The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost Hopf rings

We describe a Hopf ring structure on the direct sum of the cohomology groups $\bigoplus_{n \geq 0} H^* \left( W_{B_n}; \mathbb{F}_2 \right)$ of the Coxeter groups of type $B_n$, and an almost-Hopf ring structure on the direct sum of the cohomology groups $\bigoplus_{n \geq 0} H^* \left( W_{D_n}; \mathbb{F}_2 \right)$ of the Coxeter groups of type $D_n$, with coefficient in the field with two elements $\mathbb{F}_2$. We give presentations with generators and relations, determine additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by De Concini and Salvetti and in terms of their restriction to elementary abelian 2-subgroups.

The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost-Hopf rings LORENZO GUERRA We describe a Hopf ring structure on the direct sum of the cohomology groups L n 0 H .W B n I F 2 / of the Coxeter groups of type W B n , and an almost-Hopf ring structure on the direct sum of the cohomology groups L n 0 H .W D n I F 2 / of the Coxeter groups of type W Dn , with coefficients in the field with two elements F 2 . We give presentations with generators and relations, determine additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by De Concini and Salvetti and their restriction to elementary abelian 2-subgroups.

Introduction
The Coxeter groups of type W B n and W D n are two infinite families of finite reflection groups. Coxeter groups are traditionally described via Coxeter diagrams, ie graphs in which each edge e has a weight m e 3. Given such an object, the associated Coxeter group has a generator s v for every vertex v, with relations of the form s 2 v D 1, .s v s w / m e D 1 for every edge e D .v; w/, and .s v s w / 2 D 1 if v and w are not connected by an edge. For an exhaustive introduction to the geometry and topology of these groups we refer to Davis's book [3]. The reflection groups of type W B n and W D n are the finite Coxeter groups associated with the Coxeter diagrams in Figure 1.
The goal of this paper is to provide an effective description of the mod 2 cohomology of these groups. Other authors have previously computed these cohomology groups. Most notably, Swenson, in his thesis [18], adapted techniques used by Hu'ng [12] and Feshbach [5], stemming from the analysis of the restriction maps to elementary abelian 2-subgroups, to compute generators and relations for the mod 2 cohomology algebra of a finite reflection group. However, his presentation is involved and intrinsically recursive. Borrowing ideas from Giusti, Salvatore and Sinha [7; 9], we exploit additional t 0 t 1 t 2 t 3 t n 2 t n 1 s 0 s 1 s 2 s n 2 s n 1 Figure 1: Diagrams of type D n (left) and B n (right).
structures to provide a simpler description of the cup product. Our approach also has the advantage of being more easily readable from the well-known chain-level geometric and combinatorial description of a resolution for Coxeter groups by De Concini and Salvetti [4].
The sequences of Coxeter groups of type B and D have standard embeddings that are, in a certain sense, compatible. The homomorphisms induced by these maps on mod 2 cohomology define a coproduct . The cohomology transfer maps associated with them determine a productˇ. There is also a canonical embedding of W D n into W B n as an index-2 subgroup, which induces an involution ÃW H .W D n I F 2 / ! H .W D n I F 2 /.
In the B case, the resulting structure is modeled on that of the symmetric groups, the Coxeter groups of type A, as described by Giusti, Salvatore and Sinha [7] (mod 2) and by the author [10] (modulo odd primes). Together with the usual cup product , these maps form a ring in the category of F 2 -coalgebras, ie a Hopf ring over F 2 . More explicitly, given a ring R, a (graded) Hopf ring over R is a graded R-module with a coproduct and two products,ˇand , such that .A; ;ˇ/ is a Hopf algebra, with an antipode S; .A; ; / is a bialgebras over R; In the D case, ,ˇand satisfy the last two axioms in the definition of a Hopf ring, and and form a bialgebra. However, as we will explain later, andˇdo not form a bialgebra. We call this weaker structure an almost-Hopf ring over F 2 . Due to this fact, the study of the cohomology of W D n , with the cup product, the transfer product, and the coproduct, is more complicated. The reader will find similarities between the cohomology of W D n and that of the alternating groups, as described by Giusti and Sinha [9]. Such structures stem from the seminal work of Strickland and Turner [17], in which the authors discovered a Hopf ring structure on the cohomology of symmetric groups, even with generalized cohomology theories.
The main results of this paper are Theorems 5.9 and 5.15, stated in Section 5.2, consisting of a presentation in terms of generators and relations of the mod 2 cohomology of the Coxeter groups of type B n as a Hopf ring and of type D n as an almost-Hopf ring respectively. We provide here self-contained statements for clarity and reference.
Theorem 5.9 (main theorem for type B) The Hopf ring L n 0 H .W B n I F 2 / over F 2 admits a presentation with two families of generators, k;n 2 H n.2 k 1/ .W B n2 k I F 2 / for k 0 and n > 0, and ı n 2 H n .W B n I F 2 / for n > 0, and the following relations: . k;m / D P i Cj Dm k;i˝ k;j ; k;nˇ k;m D nCm n k;nCm ; .ı n / D P kClDn ı k˝ıl ; ı nˇım D nCm n ı nCm ; the cup product, , of classes in different components is 0; 0;n is the -unit of H .W B n I F 2 /.
The generators are explicitly characterized, both combinatorially at the cochain level (see Definition 5.1) and geometrically, as suitable Thom classes (see Proposition 5.3). The classes k;n and the relations among them arise from the presentation of the mod 2 cohomology of the symmetric groups as a Hopf ring. The only new generators are ı n and their behavior is governed by the third and fourth relations above.
The almost-Hopf ring constructed from the cohomology rings of the Coxeter groups of type D is more complicated. The relations are intricate, and the behavior of generators is more easily understood with the aid of a "polarized" basis B C t B t B 0 (see Proposition 5.22). For instance, the bialgebra axiom forˇand is replaced with a different compatibility identity involving the projection p C onto the addend .Span.B C /˝A˝3 D /˚.Span.B 0 /˝Span.B C /˝A˝2 D /: .xˇy/ D .ˇ˝ˇ/ .p C /..x/˝.y// for all x; y; where is the transposition of the second and third factors. Nevertheless, this surrogate axiom can be expressed directly in terms of the generators, without explicit reference to the additive basis (see Proposition 5.14).
The main presentation theorem in this regard is the following.
Theorem 5.15 (main theorem for type D) The almost-Hopf ring structure over F 2 of L n 0 H .W D n I F 2 / extends uniquely to a graded almost-Hopf ring structure with components on the F 2 -vector space F 2 1 C˚F 2 1 ˚L n 1 H .W D n I F 2 / such that 1 Cˇ_ D id, 1 ˇ_ D Ã, 1 C 1 C D 1 C , 1 1 D 1 , and 1 C 1 D 0; .1˙/ D 1 C˝1˙C 1 ˝1 ; .x/ D 1 C˝x C 1 ˝Ã.x/ C .x/ C Ã.x/˝1 C x˝1 C for all x in L n 1 H .W D n I F 2 /, where is the reduced coproduct in L n 0 H .W D n I F 2 /.
This almost-Hopf ring admits a presentation with two families of generators, C k;n 2 H n.2 k 1/ .W D n2 k I F 2 / for k; n > 0; ı 0 nWm 2 H n .W D nCm I F 2 / for n ¤ 1 and m 0; together with 1 . The compatibility identity above and the following list of equalities provide a complete set of relations, where 1 C is theˇ-unit: .  for all k > 0 and m; n 0 with n ¤ 1.
In this case, too, the generators are explicitly described (see Definitions 5.4 and 5.5).
The relations are spread out in a few lemmas to prove the identities concerning coproduct, transfer product, and cup product separately. Building on these core theorems, we also describe convenient additive bases for the cohomology of these groups, with a graphical description via skyline diagrams similar to that obtained for the symmetric group in [7], and compute the Steenrod algebra action. Our formulation of the cohomology of W B n and W D n yields without additional effort many features of these cohomology algebras. For instance, Hepworth's homological stability results [11] in these particular cases follow directly.
We obtain our presentation via three technical tools. First, we exploit De Concini and Salvetti's geometric combinatorial model to realize such (almost) Hopf rings structures at the cochain level. Specializing their construction to the families of groups of our interest, we observe that a resolution for W B n is obtained from the symmetrized version of the planar level trees used by Giusti and Sinha [9] for the symmetric groups. The cohomology of W D n is governed by an oriented version of these objects. We describe cochain representatives of the structural maps in detail. Our treatment follows the paper cited above closely. However, we note that while the transfer product is realized very similarly to the † n case, coproducts are more complicated and require the combinatorial operation of "pruning" symmetric planar level trees. This cochain-level description allows us to quickly retrieve some of our relations and give a more geometric flavor to our generating classes. For instance, they can be interpreted as Thom classes in a suitable sense.
Second, we use the existence of well-behaved maps between W B n , W D n and † n . These homomorphisms preserve parts of our structures. Therefore, we exploit them to build our presentations on the known result for the cohomology of the symmetric groups. We provide a cochain-level description of these morphisms, and we determine both their action on generators and their relations to the coproduct and transfer product.
Third, we reconcile with Swenson's approach, and we investigate restrictions to elementary abelian 2-subgroups. The mod 2 cohomology of finite reflection groups is known to be detected by this family of subgroups. We effectively compute the action of these restriction maps on our additive bases. The multiplicative structure on the cohomology of (the invariant subalgebras of) such subgroups is known. Thus, these calculations allow us to deduce cup product relations that would be otherwise difficult to obtain.
We organize the paper as follows. After describing the structures on the cohomology of W B n and W D n in Section 2, we devote the following two sections to developing our geometric tools. In Section 3, we review De Concini and Salvetti's construction, and we specialize it to W B n and W D n . In Section 4, we investigate the combinatorics of pruning operations, and we retrieve cochain-level representatives of our structural the normal subgroup generated by s 0 . We observe that this provides an isomorphism between W B n and the wreath product F 2 o † n , a semidirect product of F n 2 and † n . Therefore, the inclusions † n † m ! † nCm extend naturally to monomorphisms W B n W B m ! W B nCm . These inclusions are associative and commutative up to conjugation.
Let A B D L n 0 H .W B n I F 2 /. We define a coproduct and two products, andˇ, on A B in the following way: is induced by the obvious monomorphisms W B n W B m ! W B nCm ; ˇis induced by the cohomology transfer maps associated with these inclusions; is the usual cup product.
Due to the associativity and the commutativity of the natural inclusions, these morphisms define an almost-Hopf ring structure. This is a general fact, as noticed in [9]. In this case, however, A B is a full Hopf ring.
Proposition 2.1 A B , with these structural morphisms, is a Hopf ring.
Proof The almost-Hopf ring axioms hold by [9,Theorem 2.3]. It remains only to prove that .A B ; ;ˇ/ forms a bialgebra. This claim follows from the fact -compare with [7, Section 3] -that this diagram is a pullback of finite coverings for all n; m 2 N, where indicates the projections.
We remark that, since A B with andˇis a conilpotent bialgebra, the existence of the antipode comes for free. This antipodal morphism does not play a role in our treatment; thus, we will not discuss it further.
Similarly, we can construct an additional almost-Hopf ring structure on the cohomology of the Coxeter groups of type D n . Indeed, on the direct sum A D D L n 0 H .W D n I F 2 /, we can define a coproduct and two productsˇand as done for A B . However, these do not make A D a full Hopf ring because, as we will see later, .A D ; ;ˇ/ fails to be a bialgebra.
With essentially the same proof used for A B , we can prove the following easy proposition, which follows again from [9, Theorem 2.3].
Proposition 2.2 A D , with the coproduct and the two products defined before, is an almost-Hopf ring over F 2 .
As we remarked in the introduction, there is a similar result for the mod 2 cohomology of the symmetric groups, obtained by Giusti, Salvatore and Sinha in [7]. We recall their statement here because we will build our computations upon it. Theorem 2.3 [7, Theorems 1.2 and 3.2] A † D L n 0 H . † n I F 2 /, together with a coproduct W A † ! A †˝A † induced by the obvious inclusions † n † m ! † nCm , a productˇW A †˝A † ! A † given by the transfer maps associated with these inclusions, and a second product W A †˝A † ! A † defined as the usual cup product, is a Hopf ring over F 2 .
A † is generated , as a Hopf ring, by classes k;n 2 H n.2 k 1/ . † n2 k I F 2 / for k 0 and n 1. The coproduct of these classes is given by the formula There are no more relations between these classes.
The unit of the algebra H . † n I F 2 / under the cup product is 0;n 2 H 0 . † n I F 2 /. For this reason, we will often denote it with the symbol 1 n throughout the paper.
Given a finite reflection group G Ä Gl n .R/, there is a natural hyperplane arrangement A G in R n associated with G, whose hyperplanes are the fixed points sets of reflections in G. The choice of a fundamental chamber C 0 of A G gives rise to a Coxeter presentation .G; S / for G, whose set of generators S is composed by reflections with respect to hyperplanes that are supports of a face of C 0 . Every finite Coxeter group arises this way.
For any F Â R n , we can define A F gives rise to a stratificationˆ.A F / of R n , in which the strata are the connected components of sets of the form L n The strata inˆm are defined as sets of the form De Concini and Salvetti construct a regular G-equivariant CW-complex X Â Y that is "dual" to the stratificationˆ1, in the sense that for every stratum F 2ˆ1 of codimension d , there exist a unique d -dimensional cell in X that intersects F , and they intersect transversally in a single point. For m < 1, the intersection X .m/ of X with Y .m/ is a subcomplex of X whose cells are dual to strata inˆm. This construction is done equivariantly, in the sense that for every stratum F 2ˆ1 and every g 2 G, if ' W D d ! X is the cell dual to F in X, then .g:_/ ı ' W D d ! X is the cell dual to g:F . The authors then show that X is a G-equivariant strong deformation retract of Y . Since Y is contractible and G-free, the quotient X=G is a cellular model for the classifying space B.G/ and the cellular chain complex C G D C CW .X / is a ZOEG-free resolution of Z.
The strata ofˆ1 have a more compact combinatorial description in terms of the Coxeter presentation. For every s 2 S generating reflection for G, we let H s be the hyperplane fixed by s. H s divides the space R n into two semispaces, H C s and H s . We let H C s be the semispace that contains the chosen fundamental chamber C 0 . To a flag D .S Ã 1 Ã 2 Ã Ã k D ¿/ of subsets of S we can associate a stratum F ofˆ1 such that x D .x 1 ; : : : ; x n / 2 .R 1 / n belongs to F if and only if ..x 1 / r ; : : : ; .x n / r / 2 H s if s 2 r ; ..x 1 / r ; : : : ; .x n / r / 2 H C s if s 2 r 1 n r is satisfied for every s 2 S and every r 1. Thus, to a couple .; g/, where is a flag as before and g 2 G, we can associate the stratum g:F obtained from the above F by applying g. This construction yields an algebraic-combinatorial description of the cellular chain complex of X. The main theorem of De Concini and Salvetti's paper is the following.
The function described above is a bijection between this set and the set of strata in Let c.; / be the cell dual to the stratum corresponding to .; /. The boundary homomorphism in C CW .X / is given by the formula where˛is an integer number easily computed in terms of , i , , We remark that in the case of Coxeter groups of type B or D, minimal coset representatives are explicitly known. For a complete description, we refer, for instance, to [14].

