ) −→ κ X ( G ) −→ κ X ( H ) −→ δ X ( F ) −→ δ X ( G ) −→ δ X ( H ) −→ . (7.2)69 n addition, for Y another object of G , the functors τ X and τ Y commute up to natural isomorphismand they commute with limits and colimits; the difference functors δ X and δ Y commute up tonatural isomorphism and they commute with colimits; the functors κ X and κ Y commute up tonatural isomorphism and they commute with limits; the functor τ X commute with the functors δ X and κ X up to natural isomorphism. Notation 7.2
We respectively denote the iterations τ X · · · τ X τ X | {z } k times and δ X · · · δ X δ X | {z } k times by τ kX and δ kX .Then, we can define the notions of strong and very strong polynomial functors using Proposition7.1. Namely: Definition 7.3
We recursively define on d ∈ N the categories of strong polynomial functors P ol strd ( hG , Mi , A ) and very strong polynomial functors VP ol d ( hG , Mi , A ) , both of degree lessthan or equal to d , to be the full subcategories of Fct ( hG , Mi , A ) as follows:1. If d < , P ol strongd ( hG , Mi , A ) = VP ol d ( hG , Mi , A ) = { } ;2. if d > , the objects of P ol strd ( hG , Mi , A ) are the functors F such that the functor δ X ( F ) is an object of P ol strongd − ( hG , Mi , A ) ; the objects of VP ol d ( hG , Mi , A ) are the objects F of P ol d ( hG , Mi , A ) such that κ ( F ) = 0 and the functor δ ( F ) is an object of VP ol d − ( hG , Mi , A ) .For an object F of Fct ( hG , Mi , A ) which is strong (respectively very strong) polynomial of degreeless than or equal to n ∈ N , the smallest natural number d n for which F is an object of P ol strd ( hG , Mi , A ) (respectively VP ol d ( hG , Mi , A ) ) is called the strong (respectively very strong)degree of F .Very strong polynomial functors turn out to be very useful for homological stability problems: forinstance Randal-Williams and Wahl [RW17] prove homological stability results for several familiesof groups, including surface braid groups, loop braid groups and mapping class groups of surfaces,given by very strong polynomial functors. We develop this point in §8.3.1.Finally, we recall useful properties of the categories associated with strong and very strong poly-nomial functors. Proposition 7.4
Let d be a natural number. The category P ol strd ( hG , Mi , A ) is closed un-der the translation functor, under quotient, under extension and under colimits. The category VP ol d ( hG , Mi , A ) is closed under the translation functor, under normal subobjects and underextension.Let F be an object of Fct ( hG , Mi , A ) . Then, F is an object of P ol strong ( hG , Mi , A ) if andonly if it is the quotient of a constant object of Fct ( hG , Mi , A ) . Moreover, F is an object of VP ol ( hG , Mi , A ) if and only if it is isomorphic to a constant object of Fct ( hG , Mi , A ) .Finally, we assume that there exists a finite set E of objects of the category hG , Mi such that forall objects m of hG , Mi , m is isomorphic to a finite monoidal product of objects of E . Then, anobject F of Fct ( hG , Mi , A ) belongs to P ol strd ( hG , Mi , A ) (respectively to VP ol d ( hG , Mi , A ) ) ifand only if δ e ( F ) is an object of P ol strongd − ( hG , Mi , A ) (respectively κ e ( F ) = 0 and δ e ( F ) is anobject of VP ol n − ( hG , Mi , A ) ), for all objects e of E . We deal here with the concept of weak polynomial functor, introduced in [DV19, Section 2] forthe category
Fct ( S, A ) where S is a symmetric monoidal category where the unit is an initialobject and A is a Grothendieck category, and extended in [Sou18, Section 4] when S is pre-braidedmonoidal. We review the definition and properties of weak polynomial functors, which extendsverbatim to the present larger setting from those of [Sou18, Section 4].Let F be an object of Fct ( hG , Mi , A ) . The subfunctor P n ∈ Ob( β ) κ X F of F is denoted by κ ( F ) . Let S ( hG , Mi , A ) be the full subcategory of Fct ( hG , Mi , A ) of the objects F such that κ ( F ) = F .We have the following fundamental properties: 70 roposition 7.5 The category S ( hG , Mi , A ) is a thick subcategory of Fct ( hG , Mi , A ) and it isclosed under colimits. The thickness property of Proposition 7.5 ensures that we can consider:
Definition 7.6
Let St ( hG , Mi , A ) be the quotient category Fct ( hG , Mi , A ) / S ( hG , Mi , A ) . The canonical functor associated with this quotient is denoted by π hG , Mi : Fct ( hG , Mi , A ) → Fct ( hG , Mi , A ) / S ( hG , Mi , A ) , it is exact, essentially surjective and commutes with all colimits(see [Gab62, Chapter 3]). The right adjoint functor of π hG , Mi is denoted by s hG , Mi : Fct ( hG , Mi , A ) / S ( hG , Mi , A ) → Fct ( hG , Mi , A ) and called the section functor (see [Gab62, Section 3.1]).The following proposition recalls the induced translation and difference functors on the category St ( hG , Mi , A ) . Proposition 7.7
