On the Motivic Class of the Moduli Stack of Twisted $G$-Covers

We describe the class, in the Grothendieck group of stacks, of the stack of twisted $G$-covers of genus $0$ curves, in terms of the loci corresponding to covers over smooth bases.


Introduction
Moduli stacks of twisted stable maps have been introduced in the framework of Gromov-Witten theory for orbifolds and for Deligne-Mumford stacks, independently by Chen-Ruan [16] and by Abramovich-Graber-Vistoli [2,3].In the case where the target is the classifying stack BG of a finite group G, a twisted stable map is called a twisted G-cover [1].The moduli stack B bal g,n (G) of balanced twisted stable maps to BG is isomorphic to the stack of admissible G-covers, which has been extensively studied due to its several applications (see, e.g., [1,23,31,32,44]).
The present work originates from [13,14,15] where the authors studied the classification problem for the connected components of moduli spaces of Galois covers of smooth curves.The results obtained there give a complete description of such components for any genus of the base curves when the Galois group is the dihedral group, in the general case one only has a complete classification when the genus of the base curves is large enough.In this article, we consider the compactifications of moduli spaces of Galois covers of curves given by admissible covers, restricting our attention to the case where the genus of the base curves is 0. The aim is to investigate how the boundary can be used to obtain information about the geometry of the moduli spaces.Our main result (Theorem 6.4) gives a description of the class of B bal 0,n (G) in the Grothendieck group of stacks, i.e., of the motivic class of B bal 0,n (G).This description is obtained by stratifying the moduli stack as the disjoint union of locally closed substacks B bal (G, τ ), which correspond to the G-covers of a given combinatorial structure τ , where τ is a gerby tree (cf.Definition 5.4).The main technical step is Proposition 6.3, where we show how these strata are obtained by gluing together the loci in B bal 0,n (G) corresponding to covers over smooth bases.In order to keep track of the actions of the symmetric groups on the marked points, these gluing constructions correspond to two basic operations, the Day convolution (Definition 3.2) and the composition (Definition 3.7).As a result, we obtain a formula (6.7) that expresses the motivic class of B bal 0,n (G) recursively in terms of the loci B k , for k ≤ n, corresponding to covers over smooth bases (cf.Remark 6.5).Here we follow the strategy of [27] (where the authors compute the Betti numbers of the moduli spaces of stable maps to projective spaces), further extended in [8] to the case of moduli spaces of stable maps to Grassmannians.Over the field of complex numbers, by applying the Poincaré (or Serre) characteristic [10,27], our formula allows to compute the Betti numbers of B bal 0,n (G) in terms of the cohomology of B k , as a representation of the symmetric group S k , k = 3, . . ., n, and the decomposition of B k as union of the open and closed substacks B c := B bal 0,c (G) ∩ B k , where c = (c 1 , . . ., c k ) varies among the sequences of k conjugacy classes of G, and B bal 0,c (G) is the full sub-category of B bal 0,k (G) whose objects are the twisted G-covers whose evaluation at the marked points is equal to c (cf. Definition 2.9).Notice that B c is determined by B k , as a stack over Īµ (BG) k with respect to the evaluation morphism.We will address the problem of computing effectively these invariants of B n in future works.
The structure of the article is as follows.In Section 2, we recall the definition and collect basic properties of B bal 0,n (G).In Section 3, we define the Day convolution and the composition, and we prove their main properties that will be used.In Section 4, following the definition of the Grothendieck group of stacks from [22], we introduce the Grothendieck group of S-modules, over which there is an induced composition operation.In Section 5, we define the notion of gerby tree τ (following [4]) and we show that the locus in B bal 0,n (G) of covers of a fixed combinatorial type τ is locally closed.In Section 6, we prove our main results, Proposition 6.3 and Theorem 6.4.In the last Section 7, we show how these results can be used to compute the Hodge-Grothendieck characteristic of B bal 0,n (G) in terms of B k for k = 3, . . ., n.

Conventions
The symbol N denotes the set of nonnegative integers.For each n ∈ N, the symmetric group on n letters is denoted by S n .We fix a universe, and we use the term "small" in reference to the chosen universe.As usual, the category of small categories is denoted by Cat, the category of small sets is denoted by Set, and 1 denotes the terminal category.For all (enriched) categories C and D, D C denotes the (enriched) category of (enriched) functors from C to D. A 2-category is a Cat-enriched category, which is sometimes referred to as a strict 2-category.
In order to avoid set-theoretic issues when considering toposes, by a stack over a base scheme T we mean a stack in groupoids over a suitable small sub-site of the big fppf site of T (constructed as in [6, Section IV. 2

.5]).
For what concerns group actions on stacks, we follow [45], in particular, according to [45, Definition 2.1] the actions that we will consider are all strict.This degree of generality is sufficient for our purposes, even though in several places one could allow more general actions.