Alexander duality and Fox-Neuwirth complexes
We recall an alternative description of C G . This description has been exposed in [8], where it is investigated in much detail in the A n case. As observed in that paper, for every 1 Ä m Ä 1, the strata ofˆm are the interiors of cells in a G-equivariant cell structure on the Alexandroff compactification .
Denote its augmented (G-equivariant) cellular chain complex with the symbol f FN m G . Its cells are the closures e.F / of strata F 2ˆm (together with the basepoint f g) and, from the construction of X .m/ as a CW-complex dual toˆm -details in [4] e.F / is contained in the boundary of e.F 0 / if and only if the cell of X dual to F contains the cell dual to F 0 in its boundary. This fact implies that the complex f FN m G is, up to a shift of degrees, the dual of C CW .X .m/ /, at least modulo 2 (in general, there are differences in some signs due to orientations). Explicitly, the closure in f FN m G of a stratum of dimension d correspond to the dual of a chain in C CW .X .m/ / of dimension nm d . In the remaining sections of this paper, we will always implicitly assume this shift, and we will grade f FN m G to match the corresponding dimension of the dual cell.
In particular, f FN m G calculates the cohomology of Y .m/ and is therefore acyclic up to dimension nm 2. Alternatively, we can see this, as explained in [8], by observing that the Atiyah duality theorem implies that the Spanier dual of Y .m/ is .Y .m/ / C . Passing to the limit for m ! 1, we obtain an acyclic F 2 -complex f FN G˝F2 , dual to C CW .X /˝F 2 , for which a basis fe.S /g S 2ˆ1 is parametrized by strata inˆ1. The degree of e.S/ as a cochain of X is equal to the codimension of F . This is an equivariant cochain model for E.G/. In particular, the quotient FN G˝F2 D f FN G =G˝F 2 calculates z H .GI F 2 /. In the following, when we need to stress the Coxeter group G involved, we will use the heavier notationˆ1 ;G instead ofˆ1.
This description of the cochain complex FN G calculating the cohomology of G fits particularly well with a chain-level interpretation of duality via intersection theory that we will occasionally use in proofs and that we briefly recall here. Given a manifold X and an immersion i W W ! X of a codimension d manifold in X, we say that a smooth singular chain in X is transverse to i if, for every simplex W k ! X of the chain, is transverse on every face of k and subface, in the sense of manifolds with corners. It can be proved that the subcomplex consisting of chains that are transverse to i is chain equivalent to the full one. To every d -dimensional singular simplex W d ! X transverse to i we can associate the element W . / 2 F 2 given by the mod 2 cardinality of 1 .W /. This procedure defines a cochain dual in the complex dual to the chain complex of singular chains transverse to i . If i is a proper embedding, W is a cocycle and defines a cohomology class. The most important constructions in cohomology can be understood geometrically using this model. In particular, if f W Y ! X is transverse to i , then f # . W / D f 1 .W / . The reader will find a complete reference of this approach to cohomology in [6].
In our particular context, each stratum S 2ˆ1 defines such a cochain S . We understand the coboundary of S as @.S / , so we can identify f FN G , at least modulo 2, with the cochain complex spanned by S for strata S 2ˆ1. Suppose W Â Y .1/ G is a proper submanifold of codimension d obtained as a union of strata. In that case, its associated cochain W is the sum of S for strata S Â W of minimal codimension, and ı. W / D 0. If, in addition, the action of G preserves W , then, passing to the quotient, its image W Â Y .1/ G =G defines a Thom class represented in FN G by the sum of strata contained in W . This construction is made precise in [7,Definition 4.6].