Let x be an object of X . The translation functor τ X and the difference functor δ X of Fct ( hG , Mi , A ) respectively induce an exact endofunctor of St ( hG , Mi , A ) which commutewith colimits, respectively again called the translation functor τ X and the difference functor δ X . Inaddition:1. The following relations hold: δ X ◦ π hG , Mi = π hG , Mi ◦ δ X and τ X ◦ π hG , Mi = π hG , Mi ◦ τ X .2. The exact sequence ( (7.1) ) induces a short exact sequence of endofunctors of St ( hG , Mi , A ) : −→ Id i X −→ τ X ∆ n −→ δ X −→ . (7.3)
3. For another object X ′ , the endofunctors δ X , δ X ′ , τ X and τ X ′ of St ( hG , Mi , A ) pairwisecommute up to natural isomorphism. Definition 7.8
We recursively define on d ∈ N the category of polynomial functors of degree lessthan or equal to d , denoted by P ol d ( hG , Mi , A ) , to be the full subcategory of St ( hG , Mi , A ) asfollows:1. If d < , P ol d ( hG , Mi , A ) = { } ;2. if d > , the objects of P ol d ( hG , Mi , A ) are the functors F such that the functor δ ( F ) isan object of P ol d − ( hG , Mi , A ) .For an object F of St ( hG , Mi , A ) which is polynomial of degree less than or equal to d ∈ N , thesmallest natural number n d for which F is an object of P ol d ( hG , Mi , A ) is called the degree of F . An object F of Fct ( hG , Mi , A ) is weak polynomial of degree at most d if its image π hG , Mi ( F ) is an object of P ol d ( hG , Mi , A ) . The degree of polynomiality of π hG , Mi ( F ) is called the (weak)degree of F .Finally, let us recall some useful properties of the categories of weak polynomial functors. Proposition 7.9
Let d be a natural number. As a subcategory of St ( hG , Mi , A ) , the category P ol d ( hG , Mi , A ) is thick and closed under limits and colimits. Furthermore, there is an equivalenceof categories A ≃ P ol ( hG , Mi , A ) .Finally, we assume that there exists a finite set E of objects of the category hG , Mi such that forall objects m of hG , Mi , m is isomorphic to a finite monoidal product of objects of E . Let F be anobject of St ( hG , Mi , A ) . Then, the functor F is an object of P ol d ( hG , Mi , A ) if and only if thefunctor δ e ( F ) is an object of P ol d − ( hG , Mi , A ) for all objects e of E . uotient categories: A fundamental reason for the notion of of weak polynomial functors tobe introduced in [DV19] is that, contrary to the category P ol strongd ( hG , Mi , A ) , the category P ol d ( hG , Mi , A ) is localizing . This allows to define the quotient categories P ol d +1 ( hG , Mi , A ) / P ol d ( hG , Mi , A ) . A refined description of the category P ol strongd ( hG , Mi , A ) is out of reach generally speaking evenfor small d . On the contrary, understanding the quotient categories P ol d +1 ( hG , Mi , A ) / P ol d ( hG , Mi , A ) is more attainable: for example, when G = M = F B (the category of finite sets and bijections)[DV19, Proposition 5.9] gives a general equivalence of these quotients in terms of module categories.Hence these quotients provide a new classifying tool to study polynomial functors and thereforerepresentations of families of groups: this will be illustrated in §8.3.2.
In this section, we study the (very) strong and weak polynomiality of some homological func-tors. Indeed, we prove that the Lawrence-Bigelow functors introduced in §4.1 and the Moriyamafunctors defined in §4.3.2 are both very strong and weak polynomial. The use of Borel-Moorehomology and of the free generating sets studied in §6 are fundamental tools to establish theseproperties. Consequently, we prove homological stability results for braid groups with coefficientsin the Lawrence-Bigelow representations and for mapping class groups with coefficients in theMoriyama representations. Also we gain a better understanding of the quotient categories forweak polynomial functors associated to these families of groups.
The construction of §4.1 and identifications of §5.1 introduced the m th Borel-Moore Lawrence-Bigelow functor LB BMm : h β , β i → Z [ A m ] - Mod for each natural numbers m > , where A = Z and A m = Z if m > . This functor encodes theLawrence-Bigelow representations (using Borel-Moore homology) of the braid groups { B n } n ∈ N . For the remainder of §8.1, we fix the natural numbers m > . The aim of this section isto prove the following polynomiality results for this functor.