The stack of twisted G-covers
In this section, we recall the definition of the stack of balanced twisted G-covers of n-pointed genus 0 curves, B bal 0,n (G), following [1], to which we refer for further details and properties of this stack.
Let k be a fixed algebraically closed field of characteristic 0. By an algebraic k-stack we mean an algebraic stack of finite type over k which has affine stabilizers.The 2-category of algebraic k-stacks is denoted by AlgSt k , while St k denotes the 2-category of stacks over k.
Let G be a finite group.Our aim is to study the class of B bal 0,n (G) in the Grothendieck ring of algebraic k-stacks K 0 (AlgSt k ).
Roughly speaking, the objects of B bal 0,n (G) are twisted stable maps from twisted curves to BG, the classifying stack of G. Notice that in the second case the fiber over t = 0 has a node in U , which is locally smoothable if and only if a ≡ −1 (mod r).A twisted curve is called balanced if at every node it is presented as above with a ≡ −1 (mod r).
where (1) C → T is a proper twisted curve over T ; (2) Σ = i∈N Σ i , where the Σ i ⊂ C, for i ∈ N , are disjoint closed substacks contained in the smooth locus of C → T ; (3) for every i ∈ N , Σ i → T is an étale gerbes (see [2,Section 3.3] for a definition); (4) the morphism C → C exhibits C as the coarse moduli space of C; (5) for every i ∈ N , let |Σ i | ⊂ C be the image of Σ i (called the i-th marking in C), and let C gen the complement of the nodes and the markings in C, then the morphism C → C is an isomorphism over C gen .
A twisted nodal N -pointed curve over T is called balanced if C → T is a balanced twisted curve.For a positive integer n, by a (twisted nodal) n-pointed curve we mean a (twisted nodal) N -pointed curve where N = {1, . . ., n}.
Remark 2.3.In the previous definition, one should think of Σ as the marked points of the twisted curve, indeed we will refer to Σ i as the i-th marked point.The "gerby" structure of the marked points arise naturally from Definition 2.1, according to which, étale locally, Σ i → T is isomorphic to [Spec(A[z]/(z))/µ r ] → Spec(A), where µ r acts trivially.The integer r is called the index of the marked point over Spec(A).If Σ i has constant index r, then it is canonically banded by µ r [2, Theorem 4.2.1].We refer to [2, Section 4] and [40] for more details on twisted curves and their moduli.
Consider now the classifying stack BG of G, that is the category whose objects over a kscheme T are principal G-bundles P → T , and morphisms are G-equivariant fiber diagrams of such bundles: BG is a global quotient stack with presentation BG = [Spec(k)/G] and coarse moduli space Spec(k).A twisted G-cover is called balanced if the corresponding twisted curve is balanced.
Remark 2.5.To justify the name"twisted G-cover" in the previous definition, notice that the morphism C → BG corresponds to a principal G-bundle P → C.Moreover, the representability condition (iii) implies that P → T is a projective nodal curve (see [1, Lemma 2.2.1] for details), so P → C is a G-cover in the usual sense, which is ramified (at most) over the marked points |Σ i |, i ∈ N , and the nodes of C.
We have now introduced all the objects needed to define the stack we are interested in.Consider the 2-category whose objects over a k-scheme T are balanced twisted G-covers of an N -pointed curve of genus 0 over T , 1-morphisms are given by fiber diagrams, and 2-morphisms are given by base-preserving natural transformations.As explained in [1, Section 2.1.5],this 2category is equivalent to a category, obtained by replacing 1-morphisms by their 2-isomorphism classes (see [3,Proposition 4.2.2] for the proof).The category of balanced twisted G-covers thus obtained is denoted B bal 0,N (G), respectively B bal 0,n (G) when N = {1, . . ., n}.The following result is a special case of Theorems 2.1.7,3.0.2,and of Corollary 3.0.5 in [1] (see also [3]).
Theorem 2.6.For any finite set N of cardinality #N ≥ 3, B bal 0,N (G) is a smooth Deligne-Mumford stack of dimension #N − 3 and with projective coarse moduli space.There is a morphism B bal 0,N (G) → M 0,N which is flat, proper and quasi-finite (M 0,N is the moduli space of stable N -pointed curves of genus 0).Remark 2.7.In general B bal g,N (G) is defined as the stack of balanced twisted G-covers of Npointed curves of genus g.As proved in [1,Theorem 4.3.2],this stack is isomorphic to the stack of admissible G-covers Adm g,N (G).Throughout this article, it is useful to have in mind both points of view.Notation 2.8.We will denote with B sm 0,N (G) the locus in B bal 0,N (G) of twisted G-covers of smooth N -pointed curves of genus 0, i.e., where the curves C in Definition 2.4 are smooth.Notice that The stack B bal 0,N (G) comes equipped with evaluation maps ev i N , for i ∈ N (respectively ev i n , if N = {1, . . ., n}), defined as follows.Let (f : C → BG, Σ) be an object of B bal 0,N (G) over T , the restriction of f to Σ i yields the diagram Σ i BG T.
Notice that, as usual, the group S N of bijections N → N (respectively the symmetric group S n , if N = {1, . . ., n}) acts naturally on B bal 0,N (G) by permutation of the marked points (and this action is strict in the sense of [45,Definition 1.3]).Furthermore, defining we obtain a strict S N -morphism, where S N acts on the target also by permutation.

The coarse moduli space of the stack of cyclotomic gerbes
For lack of a suitable reference, we show here that the coarse moduli space of I µ (BG), which will be denoted I µ (BG) in the article, can be identified with the set of conjugacy classes of G.
To this aim, by [2, Proposition 3.4.1]we have that I µ r (BG) is isomorphic to the rigidification of the cyclotomic inertia stack I µ r (BG) with respect to µ r , hence its coarse moduli space is isomorphic to that of I µ r (BG).By [2, Definition 3.1.1],an object of I µ r (BG) over a scheme T is a pair (ξ, α), where ξ is a principal G-bundle over T and α : (µ r ) T → Aut T (ξ) is an injective homomorphism.An arrow from (ξ, α) over T to (ξ ′ , α ′ ) over T ′ is an arrow F : ξ → ξ ′ such that the following diagram commutes: Notice that, if we fix a generator of µ r , α corresponds to an element ϕ ∈ Aut T (ξ) of order r.
Over Spec(k), every principal G-bundle ξ is isomorphic to the trivial one, which we denote ξ 0 .Furthermore, we can identify Aut(ξ 0 ) with G by associating any g ∈ G with the automorphism ϕ g : h → hg.An arrow (ξ 0 , ϕ g 1 ) → (ξ 0 , ϕ g 2 ) is given by an automorphism ϕ g : ξ 0 → ξ 0 such that ϕ g 1 = ϕ −1 g • ϕ g 2 • ϕ g .From this description, we see that where the quotient to the right is with respect to the action by conjugation.
We use this description to define certain open and closed loci in B bal 0,N (G), as follows.
Definition 2.9.Let c = (c i ) i∈N be a sequence of conjugacy classes of G, i.e., c ∈ I µ (BG) N .
Let B bal 0,c (G) be the full sub-category of B bal 0,N (G) whose objects are the twisted G-covers As explained in [3] at the end of Section 8.5, this last condition is locally constant on T , hence B bal 0,c (G) is an open and closed sub-stack of B bal 0,N (G).
3 The category of S-modules over a base In order to study the class of B bal 0,n (G) in K 0 (AlgSt k ) (actually, in a slightly different ring that we will define later), we need to translate the combinatorics of its boundary into relations within this ring.As usual the boundary of B bal 0,n (G) corresponds to twisted G-covers (f : C → BG, Σ) such that C is reducible, say C = C ′ ∪C ′′ with C ′ and C ′′ being union of irreducible components of C. The restriction of f to C ′ and C ′′ yields twisted G-covers (f ′ : C ′ → BG, Σ ′ ) and (f ′′ : C ′′ → BG, Σ ′′ ) with fewer marked points.So it is expected that the class of the boundary of B bal 0,n (G) can be related to the classes of B bal 0,k (G) with k < n.This relation is the content of our main results, Proposition 6.3 and Theorem 6.4.In this section, we develop a formalism that is needed when we express f as the gluing of f ′ and f ′′ .One complication arises here because the marked points Σ 1 , . . ., Σ n are "stacky points", which contain the information of the local monodromies of the G-cover P → C. Furthermore, we shall also take into account the stacks B bal 0,n (G), with the action of the symmetric group S n , altogether as n ≥ 3 (this is, Example 3.14).
An approach towards this direction can be recovered from the theory of colored operads, where many of the tools that we use originated.The connection with this theory has been already highlighted in [43].Here we adopt the point of view of [36] according to which an operad is a monoid for a certain tensor product • (the so called "composition", which in our context is defined in Definition 3.7) in a particular category of functors.However, we can not use directly the results therein for two reasons: firstly, the category of stacks is a 2-category, hence we need to work in the context of enriched categories; secondly, as mentioned above, the gluing maps are more complicated in our setting due to the local monodromies.Let us recall the main definitions and results, which we adapt for our purposes.Since these results are effortlessly reworked versions of well-known facts in the non-enriched case, we do not present complete proofs, but we rather indicate which classical results they are adapted from.
We consider categories enriched over a fixed cosmos (V , ⊗, 1), i.e., a symmetric monoidal closed category which is bicomplete.For simplicity, we assume that the unit 1 is also the terminal object of V .The initial object of V is denoted by ∅.By Mac Lane's coherence theorem [38, Section VII.2] (cf.also [48, Theorem 8.4.1.]),we may also assume that (V , ⊗, 1) is a strict monoidal category, i.e., the associator and the unitors are identity morphisms.Unless otherwise stated, all categorical constructions (for instance, coends and Kan extensions) will be carried out in the V -enriched setting.Such level of generality is needed because St k and AlgSt k are 2-categories, i.e., they are enriched over the cosmos (Cat, ×, 1) of small categories.A reference for all the tools of enriched category theory that we will use is [35].
Let (E, ⋄, I) be a bicomplete symmetric monoidal closed V -category and B an object of E. Our guiding example is given by where I µ (BG) denotes the stack of cyclotomic gerbes in BG.
In the following S denotes the free V -category on the groupoid n∈N S n and S B : S → E is the functor that sends n → B n and σ ∈ S n to the morphism that permutes the factors, where B n denotes B ⋄n .Definition 3.1.An S-module is a functor X : S → E, that is a family of objects X n of E, n ∈ N, endowed with an action of S n .
A morphism of S-modules X and Y is a natural transformation f : X ⇒ Y , i.e., a family of morphisms f n : X n → Y n compatible with the S n -actions.
For an S-module X, we will sometimes say that X n is the part of X in degree n.The category of S-modules will be denoted by (S − mod).
An S-module over B is a pair (X, e), where X : S → E is an S-module and e : The category of S-modules over B will be denoted by (S − mod/B).
When (E, ⋄, open) immersion, and the S n -action on Y restricts to that on X, for every n.
The inclusion of S n × S m into S n+m as Aut{1, . . ., n} × Aut{n + 1, . . ., n + m} induces a tensor product + for a V -enriched symmetric monoidal structure on S with unit 0, denoted + : S ⊗ S → S.
Given two S-modules X and Y , we consider the functor where ⊙ denotes the tensor product (also known as copower) in the V -enriched category E. In the next definition we refer to [38] for the definition and properties of coends.