The special case of Coxeter groups of type B
We conclude this section by further investigating the cases of our interest G D W B n and, in the following subsection, G D W D n . The strata ofˆm for the symmetric group † n can be described in terms of leveled trees, as shown in [8] using ideas dating back to Vassiliev [19]. A straightforward adaptation of these ideas shows that, in the case of the Coxeter groups of type B n , we can describe them in terms of symmetric leveled trees. This interpretation encodes geometrically and combinatorially the structure of W B n as a wreath product of † n with a cyclic group of order 2. Below we provide the precise definitions.
First, we observe that, since W B n is generated by a set S D fs 0 ; : : : ; s n 1 g of n fundamental reflections as described in Figure 1, the Fox-Neuwirth complex f FN W Bn has a ZOEW B n -basis fe.a/g indexed by n-tuples of nonnegative integer numbers .a 0 ; : : : ; a n 1 /.
The reflection hyperplane arrangement associated with W B n can be described as Moreover, s 0 can be identified with the reflection with respect to H 0 1 and, for every i > 0, s i with the reflection with respect to H C i;i C1 . Thus the basis element corresponding to a D .a 0 ; : : : ; a n 1 / is described as the stratum e.a/ D˚.x 1 ; : : : ; x n / 2 .
Passing to the quotient by the action of W B n , we see that FN W Bn has a Z-basis constituted by OEa 0 W W a n 1 D OEe.a 0 ; : : : ; a n 1 /.
The differential on FN W Bn is complicated, but it is combinatorially accessible via a description of its basis in terms of trees.
Definition 3.2 A signed depth-ordering is a sequence of labeled inequalities of the form D .0 < a 0 i 1 < a 1 < a n 1 i n /, where i k 2 f n; : : : ; 1; 1; : : : ; ng for all 1 Ä k Ä n, and these indices have pairwise different absolute values. By convention, we let i 0 D 0.
A planar level tree is a planarly embedded tree T satisfying the following conditions: it has a root vertex embedded in .0; 0/ and all the other vertices having their second coordinate (the "height") equal to a positive integer; two edges connected by an edge have heights whose difference is 1; the height along the unique minimal path from the root to every leaf is always increasing.
A planar level tree with labels in I is a couple .T; / defined as follows: T is a planar level tree, and is a bijective labeling of the leaves of T with elements of I .
A symmetric planar level tree is a planar level tree invariant under the reflection r along the y-axis and having an odd number of leaves.
An antisymmetric planar level tree with labels in f n; : : : ; ng is a labeled planar level tree .T; / with labels in f n; : : : ; ng such that T is symmetric, and two leaves that correspond to each other under the application of r have labels opposite to each other.
The antisymmetric planar level tree associated with a depth ordering is the antisymmetric planar level tree T , unique up to isotopy, defined by the following properties: the labels of the leaves, from left to right, are i n ; : : : ; i 1 ; 0; i 1 ; : : : ; i n ; the leaves labeled i k 1 ; i k , for 1 Ä k Ä n, are separated by a vertex of height a k but not by vertices of height less than a k .
Let k 0. The k-symmetrization S k .T / (resp. z S k .T /) of a planar level tree T (with labels in f1; : : : ; ng) is a symmetric planar level tree S (resp. antisymmetric planar level tree with labels in f n; : : : ; ng) obtained by the following procedure. Glue T from the right to a vertical linear planar level tree lying into the y-axis up to height k. Then, add the mirror image of such tree under r to obtain a symmetric planar level tree (choosing the unique antisymmetric labeling that extends the labeling of T in the labeled case).
There is a free action of W B n on the set antisymmetric planar level trees with labels in f n; : : : ; ng given by interpreting elements of W B n as signed permutations and permuting labels accordingly. We always assume that the edges of a level tree are oriented so that there is a unique oriented path from the root vertex to each leaf.
Similarly to the symmetric group case, we have the following immediate proposition.
Proposition 3.3 The function 7 ! T is a bijection between the set of signed depthorderings with n labels and the set of isotopy classes of antisymmetric planar level trees with labels in f n; : : : ; ng. Furthermore, to D .0 < a 0 i 1 < a 1 < a n 1 i n / is associated a stratum e.a/ 2ˆ1 ;W Bn , where .k/ D i k , a D .a 0 ; : : : ; a n 1 /, and this provides a W B n -equivariant additive basis of f FN W Bn labeled by signed depth-orderings or, equivalently, by isotopy classes of antisymmetric planar level trees with labels in f n; : : : ; ng. W B n acts on this basis by permuting labels. Consequently, an additive basis for FN W Bn is given by symmetric planar level trees with 2n C 1 leaves.
We observe that we can use Proposition 3.3 to reinterpret operations on (symmetric) level trees in terms of n-tuples or (signed) depth-orderings. For instance, the k-symmetrization of trees provides a linear map S k W FN † n ! FN W Bn that we can interpret as OEa 1 W W a n 1 7 ! OEk W a 1 W W a n 1 .
We can now describe the differential in terms of this basis.
Definition 3.4 [8] Let .T; / be a planar level tree. Let v be an internal vertex. Let E.v/ be the set of edges whose source vertex is v. The planar embedding of T induces an order on E.v/, defined from left to right. A vertex permutation of .T; / at v is another planar level tree that is isomorphic to .T; / as a labeled tree but with a different planar embedding that differs from the original one only by the ordering on E.v/.
Given a planar level tree .T; / and an internal vertex v, let .e; f /, with e < f , be a couple of adjacent edges in E.v/. Let u e and u f be the targets of e and f , respectively. Let be a shuffle of the two sets E.u e / and E.u f /. Let d e;f; .T; / be the planar level tree obtained by gluing together e and f , with common target N u, and then applying the vertex permutation that permutes the edges in E. N u/ by .
Recall that, in the A n case, the differential in f FN of the basis element corresponding to a planar level tree with labels .T; / is given by the sum over .v; / as above of d v; .T; /. Similarly, we have the following proposition, which essentially states that a symmetrization of the previous construction gives the differential in the B n case. where the sum is over couples .e; f / of adjacent edges in the positive half-plane and shuffles of the two sets of vertices incident to the target of e and f , respectively.
We can equivalently express this construction using planar level trees T with n C 1 leaves labeled by . n; : : : ; 1; 0; 1; : : : ; n/, with labels having pairwise different absolute values, such that the leftmost leaf has label 0. We recover the corresponding antisymmetric level tree as follows. We choose a representative of the isotopy class of .T; / in which the entire oriented path from the root vertex to the label 0 lies on the y-axis. Then we merge T with its mirror image along y with opposite labels. In this case, the differential is given by contracting a couple of adjacent edges and shuffling. When the result is a tree whose leftmost leaf is not labeled by 0, we replace the part of the tree belonging to the negative half-plane f.x; y/ j x Ä 0g with its mirror image in the positive half-plane, with opposite labels, and shuffle the corresponding edges in all possible ways.

The special case of Coxeter groups of type D
We now turn to the description of the complex FN W Dn . Once again, since this Coxeter group has n fundamental reflections t 0 ; : : : ; t n 1 , a ZOEW D n -basis for f FN W Dn is indexed by n-tuples of nonnegative integers a D .a 0 ; : : : ; a n 1 /.
The inclusion j n W W D n ! W B n identifies the reflection arrangement associated to W D n with the subarrangement of A W Bn composed by the hyperplanes Hi ;j , for 1 Ä i < j Ä n, and t i D s i for 1 Ä i Ä n, while t 0 is the reflection along H 1;2 . Thus the basis element of f FN W Dn corresponding to a is described as the stratum e.a/ D˚.x 1 ; : : : ; x n / 2 .
Passing to the quotient by the action of W D n , we see that FN W Dn has a Z-basis constituted by OEa 0 W W a n 1 D OEe.a 0 ; : : : ; a n 1 /.
The complex f FN W Bn =j n .W D n / also calculates the cohomology of W D n . Therefore, there is a cochain equivalence ' W FN W Dn ! f FN W Bn =j n .W D n / between the two resolutions. In the subsequent section, we compute an explicit formula for ' that we will use to perform cochain-level computation in the following sections. For instance, we will prove the relations for coproduct of transfer products of Hopf ring generators by mapping them to f FN W Bn =j n .W D n /, where their expressions are closer to the B n case. As a notational convention, we denote this cochain complex by FN 0 W Dn .
First, we observe that OEW B n W j n .W D n / D 2; thus j n .W D n / is a normal subgroup of W B n . The two cosets of j n .W D n / in W B n are represented by the identity and s 0 , the only fundamental reflection of W B n that is not contained in j n .W D n /. Thus, given a ZOEW B n -basis B for f FN W Bn , the classes of x and s 0 :x, where x 2 B, provide a Z-basis for FN 0 W Dn . Let B be the basis defined above in terms of n-tuples or equivalently of symmetric planar level trees, parametrized by n-tuples of nonnegative integers a D .a 0 ; : : : ; a n 1 /. We denote by OEa 0 W W a n 1 C and OEa 0 W W a n 1 the cochains in FN 0 W Dn arising from the basis element corresponding to a and s 0 a.
The complex FN 0 W Dn also has a description in terms of trees.
Definition 3.6 Let T be a symmetric planar level tree with 2n C 1 leaves. An orientation of T is the choice of an element of L= , where L is the set of antisymmetric labelings of T with labels in f n; : : : ; ng, and is the equivalence relation defined by Note that if two antisymmetric labelings of a symmetric planar level tree T differ by a permutation 2 † f n;:::;ng , then must fix 0 and act as a signed permutation on f n; : : : ; 1; 1; : : : ; ng. Hence, an orientation of T is the choice of an antisymmetric labeling up to the action of j n .W D n /. Since the index OEW B n W j n .W D n / is 2, there are two possible orientations for a symmetric planar level tree T , determined by the parity of the number of negative labels of leaves in the positive half-plane. In particular, we can identify an orientation O with a sign C or , corresponding to labelings with an even or odd number of positively labeled leaves in the positive half-plane, respectively.
Moreover, from the fact that Alt.2n C 1/ is normal in † 2nC1 , it follows that if T is a symmetric planar level tree, is a labeling of T and .T / is a vertex permutation of T at a vertex v, then the orientation of the permuted labeled tree .T; / only depends on the orientation determined by . Therefore, the rule for the differential in f FN W Bn induces a formula for the differential in FN 0 W Dn in terms of trees. Hence, we have the following description. Proposition 3.7 FN 0 W Dn can be described as the cochain complex having additive basis indexed by oriented symmetric planar level trees with 2n C 1 leaves, with differential induced by the symmetric tree differential in f FN W Bn by keeping track of orientations.
The reader is encouraged to compare this description with the notion of "charged" configuration used in [9].

Geometry and combinatorics: chain-level formulas
We devote this section to developing some formulas that will allow us to perform calculations at the (co)chain level. These computations will be needed at points, especially when retrieving relations. We first compute some connecting maps between the Fox-Neuwirth complexes of Coxeter groups of type A, B and D. Then, we provide cochain representatives of the structural maps of our almost-Hopf ring structures.

The connecting homomorphisms
As f FN W Dn and f FN 0 W Dn are both free resolutions of Z as a ZOEW D n -module, they need to be W D n -equivariantly cochain equivalent. We begin by providing a formula for an explicit equivalence ' relating the two models FN W Dn and FN 0 W Dn .
if a 0 D a 1 ; OEa 1 W a 0 W a 2 W W a n 1 if a 0 > a 1 ; induced by the inclusion Y .
This implies that ' has the desired form.
We also consider the following group homomorphisms: the standard inclusion j W † n ! W B n already considered in the previous section; the involution c s 0 W W D n ! W D n given by conjugation by s 0 , the unique generating reflection of W B n that does not belong to W D n , that fixes t i for 2 Ä i < n and switches t 0 and t 1 ; the two inclusions i C ; i W † n ! W D n given, in terms of the Coxeter generators t 0 ; : : : ; t n of Figure 1, by i˙ We denote by ÃW H .W D n I F 2 / ! H .W D n I F 2 / the morphism induced by c s 0 on cohomology.
We note that the two following properties hold by construction: We compute cochain representatives of Ã in the following lemmas.
Proof Since the image under c s 0 of a fundamental reflection for W D W D n is again a fundamental reflection, for every 0 Â Â ft 0 ; : : : ; t n 1 g, the set of minimal-length coset representatives satisfies c s 0 .W 0 / D W This yields, dually, the desired We can also describe Ã in terms of FN 0 W Dn . The proof of the following lemma is straightforward.
In terms of oriented symmetric planar level trees, the map Ã 0 # acts on .T; O/ by replacing O with the opposite orientation.
The following identity is also proved by direct inspection.