Theorem 8.1
The Lawrence-Bigelow functor LB BM : h β , β i → Z [ A ] - Mod is strong polynomialof degree and weak polynomial of degree . If m > then the Lawrence-Bigelow functor LB BMm : h β , β i → Z [ A m ] - Mod is both very strong and weak polynomial of degree m . Remark 8.2
Recall from Remark 5.6 that LB BM ⊗ C is the reduced Burau functor Bur and that LB BM ⊗ C is the Lawrence-Krammer functor LK . Hence the result of Theorem 8.1 for strongpolynomiality recovers those of [Sou19, Section 3.3], i.e. that Bur is strong polynomial of degree and Lawrence-Krammer is very strong polynomial of degree .The monoidal product ♮ : β × β → β being defined by the usual addition for the objects, anyobject of braid groupoid β is isomorphic to a finite sum of . Hence by Propositions 7.4 and 7.9,it is enough to study the natural transformation i to prove the polynomiality results. Our goal isto study the kernel and cokernel of the natural transformation i LB BMm .Following Proposition 7.1, the natural transformation i LB BMm is defined for all natural numbers n by the morphism (cid:16) i LB BMm (cid:17) n = LB BMm ( ι ♮id n ) = LB BMm ([1 , id n +1 ]) , and fits into the following exact sequence in the category Fct ( h β , β i , Z [ A m ] - Mod) : / / κ LB BMm / / LB BMm i LB BMm / / τ LB BMm ∆ LB BMm / / δ LB BMm / / . (8.1)72ollowing §6.2, the use of Borel-Moore homology allows to describe the morphism LB BMm ([1 , id n +1 ]) in terms of sets of partitions of copies of the group ring Z [ A m ] . More precisely, recall from §6.2.1that for all natural numbers n > LB BMm ( n ) ∼ = M ω ∈P m ( n − Z [ A m ] ω and LB BMm (0) = 0 and therefore that the morphism LB BMm ([1 , id n +1 ]) is equivalent to the injection of Z [ A m ] -modules ι P m ( n ) \P m ( n − ⊕ id P m ( n − : M ω ∈P m ( n − Z [ A m ] ω ֒ → M ω ∈P m ( n ) Z [ A m ] ω . induced by the injective map P m ( n − ֒ → P m ( n ) sending ( ω , . . . , ω n − ) to (0 , ω , . . . , ω n − ) if n > and the trivial morphism if n = 0 and n = 1 .There are then two possible recursive ways to prove Theorem 8.1: either we can work on thedimensions of the difference functor with respect to the sets of partitions as detailed in §8.1.1, orwe can establish a key original relation between the difference functor of the m th Lawrence-Bigelowfunctor and a translation of the ( m − th one as shown in §8.1.2. This first proof consists in studying the partitions subsets which index the number of copies of Z [ A m ] in the direct sum of the successive difference functors of LB BMm . Let k be a natural number.For each natural number n , we consider the set P δ k m ( n ) := { ( ω , . . . , ω k + n − ) ∈ P m ( k + n − | ∀ i ∈ { , . . . , k } , ω i m } , with the convention that it is the empty set if n + k . Then there is a canonical injection P δ k m ( n ) ֒ → P δ k m ( n + 1) defined by sending ( ω , . . . , ω k + n − ) to (0 , ω , . . . , ω k + n − ) which induces acanonical bijection P δ k +1 m ( n ) ∼ = P δ k m ( n + 1) \ P δ k m ( n ) . (8.2)This injective set map also defines an injective Z [ A m ] -module morphism P km,n : M ω ∈P δkm ( n ) Z [ A m ] ω ֒ → M ω ∈P δkm ( n +1) Z [ A m ] ω for n > and the trivial morphism P km, . These sets describe the successive difference functors ofthe m th Lawrence-Bigelow functor: Proposition 8.3
For all natural numbers n , there is an isomorphism of Z [ A m ] -modules δ k LB BMm ( n ) ∼ = M ω ∈ P δ k m ( n ) Z [ A m ] ω and the morphism (cid:16) i (cid:16) δ k − LB BMm (cid:17)(cid:17) n is equivalent to the injection P k − m,n for all natural numbers k m + 1 if m > and for k = 1 if m = 1 .Proof. We proceed by induction on k . For k = 1 , we already know from §6.2.1 that (cid:16) i LB BMm (cid:17) n is equivalent to the injection P m,n and it follows from the universal property of the cokernel that δ LB BMm ( n ) ∼ = M ω ∈ P δ m ( n ) Z [ A m ] ω . Now we assume that the results of Proposition 8.3 are true for