Definition 3.2 (Day convolution
).For all S-modules X and Y , the convolution X * Y is the functor S → E defined by the coend of (3.1): 1.There is a V -natural isomorphism where Lan + denotes the left Kan extension along + : 2. For S-modules X and Y , and for k ∈ N, the part of X * Y in degree k has the following explicit formula (cf.[36, formula (2.2)]): where Sh(m, n) is the set of (m, n)-shuffles in S k , i.e., the permutations σ ∈ S k such that Proof .We identify X * Y with the left Kan extension Lan , which we denote with the same symbol e ⋄ ε.
Proof .The associativity of the convolution is proved in [37, Proposition 6.2.1] (see [17,Section 3] for the original proof).To see that (e ⋄ ε) Then the claim follows from the properties of the left Kan extension as in the proof of the previous proposition.■ In particular, for a fixed functor Y : S → E, Day convolution determines a V -functor S⊗S op → E given by (m, n) → Y * n (m).This allows us to define the composition of an S-module over B and a rooted S-module over B. Definition 3.6.A rooted S-module is a pair (W, ε), where W is an S-module, ε : W ⇒ ∆ B is a natural transformation and ∆ B is the constant functor on B. A rooted S-module over B is a triple (W, ε, ε) such that (W, ε) is an S-module over B and (W, ε) is a rooted S-module.
Let (X, e) be an S-module over B, and let (W, ε, ε) be a rooted S-module over B. Let us consider the functor where the morphism W * n (m) → B n is the composition of the map induced by the projection followed by the morphism induced by ε 7 (composition).Given an S-module over B, (X, e), and a rooted S-module over B, (W, ε, ε), the composition product (or plethysm) X • W is the V -functor S → E defined by the coend of (3.3): Proposition 3.8.Let (X, e, ẽ) and (W, ε, ε) be rooted S-modules over B. Then ε induces, on X • W , a structure of S-module over B, which will be denoted Proof .Let F be the functor defined in (3.3).For any n ∈ N, ẽ gives a natural transformation α n : F (n, n) ⇒ ∆ B such that the following diagram commutes for every σ ∈ S n : By the properties of the coend, this gives a natural transformation The definition of ε ′ is similar, it uses the natural transformation F (n, n) ⇒ S B induced by ε as in Proposition 3.4.■ Definition 3.9.Let (Y, e) be an S-module over B and let us suppose that B has an involution ι.
We define (DY, ε, ε) to be the rooted S-module over B such that DY n := Y n+1 with the action of S n seen as the permutations of S n+1 that fix n + 1, ε n : DY n → B n is given by the first n components of e n+1 and εn : DY n → B denotes the last component of e n+1 composed with ι.
Let B as in the previous definition with an involution ι.By I 1 , we denote the rooted S-module over B concentrated in degree one where For later use, we define I 2 to be the S-module over B concentrated in degree two where e 2 = (Id B , ι) and the symmetric group S 2 acts via the involution ι.Notice that From now on, we restrict our attention to the case where (V , ⊗, 1) = (Cat, ×, 1) and (E, Proposition 3.10.The V -category of rooted S-modules over B, together with the composition product • and the unit I 1 , forms a monoidal V -category.
To prove the proposition, we follow Kelly's arguments in [36,Section 3].The next Lemma is needed.
where in the second and sixth isomorphisms we have used the fact that for R i , S i be B k i -modules; in the third isomorphism we have used the definition of * and the property that coends commute with fiber products in the case of stacks; the fifth isomorphism follows from Yoneda's lemma and the last one follows from the definition of Day convolution.■ Proof of Proposition 3.10.Let (X, e, ẽ), (Y, ε, ε), (Z, e, ẽ) be rooted S-modules over B. Then As in the proof of Proposition 3.8, the structure of rooted S-module on X • (Y • Z) is given by the natural transformations induced by ẽ.On the other hand, the structure of rooted S-module on (X • Y ) • Z is given by the natural transformations induced by the structure of rooted S-module on X • Y (which in turn is induced by ẽ).We see, from the explicit isomorphism X • (Y • Z) ∼ = (X • Y ) • Z defined above, that these two structures of rooted S-modules coincide.
Concerning the structures of S-modules over B, the one on (X • Y ) • Z is given by the natural transformations induced by e, and the one on X • (Y • Z) is given by the natural transformations induced by the structure of S-module over B of Y • Z (which in turn is induced by e).We see that, under the isomorphism • Z defined at the beginning of the proof, these two structures of rooted S-modules coincide.
The fact that I 1 is a unit for • follows from the definition (3.3) of the functor F whose coend is X • I 1 and from the following equalities: As n varies, the stacks B bal 0,n (G), with the evaluation maps ev n : B bal 0,n (G) → B n and the S nactions given by permutation of the marked points on B bal 0,n (G) (respectively by permutation on B n ), yield an S-module over B in AlgSt k (cf.Definition 3.1), which we denote by (B, e) and we refer to it as the S-module of balanced twisted G-covers over genus 0 curves.Moreover, let B sm 0,n (G) be the locus in B bal 0,n (G) of twisted G-covers of smooth n-pointed curves of genus 0 (cf.Notation 2.8), and let ev n| : B sm 0,n (G) → B n be the restriction of the evaluation map.The S n -action on B bal 0,n (G) restricts to an action on B sm 0,n (G) such that ev n| is equivariant.In this way we obtain an open sub-S-module of (B, e) that will be denoted by (B, e).
In order to explain better the statement of our main theorem (Theorem 6.4), we spell out the details of the previous constructions in the particular cases that will be relevant for equation (6.7).