Lemma 4.4
The following diagram commutes: The formulas for the other connecting maps follow from a general remark.
Lemma 4.5 Let G be a Coxeter group, with Coxeter generators S D fs 0 ; : : : ; s n g and H Ä G be a parabolic subgroup, generated by a subset T D fs i 0 ; : : : ; s i m g Â S. The inclusion H ,! G is represented at the chain level by the chain map C H ! C G given by c.; / 7 ! c.; /, for flags Dually, it is represented at the cochain level by the cochain map FN G ! FN H given by OEe.a 0 ; : : : ; a n / 2 C H 7 ! OEe.a i 0 ; : : : ; a i m / if a j D 0 for all j … fi 0 ; : : : ; i m g; 0 otherwise: Proof Since the inclusion of parabolic subgroups preserves minimal coset representatives, the De Concini-Salvetti boundary formula of Theorem 3.1 implies that the given linear morphism C H ! C G is an H -equivariant chain map. Dualizing this yields the cochain formula between Fox-Neuwirth complexes.
As particular cases of this lemma, we retrieve cochain formulas for our connecting homomorphisms: The following statements are true.

Structural morphisms: A B
We want to describe the almost-Hopf ring structures presented in Section 2 in our geometric context. We begin with the coproduct map in A B . In contrast with the symmetric group case, the cochain-level map inducing the coproduct is relatively complicated. Its underlying combinatorics is built upon elementary steps that we, mindful of the botanic analogy, suggestively call "prunings".
Definition 4.7 Let T be a planar level tree. An elementary k-pruning of T is a planar level tree T 0 obtained by the following procedure. Choose an internal vertex v of T of height k, and consider on E.v/ the order induced by the planar embedding. Let 1 Ä l < jE.v/j, consider the l biggest elements e 1 ; : : : ; e l of E.v/ with respect to this order, and let v 0 i be the target of e i . T 00 is the subtree of T spanned by v and all vertices that can be reached from one of the v 0 i through an oriented path. T 0 is the complementary subtree of T 00 in T . We call the planarly embedded subtree T 00 the scrap of the elementary k-pruning. An elementary k-pruning is said to be minimal if l D 1. A k-pruning is a couple .T 0 ; T 00 / constructed as follows: T 0 is obtained from a sequence of elementary k-prunings performed on pairwise different vertices v 1 ; : : : ; v j of T , with scraps T 00 1 ; : : : ; T 00 j ; T 00 is a planar level tree obtained by joining these scrap subtrees along a vertex w of height k and performing a vertex permutation at w that shuffles the edges of the scraps.
Let T be a symmetric planar level tree. An elementary symmetric k-pruning of T is a tree T 0 obtained as follows. Apply to T an elementary (nonsymmetric) k-pruning whose scrap T 00 does not contain the central leaf belonging to the y-axis. Then, remove the image of the subtree of T 00 under the reflection r along the vertical axis. T 00 is called the scrap of the elementary symmetric pruning. An elementary symmetric k-pruning is said to be minimal if it is obtained from a minimal elementary k-pruning. A symmetric k-pruning is a couple .T 0 ; T 00 /, where T 0 is obtained from a sequence of elementary k-prunings performed on pairwise different vertices of T , with scraps T 00 1 ; : : : ; T 00 j ; T 00 is a nonsymmetric planar level tree obtained by joining the scrap subtrees to a vertex w of height k and performing a vertex permutation at w that shuffles the edges of the scraps.
We note that elementary k-prunings at different vertices commute, both in the symmetric and nonsymmetric cases. Hence, a k-pruning or symmetric k-pruning is uniquely determined by the set of elementary k-prunings that compose it, independently of the order in which they are performed.
There is also an alternative way to define (symmetric) k-prunings in terms of minimal k-prunings instead of elementary ones. A (symmetric) k-pruning is obtained by performing a sequence of minimal elementary (symmetric) k-prunings, not necessarily at pairwise different vertices, and then joining the scraps at a vertex of height k without shuffling the edges.
We now consider three linear morphisms that we will need to define the cochain-level coproduct map: the k-pruning map that maps a symmetric planar level tree T to the sum P T 0˝S k .T 00 / over all the possible symmetric k-prunings .T 0 ; T 00 / of T ; the minimal k-pruning map that maps a symmetric planar level tree T to the sum P T 0˝S k .T 00 / over all the possible minimal elementary symmetric k-prunings .T 0 ; T 00 / of T ; The map P k is exemplified in Figure 3. We understand C at the level of symmetric planar level trees as the function given by the following procedure. Take a couple of such objects .T; S/. Cut S along its central vertical axis. Finally, glue the right piece of S onto the right side of T and the left part onto its left side to obtain a new symmetric planar level tree. We remark that these linear morphisms are degree-preserving, but they are not chain maps.
In the A n case, we can define a similar k-pruning map P 0 k by summing all nonsymmetric k-prunings. For k D 0, P 0 0 is a chain map, and it is shown in [8] to induce the coproduct in cohomology. This statement is not true in the B n case because the differential of an antisymmetric planar level tree with labels behaves badly near the central "trunk" labeled 0. Nevertheless, at each level k, away from this central stem, this is essentially true. For this intuitive reason, we must define our cochain-level coproduct map differently: prune a symmetric planar level tree at every level and tensor it with a symmetric planar level tree whose principal k-blocks, as defined below in Definition 4.10, are the scraps of the performed prunings. To prove this statement, we need some preliminary calculations. Suppose that a symmetric planar level tree T corresponds to OEa 0 W W a n 1 2 FN W Bn . In that case, consider the set of couples of adjacent edges .e; f /, with e < f , in T having the same source vertex and belonging to the positive half-plane f.x; y/ j x 0g. This set in bijective correspondence with f0; : : : ; n 1g, and the height of the common vertex of the couple .e; f / corresponding to i is a i . This bijection is explicitly given by counting the leaves in the positive half-plane that lie on the left of e. For 0 Ä i Ä n 1, Lemma 4.8 Let T be a symmetric planar level tree corresponding to OEa 0 W W a n 1 . Let m k be the smallest index such that a m k D k. Let I be the trivial symmetric planar level tree. Then the following statements are true: (1) the pruning maps and the differential satisfy the equality (3) for all a D OEa 0 W W a n 1 and b D OEb 0 W W b m 1 with b 0 < minfa 0 ; : : : ; a n 1 g, and the latter also holds for i D n if b 0 < minfa 0 ; : : : ; a n 1 g 1; Proof The statements from (2) to (5) are easy. Regarding (2), if a i < k for all i, T has no vertex of height k with more than one outgoing edge. Thus the only possible symmetric k-pruning is the trivial one. Regarding (3), the bijection ' W f0; : : : ; n 1g t f0; : : : ; m 1g ! f0; : : : ; n C m 1g that shifts elements of f0; : : : ; m 1g by n yields a bijection between pairs .e; f / of adjacent edges of the symmetric planar level tree T corresponding to C.a˝b/ and those of the symmetric planar level trees T 0 and T 00 corresponding to a and b respectively. If b 0 < minfa 0 ; : : : ; a n 1 g, then for all i 2 f0; : : : ; n C m 1g, with the only possible exception of i D n, this bijection preserves where v e i and v f i are the target vertices of the corresponding pair of edges .e i ; f i /. The edges in E.v e i / and E.v f i / of the corresponding pair come either both from T 0 or both from T 00 . Hence d i C.a˝b/ D d ' 1 .i / a˝b. If b 0 < minfa 0 ; : : : ; a n 1 g 1 the same is also true for the edges e n and f n , so the equality is satisfied also in this case. Statement (4) is immediate from the definition of k-prunings and the combinatorics of shuffles, and (5) is obvious.
On the contrary, (1) is more complicated and requires a more detailed proof. As a notational convention, let d h l D P i Wa i Dl d i , the sum of the contributions to the differential coming from vertices at height k. We compare d h l P k .T / with P k d h l .T /. We consider different cases depending on the difference between k and l.
If l > k, d h l is computed by gluing together a pair .e; f / of adjacent edges of height bigger than k (and its mirror pair) and performing a shuffle at the new target vertex. These operations only change a connected subtree whose vertices all have height bigger than k, and, by construction, k-prunings commute with such operations.
T 00 //, where the sum is over symmetric k-prunings .T 0 ; T 00 / of T and pairs of adjacent edges .e; f / in the positive half-plane with a common source vertex of height k in T 0 or S k .T 00 /. We also note that S k .T 00 / has a unique vertex w of height k. There is an obvious that maps an edge to its image in T 0 (if it is not pruned away) or in S k .T 00 / (if it is), and that arises from the fact that, for elementary prunings, The edge e 0 is the unique edge belonging to the central vertical stem whose source vertex is w. Moreover, this bijection preserves the properties of belonging to the positive and negative half-plane. Therefore, we can write the summation above expressing d h k P k .T / as the sum of three pieces: The first piece is the sum of the terms corresponding to .e; f / such that .e; f / come from adjacent edges in T . These terms correspond to symmetric k-prunings of d S e;f; .T /, for shuffles at the common vertex of e and f . Hence, their sum The second piece is the sum of the terms corresponding to .e; f / in S k .T 00 / such that e ¤ e 0 and .e; f / do not come from adjacent vertices of T . Under this condition, the symmetric vertex permutation .S k .T 00 // of S k .T 00 / at w that switched the positions of e and f still produces a shuffle of the scraps of the elementary prunings involved in .T 0 ; T 00 /. Every tree in d .e;f / .S k .T 00 // cancel out with a tree in d .f;e/ . .S k .T 00 ///. Hence, this second piece is 0.
The third piece is given by the terms corresponding to .e; f / with e D e 0 . These terms yield .id˝d 0 /P k .T /.
Finally, we deduce that d h where the sum is taken over couples .e; f / of adjacent edges in T whose common source v has height k 1, and symmetric k-prunings .T 0 ; T 00 / of trees in d .e;f / .T /. Let v e and v f be the targets of e and f , respectively. By construction, suitably shuffled. Let A be the set of edges removed by the corresponding elementary symmetric prunings at N v and at r. N v/, the mirror vertex of N v (which might coincide). We retrieve symmetric k-prunings for which E.v e / 6 Â A and E.v f / 6 Â A from symmetric k-prunings .T 0 ; T 00 / of T by applying d .e;f / to T 0 . Now assume that v is not on the central stem of the tree. If e ¤ min.E.v//, it is the successor of an edge g 2 E.v/, and the terms of P k d e;f .T / for which E.v e / Â A cancel out with the terms of P k d g;e .T / for which E.v e / Â A. Similarly, all the terms for which E.v f / Â A and f ¤ max.E.v// cancel out. The only remaining terms are those in which we remove an entire subtree corresponding to min.E.v// -which is the mirror image of max.E.r.v//. If v belongs to the central axis, we must slightly modify the argument to take into account only edges in the positive half-plane and shows that the surviving terms are those in which an entire subtree stemming from max.E.v// is removed. The sum of all these elements is exactly equal to the correcting term .id˝d m k 1 /.id˝C /.P k˝i d/P min k 1 .T /. We deduce that If l < k 1, since k-prunings only depend on the part of the tree above height k and d h l does not change it, the same argument used for l > k shows that Combining the equalities obtained in these cases yields (1).
We are now ready to construct a cochain representative of the cohomological coproduct map be the linear maps defined recursively by the formulas (3) represents the cohomology coproduct map at the cochain level.
Proof (1) Let a 2 FN W Bn and let m D maxfa 0 ; : : : ; a n 1 g. Statement (2) of Lemma 4.8 guarantees that k .a/ D m .a/ for all k > m. Thus, the sequence f k g 1 kD0 stabilizes and consequently has a limit.
(2) We first observe that Lemma 4.8 (4) and (5) imply that for all k 0, with the convention that 1 .T / D T˝I . Combining this remark with Lemma 4.8 (3) and (5), we obtain that, for all k 1, We use this to prove by induction on k that k d D d k C .id˝d 0 /. k k 1 /. For k D 0 this identity is the content of the first statement of Lemma 4.8. For k > 0, we deduce from the identity above and the previous lemma that To justify the last equality, we observe that .P k˝i d/ k 2 is a sum of terms of the form c˝a˝b with b 0 < minfa i g 1, and we apply the stronger clause of Lemma 4.8 (3).
Now the identity d D d follows by passing to the limit, and using that the sequence f k g 1 kD0 stabilizes.
(3) Consider the dg-module U over F 2 with basis given by symmetric planar level trees with antisymmetric labels in any finite subset I Â N, not necessarily f n; : : : ; ng, with the symmetric tree differential. Note that L n 0 f FN W Bn˝F 2 embeds in U in the obvious way. We observe that the linear maps P k , P min k and C lift to linear maps z P k ; z P min k W U ! U˝U and z C W U˝U ! U . z P k and z P min k are still defined via prunings, but we additionally keep track of the labels of the subtrees involved. We compute z C on T 0˝T 00 by splitting T 00 symmetrically along the vertical axis, keeping labels, and symmetrically attach the two parts to T 0 to obtain a new basis element of U . Lemma 4.8 still holds for this labeled version of the morphisms by the same proof. Consequently, there is a labeled version z W U ! U˝U of , constructed recursively via finite approximations z k , that still commutes with the differential. Note that we can also embed into U˝U by keeping the labels of trees in f FN W Bn and relabeling trees in f FN W Bm via the bijection f0; : : : ; m 1g ! fn; : : : ; n C m 1g that raises numbers by n. There is also a projection U U ! f FN W Bn˝f FN W Bm˝F 2 that maps every basis element of U˝U that does not belong to f FN W Bn˝f FN W Bm˝F 2 to 0. By induction, we easily see that restricting each z k for all k (and, consequently, z ) to f FN W Bn˝f FN W Bm˝F 2 and composing with this projection we obtain linear maps and satisfy the same formal relation with respect to the differential. By identifying FN W Bn with the invariant subspace . f FN W Bn / W Bn , the limit map z restricts to , which is thus a cochain-level realization of the coproduct map.
We now turn our attention to the transfer map. We need a preliminary definition.
Definition 4.10 (partially from [8]) Let a D OEa 0 W W a n 1 2 FN W Bn be as defined above. In what follows, we assume, by convention, that a 1 D a n D 0. We say that the chain We say that a k-block OEa i W W a j of a is principal if, in addition, min 0Är<i a r D k. We denote by PBl k .a/ the tuple of the principal k-blocks of a, ordered from left to right.
For example, the basis element a D OE3 Note that a basis element a is uniquely determined by fPBl k .a/g 1 kD0 , the collection of its principal blocks. To retrieve a from these data, we can use the following procedure. First, for all k 0, add an entry equal to k before each principal k-block and concatenate all such tuples to obtain a k . Then, we obtain a as the concatenation of : : : ; a k ; a k 1 ; : : : ; a 0 . This sequence is necessarily finite because for k > max n 1 i D0 a i , PBl.a/ is the empty 0-tuple. With this method, we can construct a basis element a from an eventually empty collection of tuples fB k g, where the entries of B k are tuples of natural numbers strictly bigger than k.
We also observe that k-blocks can be retrieved from the corresponding symmetric level tree T . They are given by the connected components of T \f.x; y/ 2 R 2 j x 0; y > kg. Interpreted this way, a k-block is principal if and only if it does not intersect the central vertical axis but is contained in the .k 1/-block intersecting it. Proposition 4.11 Given a 2 FN W Bn , b 2 FN W Bm and k 0, let n a;k and n b;k be the lengths of PBl k .a/ and PBl k .b/, respectively. Given a sequence D f k g 1 kD0 of permutations k 2 † n a;k Cn b;k , define .a; b/ as the unique basis elements of FN W B nCm such that, for all k 0, PBl k . .a; b// D k .PBl k .a/; PBl k .b//, where .PBl k .a/; PBl k .b// is the concatenated .n a;k Cn b;k /-tuple and k acts on .n a;k Cn b;k /tuples by permuting the entries. LetˇW FN W Bn˝F N W Bm˝F 2 ! FN W B nCm˝F 2 be the homomorphism that maps a˝b to the sum P .a; b/ over sequences of permutations D f k g 1 kD0 such that k is a .n a;k ; n b;k /-shuffle for all k 0. Informally, aˇb is the sum of basis elements whose principal k-blocks are obtained by shuffling the principal k-blocks of a and b. This defines a morphism of complexes that induces the transfer product in cohomology.
Proof The reflection arrangement of W B n W B m , with its product reflection action We can explicitly obtain such inclusion by splitting a configuration of n C m points into the two subconfigurations consisting of its first n points and its last m points, respectively, and relabeling the indices of the second one. This map is a .W B n W B m /equivariant homotopy equivalence.
Therefore, passing to quotients, this yields a map Moreover, the obvious quotient map Let x D OEa 0 W W a n 1 ˝OEb 0 W W b m 1 be a basis element for the Fox-Neuwirth complex FN B n˝F N B m˝F 2 . Let be a smooth singular simplex transverse to our strata. By construction, the evaluation of OEa 0 W W a n 1 ˇOEb 0 W W b m 1 on is the sum of the evaluations of x on . Q /, as Q varies among all liftings of . A direct calculation shows that some . Q / intersects the stratum corresponding to x if and only if intersects some stratum e.c/, where the k-principal blocks of c are obtained by shuffling the k-principal blocks of a and b.
We conclude the treatment of the structural maps on the cohomology of W B n with some potentially helpful remarks. Since we will not use these facts in this paper, we will not provide complete statements nor proofs of these last claims. Nevertheless, it should be straightforward, although notationally heavy, to fill in the details.
Remark 4.12 (1) The transfer and coproduct maps commute already at the cochain level. To see this, you can observe that, by construction, .a/ is a sum of tensors a 0˝a00 , where PBl k .a 00 / is given by the leftovers of symmetric k-prunings of a, suitably shuffled, and that the pruning map P k itself commute withˇ.
(2) The same constructions of the coproduct map in terms of prunings and the transfer map in terms of principal block shuffles can be generalized to the cohomology with integral coefficients. In these cases, additional signs that we can compute from those appearing in Theorem 3.1 are required.