some fixed k > . Recall fromProposition 7.1 that, by the definition of the difference functor, the morphism (cid:16) i (cid:16) δ k LB BMm (cid:17)(cid:17) n is73anonically induced by the morphism τ δ k − LB BMm ([1 , id n ]) . The inductive hypothesis gives theequivalence between δ k − LB BMm ([1 , id n ]) and P k − m,n . Using the bijection (8.2), the image of P k − m,n induced by the canonical surjections P δ k − m ( n + 1) ։ P δ k m ( n ) and P δ k − m ( n + 2) ։ P δ k m ( n + 1) isthe morphism P km,n : hence (cid:16) i (cid:16) δ k LB BMm (cid:17)(cid:17) n is equivalent to the injection P km,n .Then the following diagram is commutative: κ δ k LB BMm ( n ) (cid:31) (cid:127) / / δ k LB BMm ( n ) ( i ( δ k LB BMm )) n / / ∼ (cid:15) (cid:15) τ δ k LB BMm ( n ) ∼ (cid:15) (cid:15) / / / / δ k +11 LB BMm ( n ) . L ω ∈P δkm ( n ) Z [ A m ] ω (cid:31) (cid:127) P km,n / / L ω ∈P δkm ( n +1) Z [ A m ] ω We deduce that δ k +11 LB BMm ( n ) ∼ = M ω ∈ P δ k +1 m ( n ) Z [ A m ] ω using the universal property of the cokernel and the bijection (8.2).We are now ready to prove Theorem 8.1. First we deduce from Proposition 8.3 that δ LB BM ( n ) = Z [ A ] for all n > and δ LB BM (0) = 0 : a fortiori δ LB BM ( n ) = 0 for all n > and δ LB BM (0) = Z [ A ] . Hence: δ LB BM = 0 and therefore LB BM is strong polynomial of de-gree two; δ LB BM is a stably null object of the category Fct ( h β , β i , Z [ A ] - Mod) and a fortiori LB BM is weak polynomial of degree one since δ ◦ π h β , β i = π h β , β i ◦ δ by Proposition 7.7.Let us now fix m > . We deduce from Proposition 8.3 that the morphisms (cid:16) i (cid:16) δ m LB BMm (cid:17)(cid:17) n is equivalent to the injection P mm,n for all natural numbers n . Note that the sets P δ m m ( n ) and P δ m m ( n + 1) are actually isomorphic: the injection P mm,n is therefore a bijection. Hence the naturaltransformation i (cid:16) δ m LB BMm (cid:17) is a natural equivalence between non-null objects. Again the factthat δ ◦ π h β , β i = π h β , β i ◦ δ implies that the functor LB BMm : h β , β i → Z [ A m ] - Mod is weakpolynomial of degree m . Also, for all natural numbers k such that k m , the evanescencefunctor κ δ k LB BMm is null since (cid:16) i (cid:16) δ k LB BMm (cid:17)(cid:17) n is an injection by Proposition 8.3. Then thefunctor δ k LB BMm is very strong polynomial polynomial of degree m − k which ends the proof. The key of this second method is the result of Theorem 8.5: the difference functor of the m thLawrence-Bigelow functor is isomorphic a translation of the ( m − th one. The proof ot Theorem8.1 is then a straightforward induction on the degree of polynomiality. In addition this isomorphismgives an original relation between two different Lawrence-Bigelow functors.Using the relations on Borel-Moore homology classes detailed in §9, we have m = m X k =0 k m − k (8.3)For all natural numbers n > , we define p m ( n ) : P δ m ( n ) ∼ = P m − ( n ) to be the set isomorphism ( ω , . . . , ω n ) ( ω − , . . . , ω n ) and ( p m ) to be the trivial set map. 74 otation 8.4 For consistency of the exposition we denote by LB BM : h β , β i → Z [ A ] - Mod thesubobject of the constant functor at Z [ A ] such that LB BM (0) = LB BM (1) = 0 . For convenience,we abuse the notation and write τ LB BMi for the functor τ LB BMi ⊗ Z [ A ] Z [ A ] for i = 0 , .It follows from §6.2.1 that the morphisms { p m ( n ) } n ∈ N induce Z [ A m ] -modules isomorphisms nb p m ( n ) : δ LB BMm ( n ) ∼ → τ LB BMm − ( n ) o n ∈ N . induced by forgetting a point of the configuration space in the first interval of I n . These iso-morphisms allow to uncover the following key relation between the difference functor of the m thLawrence-Bigelow functor and the ( m − th Lawrence-Bigelow functor. Theorem 8.5
The isomorphisms (cid:8)b p m ( n ) (cid:9) n ∈ N define an isomorphism b p m : δ LB BMm ∼ → τ LB BMm − in the category Fct ( h β , β i , Z [ A m ] - Mod) .Proof.