First of all, let us consider the rooted S-module over B, (I 1 DB, ε, ε).By Definition 3.9, (DB) n = B n+1 with S n acting by permutation only on the first n marked points, ε n : (DB) n → B n is given by the first n evaluation maps, and εn : (DB) n → B is ι • ev n+1 n+1 , whereas ε 1 = ε1 = Id B .Then, using the concrete expressions (3.2) and (3.3) and the definition of coend, we deduce that where, for a multi i=1 (I 1 DB) k i and the morphisms Sh(k) × (I 1 DB) k → B m are induced by the ε's.Notice that an object in the right-hand side of the above equality is given by the equivalence class (under the action of S m ) of a stable map in B m , an object in (I 1 ) k i B k i +1 for i = 1, . . ., m and a shuffle permutation, such that, for any i = 1, . . ., m, the evaluation of the stable map in B m at the i-th marked point coincides with the image under εk i of the object in (I 1 ) Geometrically this means that, if k i ≥ 2, we glue together the first stable map with that in B k i +1 , for i = 1, . . ., m, and get a balanced stable map.For more details on this, we refer to Section 6.
Concerning the operation I 2 • DB, that we will need later, we have that In particular, its objects are equivalence classes of pairs of stable maps in B k 1 +1 × B k 2 +1 and a shuffle in Sh(k 1 , k 2 ), such that the evaluations of the two stable maps at the last marked points are related by the involution ι.This means, geometrically, that we can glue together the two stable maps along the last marked points to get a balanced map.
The last operation that we will use is I 2 × B 2 DB * DB , this is the functor S → E which associates to n the following object: 4 Grothendieck groups and their composition structure Let us recall the definition of K 0 (AlgSt k ) from [22].Note that there are different versions in the literature (see [10,33,46]), all of which are mapped to by K 0 (AlgSt k ).
Definition 4.1.The Grothendieck group K 0 (AlgSt k ) is the abelian group generated by the isomorphism classes {X } of algebraic k-stacks subject to the relations (2) {E} = {A r × k X } if π : E → X is an S n -equivariant vector bundle of rank r on X , where the S n -action on A r × k X is the extension of the given one on X by the trivial action.
For our purposes we need to define the Grothendieck group associated to the category of S-modules over B, where (E, ⋄, I) = (AlgSt k , × k , Spec k) (we change the target category according to Remark 3.13).First of all, let us recall (from Definition 3.1) that a morphism F : (X, e) → (X ′ , e ′ ) (respectively an isomorphism) of S-modules over B is a natural transformation F : X ⇒ X ′ (respectively a natural isomorphism) such that the diagram is commutative.As usual we say that two S-modules over B, (X, e) and (X ′ , e ′ ), are isomorphic if there exists an isomorphism F : (X, e) → (X ′ , e ′ ) and we denote with {X, e} the isomorphism class of (X, e).By a vector bundle in the category of S-modules over B we mean a morphism π : (E, e ′ ) → (X, e) such that: for every n, π n : E n → X n is a vector bundle (of some rank r(n)) and it is S n -equivariant.
) is a vector bundle, where the S n -action on A r(n) × k X n is the extension of the given one on X n by the trivial action.
The Grothendieck group K S,r 0 (AlgSt k /B) associated to the category of rooted S-modules over B is defined similarly and we omit the details.
Notice that K S 0 (AlgSt k /B) and K S 0 (AlgSt k ) (with the products given by Day convolution) have natural structures of algebras over K 0 (AlgSt k ) induced by cartesian product.Notation 4.4.In the following, we will consider also S-modules over Spec(k), X, where the natural transformation X ⇒ S Spec(k) is the one induced by the structure morphisms X n → Spec(k), for all n, and hence we omit it.The corresponding Grothendieck group K S 0 (AlgSt k / Spec(k)) will be denoted K S 0 (AlgSt k ).For any k-stack B, there is a morphism of K 0 (AlgSt k )-algebras ) that is induced by associating any S-modules over B, (X, e), with X.We will say that two elements a, b ∈ K S 0 (AlgSt k /B) are congruent modulo B, in symbol a ≡ B b, if they have the same image under this morphism.
Let (X, e) be an S-module over B, and let (W, ε, ε) be a rooted S-module over B such that W 0 = ∅.We observe that the class {X • W, ε ′ } of the composition product (see Definition 3.7 and Remark 3.13) depends only on the classes {X, e} ∈ K S 0 (AlgSt k /B) and {W, ε, ε} ∈ K S,r 0 (AlgSt k /B).Furthermore, we have the following result, which is part of a theorem announced in [27] in the case of varieties (for a complete proof see [7,Proposition 1.3.6]).Proposition 4.5.Let K S,r 0 (AlgSt k /B) 1 be the sub-group generated by the classes {W, ε, ε} of rooted S-modules over B with W 0 = ∅.Then the composition product descends to an operation Proof .We follow the steps in [7,Proposition 1.3.6].By linearity we only need to define {X, e}•b for every S-modules over B, (X, e), and every b ∈ K S,r 0 (AlgSt k /B) 1 .First write b as {Y, ε, ε} − {Z, ε ′ , ε′ } where (Y, ε, ε), (Z, ε ′ , ε′ ) are rooted S-modules over B.Moreover, assume that (Z, ε ′ , ε′ ) is a closed sub-S-modules over B of (Y, ε, ε).Let U := Y \ Z, and let ε | , ε| the restrictions of ε, ε to U .Then we have Using the formalism of coend, we have Therefore, in this case, expanding where X i+j → B j is the composition of e i+j : X i+j → B i+j with the projection B i+j → B j to the last j-components, while the first i components of e i+j give the map to B i .Note that, for any k ∈ N, ({X, e} • b) k depends on (Y \ Z) * j n only for n < k.In particular, we have In these equations (and from now on in the proof), we have omitted the morphisms to B to simplify the notation, they are those induced by ε, ε and ε ′ , ε′ from Proposition 3.8.This suggests a recursive approach for defining {X, e} • b when b = {Y, ε, ε} − {Z, ε ′ , ε′ } and {Z, ε ′ , ε′ } is not a closed sub-S-modules over B of (Y, ε, ε).
Let us consider functors T : S × S → AlgSt k that associate, for any i, j ∈ N, a stack T (i, j) together with an S i × S j -equivariant morphism T (i, j) → B i × B j .For any such T , let us define Using this we define {X, e} Finally, we define {X, e}