Structural morphisms: A D
The coproduct and the transfer product for W D n are described geometrically, similarly to what we did for W B n . However, some complications arise. For example, we cannot repeat the proof of Proposition 4.11 as it is for FN W Dn , because, in this case, a product of strata S S 0 Â Y W Dm is not necessarily the closure of a union of strata in Y .1/ W DnCm . However, these ideas adapt well to the cochain complex FN 0 W Dn , which we will use in the following as a cochain model. We can retrieve the identities we need in FN W Dn by using the equivalence ' of Lemma 4.1.
We can now state the formulas parallel to Propositions 4.9 and 4.11 for W D n . First, we consider the following oriented versions of the pruning and concatenation maps. Given a symmetric k-pruning .T 0 ; T 00 / of a symmetric planar level tree T , let O and O 0 be orientations of T and T 0 respectively. Fix an antisymmetric labeling 0 of T 0 inducing O 0 , and an antisymmetric labeling of T inducing O such that its restriction to T 0 , seen as a subtree of T , is 0 . By keeping track of the labels of scraps, induces an antisymmetric labeling 00 on S k .T 00 / and, consequently, an orientation O 00 . Unless the k-pruning is trivial, it is always possible to find such labelings and 0 , and the resulting orientation O 00 only depends on O and O 0 . We now mimic the construction we produced for W B n to describe the coproduct. We thus consider the following maps: the positive and negative k-pruning maps given by the formula where the sum runs over all positive and negative oriented k-prunings of T , respectively; We can also define C Let be the direct limit lim !k . C k C k /.
Proposition 4.14 The oriented pruning coproduct is a cochain map and induces the coproduct W A D ! A D˝AD in cohomology.
Proof It is enough to observe that, looking at the proof of Proposition 4.9, we can obtain the map by restricting to W D n -invariants. Proof The proof is essentially the same as that of Proposition 4.11.

The almost-Hopf ring presentations
This section contains the statements of the Hopf ring presentation for A B and the almost-Hopf ring presentation for A D . We thus state our main theorems, whose proof will be postponed until Section 7 because we still need to develop some necessary algebraic machinery. In the first subsection, we construct our generators, providing cochain representatives and a geometric interpretation. In the second one, we explain our relations and state Theorems 5.9 and 5.15. We then apply these results to extract combinatorially accessible additive bases for A B and A D in Section 5.2. Finally, the last subsection is devoted to the link between all these almost-Hopf ring structures.