First, we prove that the morphisms (cid:8)b p m ( n ) (cid:9) n ∈ N define an isomorphism in the category Fct ( β , Z [ A m ] - Mod) . Recall from Proposition 7.1 that the definition of the difference functor δ LB BMm on the morphisms of h β , β i of is formally induced by τ LB BMm . We fix a natural number n > (the proof being trivial for n = 0 and n = 1 ). We consider the Artin generator σ i of B n for some i ∈ { , . . . , n − } . Using the work of §6.2.1, the morphism τ LB BMm ( σ i ) (and a fortiori δ LB BMm ( σ i ) ) is the morphism in Borel-Moore homology induced by the action of σ i +1 on theunion of intervals I n . Hence it is enough to prove that forgetting a point of the configurationin the first interval ( p , p ) before or after the twisting by σ i +1 is exactly the same operation atthe level of the Borel-Moore homology classes. Actually, a quick investigation shows that the onlynon-trivial case is for i = 1 : the property is indeed clear if i > since σ i +1 acts trivially on ( p , p ) in this case. The pictures of Figure 8.1 represent the action of σ by the Lawrence-Bigelow functoron a Borel-Moore homology class on the covering of C m ( I n ) for those on the left-hand side andof C m − ( I n ) for those on the right-hand side. The red cross in the first interval denotes anadditional configuration point of the first interval in C m ( D n +1 ) compared to C m − ( D n +1 ) : themorphism b p m ( n ) corresponds to forgetting this additional point. It follows from Relation (8.3)that the diagram of Figure 8.1 is commutative (note that one of the partitions (corresponding to k = 0 ) misses since we are in δ ). Hence b p m ( n ) ◦ δ LB BMm ( σ i ) = τ LB BMm − ( σ i ) ◦ b p m ( n ) for all i ∈ { , . . . , n − } .Now we prove that b p m is a natural transformation in Fct ( h β , β i , Z [ A m ] - Mod) by using Lemma3.6. We fix a natural number n > (the proof being trivial for n = 0 ) and recall that τ LB BMm ([1 , id n +1 ]) = LB BMm (cid:0) σ − (cid:1) ◦ LB BMm ([1 , id n +2 ]) . Recall from §6.2.1 that LB BMm ([1 , id n +2 ]) is the morphism on Borel-Moore homology induced bythe embedding of I n into I n as the n − last intervals: this amounts to sending the interval ( p i , p i +1 ) of I n +1 to the interval ( p i +1 , p i +2 ) of I n +2 for each i ∈ { , . . . , n } . Therefore there isno configuration point on the first interval ( p , p ) of the image of LB BMm ([1 , id n +2 ]) . Moreover, LB BMm (cid:0) σ − (cid:1) is the morphism in Borel-Moore homology induced by the action of σ on I n . Thepictures of Figure 8.2 represent the composite of these two morphisms by the Lawrence-Bigelowfunctor on a Borel-Moore homology class on the covering of C m ( I n ) for those on the left-handside and of C m − ( I n ) for those on the right-hand side. Again the red cross denotes an additionalconfiguration point, the morphism b p m ( n ) corresponding to forgetting it. Then we have to provethat the composite LB BMm (cid:0) σ − (cid:1) ◦ LB BMm ([1 , id n +2 ]) commutes with the operation of forgetting apoint of the configuration space in the interval ( p , p ) at the level of the Borel-Moore homologyclasses. Again Relation (8.3) gives the commutativity of the diagram of Figure 8.2 (note again thatone of the partitions (corresponding to k = 0 ) misses since we are in δ ). Hence a straightforwardinduction on the natural number k > gives that b p m ( n ) ◦ δ LB BMm ([ k, id n +1 ]) = τ LB BMm − ([ k, id n +1 ]) ◦ b p m ( n ) . Hence Relation (3.3) of Lemma 3.6 is satisfied for all natural numbers n : this ends the proof.75 p m ( n ) δ LB BM m ( σ ) τ LB BM m − ( σ ) b p m ( n ) p p p p p n p n +1 · · · p p p p p n p n +1 · · · p p p p p n p n +1 · · · p p p p p n p n +1 · · · p p p p p n p n +1 · · · k m − k p p p p p n p n +1 · · · k − m − k m X k =1 m X k =1 Figure 8.1 b p m ( n ) LB BM m ([1 , id n +2 ]) LB BM m − ([1 , id n +2 ]) LB BM m ( σ − ) LB BM m − ( σ − ) b p m ( n ) p p p p p n p n +1 · · · p p p p p n p n +1 · · · p p p p p p n +1 p n +2 · · · p p p p p p n +1 p n +2 · · · p p p p p p n +1 p n +2 · · · p p p p p p n +1 p n +2 · · · p p p p p p n +1 p n +2 · · · k m − k p p p p p p n +1 p n +2 · · · k − m − k m X k =1 m X k =1 Figure 8.276et us now prove Theorem 8.1. Using the commutation property of δ and τ of Proposition 7.1,we deduce from Theorem 8.5 that for all natural numbers k mδ k (cid:16) LB BMm (cid:17) ∼ = τ k (cid:16) LB BMm − k (cid:17) . (8.4)For m = 1 , note that τ LB BM is the subobject of the constant functor at Z [ A ] such that