Stratification
In this section, we introduce a stratification of B bal 0,n (G) according to the combinatorial type of its objects.This will be achieved using the notion of gerby tree associated to BG.
Let us first recall the following definitions.

Definition 5.1 ([11]
).A graph τ is a triple (F τ , j τ , R τ ), where F τ is a finite set, called the set of flags of τ , j τ : F τ → F τ is an involution, and Associated to every graph τ = (F τ , j τ , R τ ), there is a triple of sets (V τ , E τ , L τ ), which are defined as follows: V τ is the quotient of F τ by R τ , it is called the set of vertices of τ ; E τ is the set of orbits of j τ with two elements, it is called the set of edges of τ ; L τ is the set of fixed points of j τ , it is called the set of leaves of τ .
For each v ∈ V τ , identify the elements of the set {0 i | i ∈ F τ (v)} with v.For each edge {i 1 , i 2 } ∈ E τ , identify 1/2 i 1 with 1/2 i 2 .In this way, one obtains a set |τ | together with a surjection T → |τ |.Then the topology on |τ | is the quotient topology.Definition 5.3.A tree is a graph τ such that its geometric realization is simply connected, i.e., |τ | is connected and #V τ = #E τ +1.An N -tree is a tree τ together with a bijection l τ : L τ → N .A (N -)tree τ is stable if n(v) ≥ 3, for any v ∈ V τ .As usual, when N = {1, . . ., n} we will speak of n-trees instead of N -trees.
Let Īµ (BG) be the stack of cyclotomic gerbes in BG, let Īµ (BG) be its coarse moduli space (cf.Section 2.1), and let ι : I µ (BG) → I µ (BG) be the involution induced by the automorphisms µ r → µ r , ζ → ζ −1 , for r ≥ 1 (cf.Example 3.14).In order to simplify the notation, we will use the same symbol ι : Īµ (BG) → Īµ (BG) for the morphism induced by ι on the coarse moduli space.The following definition is similar to [4, Definition 2.14].Definition 5.4.A gerby tree τ associated to BG is given by a tree τ = (F τ , j τ , R τ ) together with a map m : F τ → Īµ (BG) such that m(i) = ι • m(i ′ ), whenever the flags i, i ′ ∈ F τ form an edge {i, i ′ } ∈ E τ .We will denote with o : F τ → Z the function that associates to any flag i ∈ F τ the integer r ∈ Z such that m(i) ∈ Īµ r (BG).A gerby N -tree is a gerby tree τ together with a bijection l τ : L τ → N .A gerby (N -)tree τ is stable, if the underlying (N -)tree is stable.
In the following, with a slight abuse of notation, for a gebry tree τ , we denote either with V τ (resp.E τ , L τ ) or with V τ (resp.E τ , L τ ) the set of vertices (resp.edges, leaves) of the underlying tree τ .Furthermore, we denote with E ′ τ (resp.E ′ τ ) the set F τ \ L τ .
In this article, we choose a representative for any equivalence class of stable gerby n-trees associated to BG and we denote with Γ0,n the set of these representatives.
We recall the following result, for which we provide a proof (following [27]) for lack of a suitable reference.
Lemma 5.6.The set Γ0,n of isomorphism classes of stable gerby n-trees associated to BG is finite.
Proof .Let τ ∈ Γ0,n .By definition, |τ | is connected and The right-hand side of the previous equation is equal to The claim follows since the isomorphism classes of gerby n-trees associated to BG, with a fixed number of flags, is finite.■ Given a gerby tree τ , in the literature there are notions of τ -marked pre-stable curves and of τ -marked twisted stable maps to BG (see, e.g., [4,Definitions 2.18 and 2.20]).In this article, we give a different definition of τ -marked twisted stable maps to BG, which is more convenient for our purposes.To this aim, for any v ∈ V τ let m(v) := (m(i)) i∈F τ (v) ∈ Īµ (BG) F τ (v) .Then consider the open and closed sub-stack Furthermore, let us define Definition 5.7.Let τ be a gerby tree associated to BG.The stack of τ -marked twisted G-covers of genus 0, which will be denoted B τ (G), is defined as the fiber product with respect to the inclusion I µ (BG) (τ ) → I µ (BG) E ′ τ and the morphism ev τ ,E .
Furthermore, we define B sm τ (G) to be the fiber product where is the open sub-stack consisting of twisted G-covers of smooth curves (see Notation 2.8).
Remark 5.8.By definition, an object of B τ (G) over a scheme T is a triple It follows from this remark that, for any stable gerby n-tree associated to BG, τ , we can apply the gluing construction of [2, Section 5 and Appendix A] that yields a morphism (5.1) Proposition 5.9.The morphism ξ τ in (5.1) is representable and proper.
Proof .To simplify the notation, in this proof we denote with e i the evaluation morphism ev i N , whenever the set N is clear from the context.
Let us choose an ordering of the edges of τ , In particular there is an ℓ such that either the vertex [i ℓ ] or [i ′ ℓ ] belongs only to the edge {i ℓ , i ′ ℓ } and we suppose, without loss of generality, that [i ′ k ] satisfies this condition.Let us proceed by induction on k ≥ 1 (if k = 0, ξ τ is the identity).For k = 1, we have that , where ěi 1 := ι • e i 1 , and ξ τ is the morphism in [2, Proposition 5.2.1 (1)].Therefore, ξ τ is representable by the aforementioned proposition.Finally, ξ τ is proper since the stacks B bal 0,n (G) are proper [3,Theorem 1.4.1], in particular, they are separated and also B τ (G) is proper.
Let us suppose that the result is true for gerby trees with k − 1 edges and let τ be a gerby n-tree with #E τ = k.Let τ1 be the gerby tree with Furthermore, let τ2 be the gerby tree with where ěi k := ι•e i k , and under this identification ξ τ is the composition of the following morphisms: where the morphism to the right is the gluing one of [2, Proposition 5.2.1 (1)].■ Furthermore, we have the following result.
To prove the previous proposition, we need a preliminary result.
Proof .We proceed by induction on the number #E τ of edges.If #E τ = 0 the claim is obvious.So, let us assume that #E τ = k + 1 and that the result holds true for trees with k edges.Let us fix a numbering of the edges, E τ = {{i 1 , i ′ 1 }, . . ., {i k+1 , i ′ k+1 }} as in the proof of Proposition 5.9, that is such that j τ (i ℓ ) = i ′ ℓ , for ℓ = 1, . . ., k + 1, and [i ′ k+1 ] belongs only to the edge {i k+1 , i ′ k+1 }.Notice that, since τ is stable, (in other words φ is the identity on [i ′ k+1 ]).Let now τ 1 be the tree defined as in the proof of Proposition 5.9, with ), and let l τ 1 be a n 1 -marking, where n 1 := #L τ 1 .Notice that (τ 1 , l τ 1 ) is a stable n 1 -tree and that the restriction of φ to F τ 1 is an automorphism of (τ 1 , l τ 1 ).By induction we conclude that φ = Id.■ Proof of Proposition 5.10.The claim follows essentially by the same proofs of Proposition 10.11 and Corollary 10.22 in Chapter XII of [5].The main ingredients in these proofs are: the existence of Kuranishi families (see [5, Chapter XVI, Section 9]), and the fact that the automorphism group of any stable gerby n-tree (τ , l τ ) is trivial (Lemma 5.11).■ The image of B sm τ (G) under ξ τ is the locus of (balanced) twisted G-covers of combinatorial type τ , which we denote B bal (G, τ ).Then we have the following corollary of Proposition 5.10.