Generators
We define certain cohomology classes that we will later prove to generate our (almost) Hopf rings. We begin with A B .
Definition 5.1 In FN W Bn , the following cochains are defined for k 0, m > 0, and n > 0: A direct calculation shows that both k;m and ı n have trivial differential, and thus define cohomology classes k;n 2 H m.2 k 1/ .W B m2 k ; F 2 / and ı n 2 H n .W B n I F 2 /, that we still denote, with a slight abuse of notation, by the same symbols. While the proof of this fact is entirely straightforward, we provide a proof for the sake of completeness.
Lemma 5.2 k;m and ı n are cocycles in FN W B m2 k˝F 2 and FN W Bn˝F 2 , respectively.
Proof k;m is represented by the symmetric planar level tree in Figure 4. We prove that d i . k;m / D 0 for all 0 Ä i < m2 k by considering different cases: If i ¤ l2 k for 0 Ä l < m, the addend d i of the differential identifies two edges adjacent in a vertex v j for 1 Ä j Ä m, and performs a vertex shuffle at the new vertex. Exactly two possible vertex shuffles yield the same tree. Hence d i . k;m / D 0.
If i D l2 k for some 1 Ä l < m, then d i . k;m / is obtained by gluing together v l and v lC1 and shuffling the outgoing edges of these two vertices. Since all these shuffles yield the same tree, and there is an even number of them -precisely 2 kC1 2 k -we have again that d i . k;m / D 0.
If i D 0, v 1 and its mirror vertex are glued to the central axis of the tree, and the corresponding outgoing edges are permuted with a symmetric shuffle. Again, there is an even number of them (precisely 2 2 k ), and thus d 0 . k;m / D 0. ı n is represented by a symmetric planar level tree with 2n C 1 leaves and a single internal vertex of height 1. The same proof used in the second case of k;m shows that d i .ı n / D 0 for all 0 Ä i < n.
Another possible point of confusion is that the symbol k;m is used in [8] to indicate a class in H m.2 k 1/ . † m2 k I F 2 /. The class we define is the image of this cohomology class of the symmetric group in H m.2 k 1/ .W B m2 k I F 2 / via the map induced by the projection W W B m2 k ! † m2 k , as we will prove later (Proposition 5.26).
We can interpret all the cohomology classes that we defined above geometrically.
: : : We observe that z X n is a proper submanifold of Y .1/ W Bn and that the Thom class of the image X n of z X n in Y .1/ =W B n is ı n . We observe that the normal bundle of X n in Y .1/ =W B n is isomorphic to Áj X n . Since restriction of vector bundles to subspaces preserve Thom classes, we deduce that, if we take Y .1/ =W B n as a model for B.W B n /, then j .l / 1 k .T .Á// D ı n , where k W .E.Áj X n /; E.Áj X n / n 0 .X n // ! .E.Á/; E.Á/ n 0 .B.Á///, l W .E.Áj X n /; E.Áj X n / n 0 .X n // ! .B.W B n /; B.W B n / n X n / is a tubular neighborhood of X n in B.Á/, and To prove this claim, we first observe that we can use a slightly different model for B.Á/. We recall that there is a tubular neighborhood z N of z X n in Y .1/ W Bn determined by an embedding of the total space of the normal bundle. Explicitly, we can define the embedding by the formula .x 1 ; : : : ; x n / . 1 ; : : : ; n / 2 z X n R n 7 ! .x 1 C 1 e 1 ; : : : ; x n C n e 1 /; where e 1 is the first element of the canonical basis of R 1 . Hence z N D˚.x 1 ; : : : ; x n / j .x i .x i / 1 e 1 / ¤˙.x j .x j / 1 e 1 / for all 1 Ä i < j Ä n; .
Note that the action of W B n preserves z N , and z N is provided with a stratification induced from that on Y N is still contractible, and therefore we can use its quotient N D z N =W B n as an alternative model for B.W B n /. In this model, the inclusion l is an isomorphism. Thus we do not need to worry about excision maps, and this simplifies the argument. The claim now follows by observing that i and kjp are homotopic. An explicit homotopy is ..x 1 ; : : : ; x n /; ; t/ 2 z N W Bn R n OE0; 1 Definition 5.4 Let n 1 and m 0. We define ı 0 nIm 2 H n .W D nCm I F 2 / as the restriction of ı nˇ1m 2 H .W B nCm I F 2 / to the cohomology of W D nCm . We also let ı 0 0Wm be the unique nonzero class in H 0 .W D m I F 2 / for all m 0.
We will require some other generators that do not arise as restrictions of cohomology classes of W B n .
Definition 5.5 Given k; m 1, we define two cochains in FN m2 k W Dn : Lemma 5.6 C k;m and k;m are cocycles.
Proof The cochain equivalence ' of Lemma 4.1 maps k ;m to The same proof used for Lemma 5.2, with the additional requirement of keeping track with orientations, shows that these cochains in FN 0 m2 k W Dn are cocycles. As ' is injective, k ;m must also be a cocycle. An alternative proof can be obtained by directly using the De Concini and Salvetti description of the boundary in C W Dn and dualizing.
A consequence of the previous lemma is that C k;m and k;m represent cohomology classes, that, once again, we denote by the same symbols with a slight abuse of notation.
To adapt our notation to Giusti and Sinha's for the alternating groups, we will refer to C k;m (resp. k;m ) for some k and m as positively (resp. negatively) charged generators, and to ı 0 nWm for some n and m as neutral generators.

Relations
This subsection is devoted to deriving algebraic relations between the generators defined above. We will mainly obtain the relations as a consequence of the results in Section 4.
We first focus on A B . We can retrieve in A B the same relations among the classes k;m that appear in Giusti, Salvatore and Sinha's Theorem 2.3.

Proposition 5.7
The following formulas hold in A B : . k;m / D X i Cj Dm k;i˝ k;j ; k;nˇ k;m D nCm n Á k;nCm : Proof We use the chain-level formulas computed in Propositions 4.9 and 4.11.
To compute the coproduct, we represent k;m by the symmetric planar level tree depicted in Figure 4. Note that P l . k;m / is trivial for l 2 and that the 0-pruning map gives P 0 . k;m / D P i Cj Dm k;i˝ k;j . Therefore it is enough to prove that P 1 . k;n / D k;n˝I , where I is the trivial symmetric level tree, for all k 0 and n > 0. Consider a 1-pruning .T 0 ; T 00 / of k;m . Every vertex v i , for 1 Ä i Ä m, as depicted in Figure 4 corresponds to a vertex u i of height 1 in T 0 . Let 2 k n i be the number of outgoing edges of u i for some integer 0 Ä n i < 2 k . We can obtain the pruning .T 0 ; T 00 / from k;m by applying a sequence of elementary 1-prunings at each vertex v i and their mirror vertices r.v i / that prunes away a i outgoing edges from v i and n i a i outgoing edges from r.v i /, for some 0 Ä a i Ä n i . Therefore, summing over all the possible shuffles of leftovers, whose number is we deduce that .T 0 ; S 1 .T 00 // appears in P 1 . k;m / with coefficient X 0Äa 1 Än 1 ;:::;0Äa m Än m This number is even unless n i D 0 for all 1 Ä i Ä m, yielding the trivial 1-pruning.
The transfer product formula follows directly from the application of the cochain-level map of Proposition 4.11, by observing that k;m has m principal 0-blocks all equal to OE1; : : : ; 1, where the entry 1 is repeated 2 k 1 times, and that it has no principal l-block for l 1. Thus k;nˇ k;m is given by a single basis element in FN W B2 k .nCm/ (representing k;nCm ) counted as many times as the number of .n; m/-shuffles, that is the binomial coefficient appearing in the equation.
We can obtain coproduct formulas for ı n via the same geometric description. The following is again a consequence of the formulas in Lemmas 4.9 and 4.11. Proof Since all the entries of ı n are equal to 1, the cochain-level coproduct map on ı n reduces to the 1-pruning map P 1 and provides the desired formula. We compute the transfer product as in Proposition 5.7, by observing that ı n has no principal l-blocks for l ¤ 1, and has n principal 1-blocks all empty.
These relations will suffice to describe A B completely. We restate here our main result, which we will prove in Section 7.
Theorem 5.9 (main theorem for type B) The Hopf ring A B is generated by classes k;n (with k 0 and n > 0) and ı n (with n > 0) with the relations described in Propositions 5.7 and 5.8, together with the following additional relations: the product of generators from different components is 0; 0;n is the -product unit of the n th component.
We now turn our attention to A D . A trick borrowed from [9, page 9] can be used to simplify the presentation of this almost-Hopf ring. We recall that there is an involution ÃW A D ! A D . We can define A 0 D to be the bigraded F 2 -vector space defined by .A 0 D / n;d D H d .W D n I F 2 / if .n; d / ¤ .0; 0/ and .A 0 D / 0;0 D F 2 f1 C ; 1 g. We can embed A D as a vector space in A 0 D by identifying the nonzero class in H 0 .W D 0 I F 2 / with 1 C C 1 .

Lemma 5.10
The following statements are true in A D : (1) Ã.xˇy/ D Ã.x/ˇy D xˇÃ.y/, Proof (1)ˇis commutative, and the following diagram induces a pullback of finite coverings at the level of classifying spaces: (2) This follows from the cocommutativity of and the commutativity of the diagram above.
(3) It is the cohomological consequence of the diagonal map being equivariant with respect to the conjugation c s 0 .