τ LB BM (0) = 0 . Hence LB BM is weak polynomial of degree and δ LB BM is the functor whichunique non-null value is δ LB BM (0) = Z [ A ] . A fortiori LB BM is strong polynomial of degree .For m > , first note that for all natural numbers l > , τ LB BM = τ l LB BM is a constantfunctor. It thus follows from the isomorphism (8.4) with k = m that LB BMm is both strong andweak polynomial of degree m .From the definition of the morphisms LB BMm ([1 , id n +1 ]) for all natural numbers n , we know that κ (cid:16)(cid:16) LB BMm (cid:17)(cid:17) = 0 . Using the isomorphism (8.4), the commutation property of κ and τ ofProposition 7.1 implies that κ (cid:16) δ k (cid:16) LB BMm (cid:17)(cid:17) = 0 for all natural numbers k m : this provesthat LB BMm is very strong polynomial of degree m . We recall that the construction of §4.3.2, identification of §5.3.2 and properties of §6.2.2 prove thatthe representations of the mapping class groups considered in [Mor07] define the m th Moriyamafunctor Mor m : (cid:10) M + ,gen , M + ,gen (cid:11) → Z - Mod . for each natural numbers m > . For the remainder of §8.2, we fix the natural numbers m > . We prove in this section that this functor satisfies polynomial properties.Let k be a natural number. For each natural number g , we consider the set Q δ k m ( g ) := S m × { ( ω , . . . , ω k +2 g − ) ∈ P m (2 k + 2 g − | ∀ i ∈ { , . . . , k } , ( ω i − , ω i ) = (0 , } , with the convention that it is the empty set if k + 2 g . Then there is a canonical injection Q δ k m ( g ) ֒ → Q δ k m ( g + 1) defined by sending ( ω , ω , . . . , ω k +2 g − ) to (0 , , ω , ω , . . . , ω k + n − ) . Itinduces a canonical bijection Q δ k +1 m ( g ) ∼ = Q δ k m ( g + 1) \ Q δ k m ( g ) . (8.5)This injective set map also defines an injective Z -module morphism Q km,g : M ω ∈Q δkm ( g ) Z ω ֒ → M ω ∈Q δkm ( g +1) Z ω for g > and the trivial morphism Q km, . These sets describe the successive difference functors ofthe m th Moriyama functor: Proposition 8.6
For all natural numbers g , there is an isomorphism of Z -modules δ k Mor m ( g ) ∼ = M ω ∈ Q δ k m ( g ) Z ω and the morphism (cid:0) i (cid:0) δ k − Mor m (cid:1)(cid:1) n is equivalent to the injection Q k − m,n for all natural numbers k m + 1 .Proof. We proceed by induction on k . For k = 1 , we already know from §6.2.2 that ( i Mor m ) g isequivalent to the injection Q m,g and it follows from the universal property of the cokernel that δ Mor m ( g ) ∼ = M ω ∈ Q δ m ( g ) Z ω . k > . Recall fromProposition 7.1 that, by the definition of the difference functor, the morphism (cid:0) i (cid:0) δ k Mor m (cid:1)(cid:1) g iscanonically induced by the morphism τ δ k − Mor m ([1 , id g ]) . The inductive hypothesis gives theequivalence between δ k − Mor m ([1 , id g ]) and Q k − m,g . Using the bijection (8.5), the image of Q k − m,g induced by the canonical surjections Q δ k − m ( g + 1) ։ Q δ k m ( g ) and Q δ k − m ( g + 2) ։ Q δ k m ( g + 1) isthe morphism Q km,g : hence (cid:0) i (cid:0) δ k Mor m (cid:1)(cid:1) g is equivalent to the injection Q km,g .Then the following diagram is commutative: κ δ k Mor m ( g ) (cid:31) (cid:127) / / δ k Mor m ( g )( i ( δ k Mor m )) g / / ∼ (cid:15) (cid:15) τ δ k Mor m ( g ) ∼ (cid:15) (cid:15) / / / / δ k +11 Mor m ( g ) . L ω ∈Q δkm ( g ) Z ω (cid:31) (cid:127) Q km,g / / L ω ∈Q δkm ( g +1) Z ω We deduce that δ k +11 Mor m ( g ) ∼ = M ω ∈ Q δ k +1 m ( g ) Z ω using the universal property of the cokernel and the bijection (8.5).From the previous properties, we deduce the following polynomiality results for the Moriyamafunctors: Theorem 8.7
The Moriyama functor
Mor m : (cid:10) M + ,gen , M + ,gen (cid:11) → Z - Mod is both very strongand weak polynomial of degree m .Proof. Note that the sets Q δ m m ( n ) and Q δ m m ( n + 1) are isomorphic: then it from Proposition 8.6that the morphisms ( i ( δ m Mor m )) n is a bijection. Hence the natural transformation i ( δ m Mor m ) is a natural equivalence between non-null objects. Since δ ◦ π h M + ,gen , M + ,gen i = π h β , β i ◦ δ , wededuce that the functor Mor m : (cid:10) M + ,gen , M + ,gen (cid:11) → Z - Mod is weak polynomial of degree m .Also, for all natural numbers k such that k m , the evanescence functor κ δ k Mor m is nullsince (cid:0) i (cid:0) δ k Mor m (cid:1)(cid:1) n is an injection by Proposition 8.6. Then the functor δ k Mor m is very strongpolynomial polynomial of degree m − k which ends the proof. Finally, we detail here some uses of the polynomiality results stated in §8.1 and §8.2.