Recursive relations
In this section, we prove our main result, Theorem 6.4, which yields recursive formulae that express each class {B bal 0,n (G)} ∈ K 0 (AlgSt k ) in terms of the classes of the open loci in B bal 0,m (G), for m ≤ n, corresponding to stable maps with smooth domain.To this aim we need some preliminary definitions and results.First of all, following [8, Definition 4.2], let us give the following definition.Definition 6.1.Let X be an algebraic k-stack.A locally closed decomposition of X is a morphism f : Y → X of algebraic k-stacks that satisfies the following conditions: the restriction of f to each connected component of Y is a locally closed immersion; f is bijective on closed points.
Let us recall that a locally closed immersion is a morphism that can be factored as j • i, where i is a closed immersion and j is an open immersion.
In the following, we will use the next result, which is well known, but we provide an elementary proof for completeness.Lemma 6.2.Let f : Y → X be a locally closed decomposition.Let Y 1 , . . ., Y m be the connected components of Y. Then the following equality holds true in K 0 (AlgSt k ): We proceed by induction on m.Let us assume m > 1, since the case m = 1 is trivial.Now, for f (Y m ) being the closure of f (Y m ), using the properties of K 0 (AlgSt k ) and the fact that f (Y m ) is open in f (Y m ), we have the following equalities: which are locally closed decompositions of the corresponding right-hand sides.By induction, we deduce that For any integer n ≥ 3, let Γ0,n be a set of representatives of the isomorphism classes of stable gerby n-trees associated to BG.Let (B, e) and (B, e) be the S-modules of Example 3.14.Then the following equalities hold true in K Sn 0 (AlgSt k ): Proof .To prove (6.1), we show that there is a locally closed decomposition where so that the claim then follows since (by (3.4) in Definition 3.7) the class in K 0 (AlgSt k ) of the target of (6.4) is the right-hand side of (6.1).Notice that for any n the set Γ0,n is finite and there are only non-zero contributions for 3 ≤ m ≤ n in (6.4).Let τ ∈ Γ0,n , let w ∈ V τ and let m = |F τ (w)|.We first construct a morphism where, with a slight abuse of notation, the S m in the left-hand side denotes the set of bijections {1, . . ., m} → F τ (w).To this aim, let θ ∈ S m be such a bijection and let ) with the projection to the factor indexed by w, we obtain a morphism B sm τ (G) → B sm 0,F τ (w) (G), which gives, via θ, the first component B sm τ (G) → B m of (6.5).The second component of (6.5) is defined as follows.Let τw be the graph (F τ \F τ (w), j τw , R τw ), where and R τw is the restriction of R τ to F τ \ F τ (w).Notice that the geometric realization in particular, the connected components of |τ w | are in bijection with the flags in F τ (w) \ L τ .For any i = 1, . . ., m such that f i ∈ F τ (w) \ L τ , let τ i w the sub-graph of τw that contains j τ (f i ) and whose geometric realization is a connected component of |τ w |.For any i = 1, . . ., m, let The marking l τ : L τ ↔ {1, . . ., n} (recall that τ is an n-tree) induces a unique marking of the leaves of τ i w such that j τ (f i ) is marked with k i + 1 and the other leaves are marked from 1 to k i preserving the order determined by l τ .This data determines a (k 1 , . . ., k m )-shuffle σ in the following way: setting k 0 = 0, for any i = 1, . . ., m, if are the indices of the first k i leaves of τ i w (i.e., those different from j τ (f i )) with respect to the marking l τ ; if Finally, for any i = 1, . . ., m such that k i > 1, the subgraph τ i w is stable, hence we have a morphism , which is the composition of the natural morphism ) with the projection to the product of factors corresponding to the vertices v ∈ V τ i w .Then the gluing construction of [2, Section 5 and Appendix A], together with the enumeration of the leaves of τ i w as described before, yields a morphisms B sm τ (G) → (DB) k i , where the indices i are such that k i > 1.In the remaining cases where k i = 1 the composition of the first component B sm τ (G) → B m in (6.5) with the evaluation ev i : B m → B yields a map B sm τ (G) → (I 1 ) 1 .In this way we define the third component of Φ w in (6.5).
By construction Φ w is S m -equivariant, hence it induces a morphism which canonically induces the morphism (6.4).
In order to prove that (6.4) is a locally closed decomposition, it suffices to show that it has the following two properties: it is bijective on closed points; (6.5) is a locally closed immersion (because the quotient morphism by a finite group is open and closed).We first prove the second property.To this aim let θ ∈ S m and let k be the restriction of Φ w , where we have adopted the same notation as before, so that k refers to the shuffle determined by θ.In the next discussion w and θ are fixed, so we omit them.Furthermore, we view where pr denotes the projection to the corresponding factor.The claim now follows from Proposition 5.10.We now prove that (6.4) is bijective on closed points.Notice that, since (6.5) is a locally closed immersion, (6.4) is injective on closed points.To see that (6.4) is surjective on closed points, notice that any object in the target can be obtained as the glueing of twisted G-covers over smooth basis.
The proof of (6.2) and (6.3) are similar to that of (6.1), so we sketch them.The right-hand side of (6.2) is the class of while the right-hand side of (6.3) is the class of where we do not pass to the quotient by S 2 .
The choice of a flag f ∈ F τ \ L τ corresponds to the oriented edge − → e = (f, j τ (f )) of τ .For any such oriented edge − → e , if we cut |τ | at the point corresponding to 1/2 f , we obtain two subtrees τ 1 and τ 2 of τ , where f ∈ F τ 1 and j τ (f ) ∈ F τ 2 .Let k 1 + 1 (resp.k 2 + 1) be the number of leaves of τ 1 (resp.τ 2 ), then k 1 + k 2 = n.As before this decomposition of τ determines a shuffle in Sh(k) and gluing the various irreducible components of the objects in B sm τ (G) according to τ 1 and τ 2 yields a morphism In this way, we obtain a locally closed decomposition that proves (6.3).At the same time, exchanging the orientation of − → e corresponds to the natural S 2 -action on the target of (6.6), so we also have a locally closed decomposition that proves (6.2). ■ Proof .Since the relation (6.7) is an equality in K S 0 (AlgSt k ), it suffices to prove that B n ∈ K Sn 0 (AlgSt k ) coincides with the part in degree n of the right-hand side, for any n ∈ N. To this aim, let Γ0,n be a set of representatives of the isomorphism classes of stable gerby n-trees associated to BG.Since B n is the disjoint union of the locally closed substacks B bal (G, τ ) (Corollary 5.12), for τ ∈ Γ0,n , we obtain the following equalities in K 0 (AlgSt k ): where, in the second equality, we have used the fact that B bal (G, τ ) and B sm τ (G) are isomorphic via the morphism ξ τ defined in the previous section.