Proposition 5.11
Write the coproduct of every element x 2 A D in A D as so is the reduced coproduct. By letting 1 1 C D 0, 1 1 D 1 , 1 C 1 C D 1 C , 1 ˇ1 D 1 C and .1˙/ D 1 C˝1˙C 1 ˝1 , the almost-Hopf ring structure on A D extends to an almost-Hopf ring structure on A 0 D such that 1 ˇx D Ã.x/, 1 x D 0 Proof Using the formulas in the statement of this proposition, we can extendˇand uniquely to two commutative products on A 0 D and to a unique cocommutative coproduct on A 0 D . The coassociativity of follows from Lemma 5.10(3). The associativity of on A 0 D is obvious. The bialgebra structure of A 0 D with and follows from the bialgebra structure on A D and (2) of the previous lemma. Moreover, the fact that the transfer product with 1 is associative follows from (1). Hopf ring distributivity with classes involving a transfer product with 1 follows again from (3) of the result referenced above.
Instead of determining a presentation for A D , we calculate a presentation for A 0 D because we can write it more concisely. For example, k;m D 1 ˇ C k;m in A 0 D ; thus the formulas for k;m arise as a direct consequence of the formulas for C k;m and the almost-Hopf ring structure of A 0 D . The two approaches are equivalent. The coproduct formula for ı 0 nWm is a consequence of Proposition 5.8, the Hopf ring properties of A B , and the fact that the restriction map W A B ! A D preserves coproducts.
Regarding transfer product, we prove the first identity using Proposition 4.15 precisely in the same way as the second part of Proposition 5.7.
Let W A B ! A D be the restriction map. For every x 2 H .W B n I F 2 / and y 2 H .W B m I F 2 /, we can prove that .x/ˇ .y/ D 0 in H .W D nCm I F 2 / with the same argument used in [9, Proposition 3.14]. Essentially, it is sufficient to observe that both the restriction H .W B n W B m I F 2 / ! H .W D n W D m I F 2 / and the transfer H .W D n W D m I F 2 / ! H .W D nCm I F 2 / factor through the cohomology of the subgroup G D W D nCm \ .W B n W B m /, and that the composition is 0 for mod 2 coefficients because W D n W D m has even index in G. In particular, nontrivial transfer products of blocks obtained by cup-multiplying neutral generators must be 0. The last relation also follows from the invariance of ı 0 nWm with respect to the involution Ã.
After these coproduct and transfer product formulas, we will also need some cup product relations. Since the Fox-Neuwirth type cell complex does not behave well with cup products, we found that it is simpler to obtain these formulas via restriction to elementary abelian subgroups. This approach is fruitful because of a detection theorem for these subgroups. We postpone the proof of the following proposition to Section 6, where we will explain this in detail. The last relation we require involves the behavior of the coproduct with the transfer product. We need a preliminary remark. Let b 2 A 0 D be an element obtained as a cup-product of positively and neutrally charged generators (ie C k;m or ı 0 nWm ), with at least one positively charged generator. Note that, by Propositions 5.12 and 5.13, .b/ can be written as a sum where 0 is the expression described above.
The proof of the analog of this proposition is done in [9] by a careful examination of certain spectral sequences. It can be done this way also for A D . Still, we decide to argue here using detection by elementary abelian subgroups that for finite Coxeter groups comes for free and leads to a shorter proof. Therefore, we postpone the proof of this proposition until the next section.
We restate our presentation theorem for A 0 D , whose proof we postpone.
Theorem 5.15 (main theorem for type D) A 0 D is generated , as an almost-Hopf ring, by the classes ı 0 nWm for n 0 and m 0, C k;m for k; m 1, and 1 defined above, under the relations described in Propositions 5.12, 5.13 and 5.14 and the relations 1 ˇ1 D 1 C , 1 1 D 1 , 1 C 1 D 0 and .1 / D 1 C˝1 C 1 ˝1 C coming from Proposition 5.11.

Additive bases
We describe here additive bases for A B and A D . In this subsection, we assume that the statements of Theorems 5.9 and 5.15 are true. They do not rely logically on the existence of such bases in A B and A D . Thus this choice does not invalidate their proof.
We begin with A B .
where m is a positive integer, 2 n divides m, and n is the maximal index such that n; m 2 n appears in b with a nonzero exponent. The profile of b is the .nC1/-tuple .t 0 ; : : : ; t n /. We also allow n D 0: in this case, b D ı t 0 m for some t 0 0. A Hopf monomial is a transfer product of gathered blocks x D b 1ˇ ˇb r . We denote by M B the set of Hopf monomials whose constituent gathered blocks have pairwise different profiles. Note that, given a possible profile .t 0 ; : : : ; t n /, for all l 1, there is a unique gathered block b in the .l2 n / th component having that profile. As a notational convention, we denote it b l;t .
We can describe elements of M B graphically. We represent k;n as a rectangle of width n2 k and height 1 2 k and ı n as a rectangle of width n and height 1. The width of a box is the number of the component to which the class belongs. Its area is its cohomological dimension. We understand the cup product of two generators as stacking the corresponding boxes on top of the other. In contrast, their transfer product corresponds graphically to placing them next to each other horizontally. The profile of a gathered block is described by the height of the rectangles of the corresponding column. Thus, we can represent every gathered block as a column made of boxes with the same width. Hence, an element of M B is a diagram consisting of columns placed next to each other, such that there are not two columns that consist of rectangles of the D˝1 C˝C 1i same height. Following the notation of Giusti, Salvatore and Sinha [7], we call these objects B-skyline diagrams or, more concisely, skyline diagrams where it is clear that we are considering the Hopf ring A B .
As in [7], the coproduct and the two products in M B have a graphical description, derived from our relations: We divide rectangles corresponding to ı n or k;n in n equal parts via vertical dashed lines. The coproduct is then given by dividing along all vertical lines (dashed or not) of full height and then partitioning the new columns into two to make two new skyline diagrams.
The transfer product of two skyline diagrams is given by placing them next to each other and merging every two columns with constituent boxes of the same heights, with a coefficient of 0 if the widths of these columns share a 1 in their binary expansion.
To compute the cup product of two diagrams, we consider all possible ways to split each into columns, along vertical lines (dashed or not) of full height. We then match columns of each in all possible ways up to automorphism and stack the resulting matched columns to build a new diagram.
We depict some examples of calculations with skyline diagrams in Figure 5. We prove this formula by induction on the number of cup-product generators constituting b i;t : for single generators ı m or k; m 2 k the formula appears in the set of relations for A B , and the induction step is a consequence of the bialgebra structure formed by and . Thus, the graphical procedure for the calculation of the coproduct is correct on single-column skyline diagrams. As a general skyline diagram represents the transfer product of its columns, the general algorithm is justified because andˇform a bialgebra.
Regardingˇ, the transfer product of two Hopf monomials corresponds to the horizontal juxtaposition of the corresponding skyline diagrams. Thus, we only need to justify the merging of columns. In formulas, this reads as follows. Fix a profile t D .t 0 ; : : : ; t n /, with t k 0 for 0 Ä k < n and t n > 0. Then Again, we prove this by induction on r D t 0 C C t n . For r D 1, gathered blocks with profile t are single generators, and the formula above is exactly our transfer product relation among them. For r > 1, the induction step is proved by combining the coproduct formula for l;.i Cj /2 n l , Hopf ring distributivity, and the fact that cup products of elements in different components is 0 to deduce that l;.iCj /2 n l .b i;tˇbj;t / D .b i;t l;i 2 n l /ˇ.b j;t l;j 2 n l /; or the analogous formula with ı i Cj in place of n;.iCj /2 n l if n D 0.
The -product algorithm above graphically encodes Hopf ring distributivity.
Finally, we prove that M B is an additive basis for A B . We consider the bigraded vector space V over F 2 with skyline diagrams or, equivalently, M B as a basis. Define linear mapsˇ; W V˝V ! V and W V ! V˝V by computing their values on basis elements via the algorithm above. Note that these maps define a Hopf ring structure on V . There is a map V ! A B that realizes every Hopf monomial as the corresponding element of A B . Since the procedures to compute the structural morphisms on M B are deduced from the Hopf ring structure of A B and the relations of Theorem 5.9, this map is a morphism of Hopf rings. We also note that V is generated as a Hopf ring by single rectangles, corresponding to k;n and ı n , and that the relations of Theorem 5.9 are satisfied in V . Since A B is presented by such generators and relations, it follows that the map V ! A B is an isomorphism.
We now construct an additive basis for A D , assuming Theorem 5.15. The first step is to identify the subalgebra of A D under the cup product generated by neutral generators. Let z B 0 be the set of Hopf monomials x 2 A B of the form x D ı a 1 k 1ˇ ˇı a r k r , ordered with a 1 > > a r and k 1 2. These correspond to skyline diagrams in which only boxes of height 1 appear and in which the highest column has width strictly bigger than 1. ; a 2 ; : : : ; a r 1 ; a r ; : : : ; a r " ƒ‚ … k r times / By identifying B 0 \ H .W B n I F 2 / with a subset of N n this way, the lexicographic ordering on N n induces a total order on B 0 . We observe that Q n i D1 .ı iˇ1n i / a i is a linear combination of elements of B 0 . In this linear combination, the maximal nonzero Hopf monomial corresponds to P n i D1 a i ; P n i D2 a i ; : : : ; a n 1 C a n ; a n . Moreover, this belongs to z B 0 if and only if a 1 D 0, ie if and only if ı 1ˇ1n 1 does not appear as a factor. Since these are all different, ı n ; ı n 1ˇ11 ; : : : ; ı 1ˇ1n 1 generate, under the cup product, a polynomial subalgebra with basis B 0 \ H .W B n I F 2 /. By Proposition 5.13, the kernel of the restriction map to H .W D n I F 2 / on this subalgebra is the ideal generated by ı 1ˇ1n 1 . Consequently, the images of elements of z B 0 in A 0 D are a basis for the cup product subalgebra generated by the elements ı 0 nWm . Since the transfer products of these elements are trivial and this subalgebra is closed under coproduct by Proposition 5.12, this is a subalmost-Hopf ring.
Definition 5. 19 We call a neutral gathered block in A D an element b 2 A 0 D obtained as the image in A 0 D of an element of the set z B 0 considered in the previous lemma. A positively charged gathered block, or simply positive gathered block, is an element of the form b D .ı 0 2 n mW0 / t 0 Q n i D1 . C k;m2 n k / t k , for some n; m 1, t k 0 for 0 Ä k < n and t n > 0. The profile of b is .t 0 ; : : : ; t n /. A negatively charged gathered block, or simply negative gathered block, is an element of the form b D .ı 0 2 n mW0 / t 0 Q n iD1 . C k;m2 n k / t k , for some n; m 1, t k 0 for 0 Ä k < n and t n > 0. The profile of b is .t 0 ; : : : ; t n /. A Hopf monomial is a transfer product of gathered blocks. Note that, given a possible profile t D .t 0 ; : : : ; t n /, for all l 1, there is a unique positively (resp. negatively) charged gathered block in the .l2 n / th component having that profile. As a notational convention, we denote it by b C l;t (resp. b l;t ). Moreover, we stress that we require that a positively charged generator and a negatively charged one do not appear in the same gathered block. This is not a restriction since, due to Proposition 5.13, a cup product of two such generators is 0, or we can write it as a transfer product of gathered blocks. Therefore Hopf monomials generate A 0 D as an F 2 -vector space.
We also define a filtration of A 0 D that we will use to extract an additive basis from this set of (linear) generators. We first compute formulas for the coproduct and transfer product of gathered blocks in A 0 D . These are essentially the charged versions of the corresponding identities in A B , except for gathered blocks involving the generators 1 ;n , for which this is true only in the graded space gr F .A 0 D / associated with the weight filtration. Complete formulas in A 0 D are complicated and can be retrieved recursively on the filtration F . Lemma 5.21 Let n 1. Let t D .t 0 ; : : : ; t n / with t k 0 for all 0 Ä k < n and t n > 0. In any almost-Hopf ring satisfying the relations of Theorem 5.15 the following statements are true for all i; j > 0: (3) and (6) are true in gr F .A 0 D / even if n D 1.
Proof (1) Recall that, by definition, k;m D 1 ˇ C k;m . Combining the link between transfer product and coproduct provided by Proposition 5.14 with the coproduct formula for 1 and C k;m , we deduce that with the convention that k ;0 D 1˙. Then, we can prove that 1 ˇb C i;t D b i;t by induction on the number of cup-product factors of the involved gathered block. If b C i;t is a single generator C k;m , the statement holds by definition. The induction step is deduced from Hopf ring distributivity and the coproduct formula derived above for k;m , using that 1 1 D 1 , 1 1 C D 0 and that the cup product of elements in different components is zero. The statement for negatively charged gathered blocks is obtained from its analog for positively charged ones by taking the transfer product of both members of the identity with 1 and using the relation 1 ˇ1 D 1 C .
(2) We begin with the case of positively charged gathered blocks b C m;t . We proceed, again, by induction on the number of generators appearing in the expression of b C m;t . If b C m;t is a single generator, then the statement holds by the coproduct identities of Proposition 5.12. The induction step follows from the fact that and form a bialgebra, and relations 1,2,3 of Proposition 5.13. For instance, for k 2, we explicitly have We only need to be careful when k 1 > 1 because C 1;l 1;l is not necessarily 0. Note that for k 2 we have by Hopf ring distributivity k ;r C 1;2 k 1 r 1;2 k 1 r D k ;r .. C 1;2 k 1 r 1 / 2ˇı0 2W0 / D 0; because the coproduct of k ;r does not have an addend x 0˝x00 with the component of x 00 equal to 2. This observation guarantees that, if n > 1, the mixed-charge terms vanish. Even if n D 1, we obtain the additional terms by applying relation 2 of Proposition 5.13 to expressions of this form, and this procedure lowers weights. Thus, the desired formula holds in gr F .A 0 D / in this case. The formulas for negatively charged gathered blocks are, once again, obtained by applying the transfer product with 1 .
(3) The formula is easily deduced from the coproduct formulas (2) by induction on the number of -product generators appearing in bi ;t . In the case n D 1, we use the obvious fact thatˇpreserves the weight filtration to deduce that the desired formula holds in the graded space.
(5) This is a combination of (1) and the relations 1˙ˇ1˙D 1 C and 1˙ˇ1 D 1 .
(6) If n > 1, it follows directly from relation (1) of Proposition 5.13. If n D 1, assume that b C i;t 2 F a and b j;u 2 F b . Relation (2) of Proposition 5.13 provides a way to write b C i;t b j;u as a product of the form ..
i;t 0 2 F a l and b i;u 0 2 F b l . By relation (5) of the same proposition, these products preserve the weight filtration. Therefore the statement is true in gr F .A 0 D /. (7) We argue as we did for (1), combining the formula given in (1) with the relation 1 ˇı nWm D ı nWm , which implies that neutral gathered blocks are invariant by the action of 1 ˇ_.
Using this lemma, we can use Hopf monomials in the additive basis for A B to construct basis elements of A D by adding charges.
Before providing a proof of this statement, we make a remark that clarifies the cumbersome identity of Proposition 5.14.
Remark 5.23 Proposition 5.22 provides a direct sum decomposition of A 0 D as an F 2 -vector space with three addends, V C , V and V 0 , defined as the linear span of B C , B and B 0 , respectively. Note that the involution Ã D 1 ˇ_ switches V C and V and fixes all elements of V 0 by Lemma 5.21. We can consider the linear projection W V ! V C defined as the identity on V C and as 0 on V and V 0 . With this notation, we can rewrite Proposition 5.14 as j /, where x 0 i , x 00 i , y 0 j and y 00 j are all fixed by Ã. But this implies that these classes belong to B 0 , and thus their transfer product is zero. Consequently, such addends do not appear in the summation. This implies that z .x˝y/ is killed by the multiplication by 2, and thus z extends linearly to a map as desired. The image of z is contained in the image of the embedding W A 0 D ,! z V that maps every x 2 M D to 2x 2 z V . We can rephrase Proposition 5.14 by saying that is the unique linear map satisfying ı .ˇ/ D z . We immediately see that this statement is equivalent to the formulation above when x or y is a charged gathered block. If both x and y belong to B 0 , then z .x˝y/ D 0 because the transfer product of two neutral gathered blocks is always zero. The general case follows by induction on the number ofˇ-factors in the Hopf monomials involved.