A first application of the previous polynomial results is their homological stability properties.Indeed Randal-Williams and Wahl [RW17] prove homological stability for several families of groups(including in particular most of those considered in §3.4) with twisted coefficients given by verystrong polynomial functors. Namely, fixing a strict monoidal groupoid ( G , ♮, , an object X of G ,a left-module ( M , ♮ ) , an object A of M , we denote by hG , Mi X,A the full subcategory of hG , Mi with objects (cid:8) X ♮n ♮A (cid:9) n ∈ N . We also denote by G n the automorphism group Aut hG , Mi (cid:0) X ♮n ♮A (cid:1) forall natural numbers n . Then: Definition 8.8
The family of groups { G n } n ∈ N is said to satisfy homological stability (with twistedcoefficients) if for any very strong polynomial functor F : hG , Mi X,A → Z - Mod of degree d , thenatural maps H ∗ (cid:0) G n , F (cid:0) X ♮n ♮A (cid:1)(cid:1) → H ∗ (cid:0) G n , F (cid:0) X ♮ n ♮A (cid:1)(cid:1) are isomorphisms for N ( ∗ , d ) n with N ( ∗ , d ) ∈ N depending on ∗ and d .78 heorem 8.9 [RW17, Theorem A] Homological stability with twisted coefficients is satisfied for: • Classical braid groups of surfaces { B n } n ∈ N using G = M = β ; • Braid groups on surfaces { B n (Σ g, ) } n ∈ N and { B n ( N c, ) } n ∈ N , using G = β and respectively M = B g, +2 and M = B c, − ; • Mapping class groups { Γ n, } n ∈ N and { N c, } n ∈ N , using G = M = M +2 and G = M = M − respectively; • Extended and non-extended loop braid groups (cid:8) LB ext n (cid:9) n ∈ N and { LB n } n ∈ N , using G = M = L β ext and G = M = L β respectively. The above framework is generalised in [Kra17] to a topological setting: more precisely, the ho-mological stability results of [RW17] are extended to families of spaces that are not necessarilyclassifying spaces of discrete groups. [Pal18] also proves homological stability stability for configu-ration spaces of path-connected subspaces on an open connected manifold, with twisted coefficientsgiven by polynomial-type functors. Homological stability is thus satisfied by the Lawrence-Bigelowfunctors and Moriyama functors by Theorem 8.9. As the representation theory of braid groupsand mapping class groups of surfaces is wild and an active research topic (see for example [BB05],[FLM01], [Fun99], [Kor02], [Mas08] or [MR12]), there are very few known examples of very strongpolynomial functors over the asssociated categories. Hence the result of §8.1 and §8.2 allow togain a better understanding of polynomial functors for braid groups of surfaces and mapping classgroups, and therefore extends the scope of twisted homological stability to more sophisticatedsequences of representations.
A first matter of interest in the notion of weak polynomiality is that it reflects more accurately thanthe strong polynomiality the behaviour of functors for large values. As Theorem 8.1 shows, the firstLawrence-Bigelow functor LB BM is strong polynomial of degree two whereas it is weak polynomialof degree one. On the contrary the further Lawrence-Bigelow functors LB BMm are strong and weakpolynomial of degree m . We also refer the reader to [DV19, Section 6] and [Dja17] for examples ofsuch phenomena.Also, the quotient categories for weak polynomial functors shed a light onto what is going onwith the successive Lawrence-Bigelow functors (see §7.2). Indeed, denoting for convenience thecategory of weak polynomial functors P ol d ( h β , β i , Z - Mod) by P ol d ( β ) , we have as a consequenceof Theorem 8.1: Corollary 8.10
The sequence of the quotient categories induced by the difference functors: · · · P ol m ( β ) / P ol m − ( β ) δ o o P ol m +1 ( β ) / P ol m ( β ) δ o o · · · δ o o gives a classification of the family of Lawrence-Bigelow functors n LB BMm o m > and the connectionbetween two functors of this family.Proof. Each Lawrence-Bigelow functor LB BMm is both weak polynomial and very strong polynomialof degree d . Therefore LB BMm ∼ = τ LB BMm in the quotient categories.