On the one hand, (6.8) yields on the other hand, identifying * ⊔ F τ with V τ ⊔ E τ ⊔ L τ via (6.9), we obtain the next equality: As F τ = (F τ \ L τ ) ⊔ L τ , the previous equality reduces to The theorem now follows directly from Proposition 6.3.■ Remark 6.5.For any n ≥ 3, the part of the right-hand side of (6.7) in degree n expresses B n ∈ K Sn 0 (AlgSt k ) in terms of I 1 , I 2 , B 3 , . . ., B n−1 , B n , considered as stacks over B (as we explain below).Therefore, by induction, the class B n can be computed knowing I 1 , I 2 , B 3 , . . ., This follows from the expressions below for For the first one, using (3.5), we have Notice that m ≥ 3 (otherwise B m = ∅) and, for the multi-index k = (k 1 , . . ., k m ), k 1 , . . ., k m ≥ 1 (otherwise I 1 DB k i = ∅).Now, the condition |k| = m i=1 k i = n implies, on one hand that on the other hand that, for every j = 1, . . ., m, For the second one, are Deligne-Mumford stacks and we consider their cohomology with rational coefficients, we can replace them with their coarse moduli spaces (cf.[9]), which are respectively projective and quasi-projective varieties, and will be denoted with the same symbols, B n and B n , in this section.Following [27] (see also [8]), let K Sn 0 (Var) be the Grothendieck group of complex S n -varieties.It is the quotient of the free abelian group on the isomorphism classes [X] of quasi-projective varieties with an action of S n by the subgroup generated by the relations The notions of S-module, S-module over B, and rooted S-module makes sense for varieties.Notice that in this case an S-module is called an S-variety in [27] and [8].The Grothendieck groups K S 0 (Var), K S 0 (Var/B) and K S,r 0 (Var/B) are defined accordingly.Moreover, we can consider the Day convolution and the composition in this case, as a consequence of Proposition 4.5, we have an operation which is K S 0 (Var/B)-linear in the first argument and such that From the results of the previous section we have the following relation in K S 0 (Var).For computations, we need to express the three summands in the right-hand side of this equation in a more explicit way.The part in degree n of the first summand is For any m = 3, . . ., n and for any sequence c = (c 1 , . . ., c m ) ∈ I µ (BG) m of conjugacy classes of G, let B c = B bal 0,c (G) ∩ B m (cf.Definition 2.9), let us denote B bal 0,c (G) with B c , and let us stress that in this section we denote with the same symbols stacks and their coarse moduli spaces.Then, using this notation, we rewrite the right-hand side of (7.2) as follows: where B k i ,c i ⊆ B k i +1 consists of covers such that the evaluation at the last marked point is c i , and ({c i }) k i = Spec(C) if k i = 1, empty otherwise.Using the same notation, for the second summand of (7.1), we rewrite (6.10) as follows: Finally, the third summand is Let us now consider the category MHS of mixed Hodge structures over Q, for which our basic references are [42] and [47] (it might be useful to consult also [27,Section 5] and [8,Section 2.2] for the purposes of the present work).Since MHS is a Q-linear rigid abelian tensor category, we can consider the associated Grothendieck group K 0 (MHS), which is isomorphic to the Grothendieck group of pure Hodge structures over Q (see, e.g., [42,Corollary 3.9]).Moreover, for any integer n ≥ 0, we denote with K Sn 0 (MHS) (respectively, with K S 0 (MHS)) the Grothendieck group of the category of functors from S n (respectively, from S) to MHS.The Hodge-Grothendieck characteristic (called the Serre characteristic in [27], and denoted HG-characteristic hereafter) yields a group homomorphism e : K Sn 0 (Var) → K Sn 0 (MHS), for any n, where e is defined on generators by • is the natural mixed Hodge structure of the compactly supported cohomology group H i c (X, Q) of the variety X, as defined in [19], [20].Actually, K S 0 (Var) and K S 0 (MHS) have a richer structure, namely that of complete composition algebra [27, Theorems 2.2 and 5.1], and e : K S 0 (Var) → K S 0 (MHS) is a morphism of complete filtered algebras with composition operations [27, Section 5].In particular, e : K 0 (Var) → K 0 (MHS) is a ring homomorphism (where the ring structures are induced by the cartesian product on K 0 (Var) and by the tensor product on K 0 (MHS)).However, notice that the composition operation of [27] is different from ours, so we can not apply their results directly.
For completeness we recall that, for every mixed Hodge structure (V, W • , F • ) over Q, its Hodge numbers are defined as and its Hodge-Euler polynomial is This induces a ring homomorphism see, e.g., [42,Examples 3.2].In particular, for any variety X, its Betti numbers are encoded in e(e([X])).
We will need the following result, which we prove in Section 7.1 (for the case where X is smooth, see, e.g., [24,Proposition 4.3]).Proposition 7.1.Let X be a quasi-projective variety with an action of a finite group G. Then where Applying e to (7.1), we obtain Using (7.3), (7.4) and (7.5), we rewrite the part in degree n of the right-hand side of the previous equation as follows: The following result holds true over any field of characteristic 0 (see [28,Theorem 4] and [30]).
Let us recall for later use that h([X]) coincides with the class in K 0 (Hot b (M)) of the weight complex W (X) of X [28, Sections 2.1 and 3.2].Hot b (M) denotes the category of bounded chain complexes in M up to homotopy, and one uses the identification of K 0 (Hot b (M)) with K 0 (M) to get a class in K 0 (M).
The cohomology of a motive is defined as follows.For any object M = (X, p) of M where p * denotes the action induced on cohomology by p, p * (α) := (pr 2 ) * (p • pr * 1 (α)), where pr i : X × X → X is the projection to the i-th component.Since p * is a morphism in MHS, H i (M ) is an object of MHS, and following [41, we define e M (M ) := i≥0 (−1) i H i (M ) ∈ K 0 (MHS).
Notice that e M is additive on the objects of M, therefore it induces a group homomorphism e M : K 0 (M) → K 0 (MHS).
It is proved in [41, Appendix A] that the HG-characteristic lifts to motives, i.e., e = e M • h.
It follows from this that e([X/G]) = e M h([X/G]) .
Corollary 5.3 in [18] implies that there is an isomorphism of weight complexes, W (X/G) ∼ = W (X) G , where W (X) G = W (X), 1 Let us now apply the functor H n to W (X/G).We obtain a complex of Q-vector spaces whose i-th cohomology group is denoted R i H n (W (X/G)), and it coincides with gr W n H i+n c (X/G, Q) [28, Section 3.1].On the other hand, where we have used the fact that H n (X, p) = Im(p * : H n (X) → H n (X)) for any motive (X, p), by definition, and that taking G-invariants is exact for vector spaces over Q.
The claim follows because the Grothendieck group of mixed Hodge structures is isomorphic to the one of pure Hodge structures (HS) via the homomorphism K 0 (MHS) → K 0 (HS) defined by sending [(V, W • , F • )] to i∈Z gr W i V [42, Corollary 3.9].
Notice that this action commutes with the G-action, hence it induces an action on HV(G, n)/G.One can use this action to describe the connected components of B sm 0,n (G).In particular, when G is abelian, by [12, Proposition 2.1], we have that π : B sm 0,n (G) → M 0,n is a trivial bundle, hence B sm 0,n (G) ∼ = M 0,n × HV(G, n) and the evaluation maps correspond to the projections HV(G, n) → G. Therefore, we obtain an expression of [B] ∈ K S 0 (Var/B) in terms of the classes [M 0,n ], for all n.Using the known results about the cohomology of M 0,n (e.g., [26,34,39]), one can determine the Betti numbers of B n , for every n, hence those of B n using the results of the previous section.