In this appendix, we give a quick proof of the relation (8.3) among Borel-Moore homology classes.In §9.1 we recall the basic facts about fundamental classes in Borel-Moore homology, and we usethis in §9.2 to prove the relation (8.3). 79 .1 Fundamental classes in Borel-Moore homology
Let X be a manifold (possibly with boundary) and let M be an orientable k -manifold, withoutboundary and with finitely many components. Then any proper embedding e : M ֒ −→ X (9.1)determines an element [ e ] ∈ H BM k ( X ) . More precisely: the homology group H BM k ( M ) is Z r , where r is the number of components of M . An orientation of M is a choice of generator for the summandcorresponding to each component, which determines a fundamental class [ M ] ∈ H BM k ( M ) . Borel-Moore homology is functorial with respect to proper embeddings, so the image of this class isthe element [ e ] mentioned above. So we should say more precisely that an orient ed k -manifold M (without boundary and with finitely many components) and a proper embedding (9.1) determinean element [ e ] ∈ H BM k ( X ) . This is additive for finite disjoint unions, i.e., if M is the disjoint unionof M , . . . , M j , then [ e ] = j X i =1 [ e | M i ] in the group H BM k ( X ) .Now let N be an oriented ( k + 1) -manifold with finitely many boundary-components and let f : N ֒ −→ X be a proper embedding. Then we have the relation [ f | ∂N ] = 0 (9.2)in the group H BM k ( X ) . By §9.1, we simply need to exhibit an oriented manifold with boundary, embedded in the (cover ofthe) m -point configuration space, such that the image of its boundary (considered with its inducedorientation) is the difference of the two sides of the relation (8.3). We may define such a manifoldas follows: m This is the space of unordered configurations of m points in the plane, such that all m points lieon one of the lines depicted in the closed half-disc above, and do not intersect any of the threepunctures. In particular, note that, if they lie on the top (straight) line, they must be partitionedinto two subsets of size k and m − k by the two little intervals between the three punctures. It isthen not hard to see that this has the desired (oriented) boundary. Remark 9.1
We note that [Ito13, Section 4] and [Ike18, Section 3.4] prove analogous relationsamong Borel-Moore homology classes.
10 Appendix: Notations and tools
We take the convention that the set of natural numbers N is the set of non-negative integers { , , , . . . } . 80e denote by Gr the category of groups and by ∗ the coproduct in this category. The trivialgroup is denoted by Gr . For a group G , the lower central series of G is the descending chainof subgroups { Γ l ( G ) } l > defined by Γ ( G ) := 0 Gr , Γ ( G ) := G and Γ l +1 ( G ) := [ G, Γ l ( G )] thesubgroup of G generated by all commutators [ x, y ] := xyx − y − with x in G and y in Γ l ( G ) .The induced canonical projection on the quotient by the l th lower central term is denoted by γ l ( G ) : G ։ G/Γ l ( G ) . In particular γ ( G ) denotes the abelianisation map of the group G . Whenthere is no ambiguity, we omit G from the notations.For a (commutative) ring R , we denote by R - Mod the category of R -modules. For M a G -module,we denote by Aut G ( M ) the group of G -module automorphisms of M . If G = Z , the we omit itfrom the notation if there is no ambiguity.We denote by Top the category of topological spaces and by Top ∗ the category of based topologicalspaces with based maps. For X a topological space, we denote by Homeo ( X ) the group of self-homeomorphisms of X . If X is a differentiable manifold, then we denote by Diff ( X ) the self-diffeomorphism group of X .Let Cat denote the category of small categories. Let C be an object of Cat. We use the abbreviation Ob ( C ) to denote the set of objects of C . If there exists an initial object Ø in the category C , thenwe denote by ι A : Ø → A the unique morphism from Ø to A . If T is a terminal object in thecategory C , then we denote by t A : A → T the unique morphism from A to T . Let Grpd denotethe subcategory of Cat of small groupoids. The maximal subgroupoid G r ( C ) is the subcategory of C which has the same objects as C and of which the morphisms are the isomorphisms of C . Wedenote by G r : Cat → Grpd the functor which associates to a category its maximal subgroupoid.For D a category and C a small category, we denote by Fct ( C , D ) the category of functors from C to D .We take this opportunity to recall some terminology about (strict) monoidal categories and modulesover them. We refer to [Mac98] for a complete introduction to these notions. A strict monoidalcategory ( C , ♮, , where C is a category, ♮ is the monoidal product and is the monoidal unit. Ifit is braided, then its braiding is denoted by b C A,B : A♮B ∼ → B♮A for all objects A and B of C .A left-module ( M , ♯ ) over a (strict) monoidal category ( C , ♮, is a category M with a functor ♯ : C × M → M that is unital (i.e. ♯ ◦ ( ι C × id M ) = id M where ι C : 0 Cat → C takes the trivialcategory Cat to the unit object of C ) and associative (i.e. ♯ ◦ ( id C × ♯ ) = ♯ ◦ ( ♯ × id M )) . Forinstance, a monoidal category ( C , ♮, is automatically equipped with a left-module structure overitself, induced by the monoidal product ♮ : C × C → C . References [AK10] B. H. An and K. H. Ko.
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