Definition 2 . 4 .
Let N be a finite set, and let T be a k-scheme.A twisted G-cover of an N -pointed curve of genus g over T consists of a commutative diagramΣ C BG C Spec(k)T such that the following conditions are satisfied:(i) (C → C → T, Σ)is a twisted nodal N -pointed curve over T ; (ii) for any i ∈ N and for |Σ i | being the image of Σ i under the morphism C → C, then (C → T, (|Σ i |) i∈N ) is a stable N -pointed curve of genus g;(iii) the morphism C → BG is representable.

Proposition 3 . 4 .
Let (X, e) and (Y, ε) be S-modules over B. Then e and ε induce a natural transformation e ⋄ ε : X * Y ⇒ S B , therefore (X * Y, e ⋄ ε) is an S-module over B.

The group K 0 (
AlgSt k ) has, in fact, a commutative ring structure, given on generators by {X }{Y} = {X × k Y}.Definition 4.2.The Grothendieck group K Sn 0 (AlgSt k ) is the abelian group generated by the isomorphism classes {X } of algebraic k-stacks with an action of S n subject to the relations (1) {X } = {Y} + {X ∖ Y} if Y is a closed S n -invariant substack of X with the restricted action, and

Finally, we associate
to every vertex v ∈ V τ (1) the subset F τ (v) ⊂ F τ of flags whose equivalence class under R τ is v, and (2) the valence n(v) := #F τ (v) of v. Definition 5.2.The geometric realization |τ | of a graph τ is the topological space constructed as follows.Let us consider the disjoint union of intervals and singletons by definition, and B bal (G, τ ) is open in the image of ξ τ because it is the complement of a closed locus, by Proposition 5.9.■ as an open substack (in Definition 5.7, we permute the factors to bring B sm 0,m(w) (G) in the first position to the left, and we embed B sm 0,m(w) (G) ⊂ B m ).Then the morphism Φ w,θ is the composition of this open inclusion followed by pr Bm × i :

Theorem 6 . 4 .
Let B, e be the S-module defined in Example 3.14 and let (B, e) be the Ssub-module given by the open locus corresponding to the stable maps with smooth domain (see Example 3.14).Then the following relation holds true:B ≡ B {B} • {I 1 } + D B + {I 2 } • D B − (I 2 ) 2 × B 2 DB * DB ,(6.7)whereB is I µ (BG) and both sides of the relation belong to K S 0 (AlgSt k /B) (see Notation 4.4).

c) S 2 −(− 1 )Example 7 . 2 .
c∈Iµ(BG) k∈N2  |k|=n e Sh(k) × B k 1 ,c × B k 2 ,ι(c) .Using Proposition 7.1, we deduce the following expressions for the three summands above: the class of the S m -invariant part of3≤m≤n c ) ⊗ H 0 (Sh(k)) ⊗ H t × m i=1 (({c i }) k i B k i ,ι(c i ) )for the first summand, where I µ = I µ (BG);H 0 (Sh(k)) ⊗ H s B k 1 ,c ⊗ H t B k 2 ,ι(c) s+t H 0 (Sh(k)) ⊗ H s B k 1 ,c ⊗ H t B k 2 ,ι(c)for the third one.If n = 3, B 3 = B 3 .As we will explain at the beginning of Section 7.2, B 3 is a finite set, which is in bijection (after the choice of a geometric basis of the fundamental groupπ 1 P 1 \ {0, 1, ∞}, p 0 ) with the set (g 1 , g 2 , g 3 ) ∈ G 3 | g 1 • g 2 • g 3 = 1 G,where G acts via simultaneous conjugation.theclass of the diagonal.Disjoint union and product of varieties can be extended to M, where they are denoted by ⊕ and ⊗, respectively.The Grothendieck group K 0 (M) is the quotient of the free abelian group on the isomorphism classes [M ] of objects M in M by the subgroup generated by elements of the form [M ] − [M ′ ] − [M ′′ ] whenever M ∼ = M ′ ⊕ M ′′ .The operation ⊗ induces a ring structure on K 0 (M).
|G| g∈G [g] is the image of W (X) under the projector1 |G| g∈G [g].Here W (X) has to be considered as an object of the category of functors from G to Hot b (M).

[γ 1
], . . ., [γ n ] generate the fundamental group π 1 (C \ {p 1 , . . ., p n }, p 0 ) and are subject only to the relation[γ 1 ] • • • [γ n ] = 1.Using this geometric basis, we can identify the set of morphisms Hom(π 1 (C \ {p 1 , . . ., p n }, p 0 ), G) with the set HV(G, n) of G-Hurwitz vectors in G of genus 0 (we use the same notation as in [15, Definition 2.1]), HV(G, n) := (g 1 , . . ., g n .10) Since DB k = ∅ for k < 2, we deduce as before that {I 2 }•D B n is determined by {I 2 } and by B k for k = 3, . . ., n−1.The same arguments apply to the summand (I 2 ) 2 × B 2 DB * DB 7 Computations in the Grothendieck group of mixed Hodge structures In this section, we work over the field of complex numbers C and derive formulae for the class, in the Grothendieck group, of the mixed Hodge structure over Q of B n .Since B n and B n