Proof of Proposition 5.22
We can write every element in an almost-Hopf ring with generators C k;n , 1 and ı 0 nWm satisfying the relations of Theorem 5.15 as a linear combination of addends in M D due to Lemmas 5.18 and 5.21. Therefore M D is a set of linear generators for A 0 D . The fact that Hopf monomials in M D are linearly independent is a byproduct of the proof of Theorem 5.15. It is nevertheless possible to provide a fully independent proof that a basis for the almost-Hopf ring with the presentation of Theorem 5.15 has an additive basis given by M D , but we will not provide it, as it would be uselessly long.

Comparison between A † , A B and A D
In this subsection, we compute the action of the connecting homomorphisms on the elements of the additive bases determined in the previous subsection.
We first start with the link between A † and A B . We recall that there are a natural injection j W † n ! W B n and a natural projection W W B n ! † n , providing linear maps linking A B and A † . We begin by analyzing the relationship between A † and A B .
Proposition 5.24 Let j W † n ! W B n and W W B n ! † n be the natural homomorphisms. The induced maps j W A B ! A † and W A † ! A B are Hopf-ring homomorphisms.
Proof It is obvious from the fact that the diagrams The following proposition is a direct consequence of Corollary 4.6 and Proposition 5.24.
Proposition 5.25 With reference to the notation of Theorem 2.3, j . k;n / D k;n and j .ı n / D 0. More generally, given a B-skyline diagram x 2 M B , j .x/ is zero if x contains a rectangle of height 1. Otherwise, it is obtained by interpreting x as a skyline diagram in A † .
We can now use our algebraic description to compute the action of on generators.
Proposition 5.26 . k;n / D k;n . For a skyline diagram x 2 A † , .x/ is obtained by interpreting x as a B-skyline diagram without rectangles of height 1.
Proof We proceed by induction on n. If n D 1, since ı j D id, is injective. Hence . k;1 / is a nonzero class in H 2 k 1 .W B 2 k I F 2 /. Thanks to Proposition 5.24, . k;1 / is primitive. From our description of A B in terms of skyline diagrams, formalized with the statement of Proposition 5.17, we see that the only nontrivial primitive of A B in the right component and cohomological degree is k;1 . For n > 1, Proposition 5.24 guarantees that preserves coproducts. Hence we inductively have that . k;n / C k;n is primitive. However, there are no nonzero primitive in that bidegree, thus . k;n / D k;n .
We now turn to A D . There is a restriction map W A B ! A D induced by the inclusions W D n ,! W B n . Moreover, we recall that we have natural injections i C ; i W † n ! W D n determining maps A D ! A † and an involution ÃW A D ! A D induced on H .W D n I F 2 / by the conjugation with s 0 2 W B n . We analyzed these maps in Section 4.1.
First, we explain the relation between C k;m and k;m and the natural maps between W D n , W B n and † n .

Proposition 5.27
For all n; k 1 and m 0, i C . C k;n / D k;n ; i C . k;n / D 0; i . k;n / D k;n ; i . C k;n / D 0; More generally, with reference to Proposition 5.22, i C (resp. i ) is zero on all neutral or negatively (resp. neutral or positively) charged Hopf monomials. We obtain the value of positively (resp. negatively) charged Hopf monomials under i C (resp. i ) by forgetting the charge to get a Hopf monomial in M B and then applying j as described in Proposition 5.25.
Proof The formulas involving k ;n are a direct consequence of Corollary 4.6 and the form of the chain representative of k;n 2 FN † n˝F 2 retrieved in [8,Definition 4.9]. To deduce that i C .ı 0 nWm / D 0, we recall that ı 0 nWm D .ı nˇ1m / and that the composition † n i C , !W D n ,! W B n is equal to j . By Proposition 5.25 j .ı nˇ1m / D 0; therefore i C .ı 0 nWm / D 0. The same is also true for i .ı 0 nWm / because i is obtained by composing i C with the conjugation with an element of W B n , whose action is trivial on elements coming from A B .
Since we identify the involution Ã with the transfer product with 1 , the following proposition is essentially a restatement of the description of the previous subsection. To complete the description of the homomorphisms connecting our structures, we need to compute the restriction W A B ! A D and transfer trW A D ! A B maps. To do this, we need to establish preliminary identities.
Proof The first statement follows from the fact that this commutative diagram induces a pullback of covering spaces at the level of classifying spaces: Regarding the second statement, since the conjugation by s 0 is an endomorphism of the covering B.W D n W D n / ! B.W D 2n \ .W B n W B n //, Moreover, the classifying space functor applied to the following square produces a diagram homotopy equivalent to a pullback of covering, where d and d 0 are diagonal maps: . These facts imply that, denoting by d the diagonal subgroups, Similarly, the last two statements follow from the diagrams below, where the vertical maps of the first one are the diagonal morphisms: The transfer of every neutral gathered block is 0, and we realize the transfer of a Hopf monomial as the transfer product of the transfer of its constituent gathered blocks.
Proof The statement for generators is a direct consequence of their definition at the cochain level. The general claim for Hopf monomials in A D follows directly from Lemma 5.29.
Proposition 5.31 . k;n / D C k;n C k;n for all n; k 1. Moreover, .ı m / D ı 0 mW0 for n 2 and .ı 1 / D 0. More generally, for every Hopf monomial x 2 M B , .x/ can be computed as follows. If x D b l;t is a gathered block with profile t D .t 0 ; : : : ; t n /, we have that The restriction of a Hopf monomial x with a constituent gathered block in z B 0 is x 0 . We calculate the restriction of a Hopf monomial x 2 z B c as follows. First, replace every constituent gathered block in x with the sum of the positively or neutrally charged elements of its restriction. Then, write the resulting linear combination as a sum of Hopf monomials in A D . Finally, add to that the negatively charged counterpart of every positively charged Hopf monomials appearing in the sum. Proof Using the cochain-level representative of k;n introduced in Definition 5.1, we immediately see that its restriction is represented in FN 0 W Dn2 k by the sum of two elements obtained by providing this cochain with the two possible orientations. These elements correspond to cochain representatives of C k;n and k;n via the cochain equivalence ' of Lemma 4.1. The formulas for ı m are a consequence of the generators' definition in A D and relation (4) of Proposition 5.13.
We conclude this section with a short description of the Gysin sequence of the double cover W D n ! W B n . In [9], Giusti and Sinha adopt the analysis of a similar Gysin exact sequence as the starting point to compute the cohomology of the alternating groups as an almost-Hopf ring. While we retrieve that as a byproduct of our algebraic description, we stress that Giusti and Sinha's approach could be used in our framework as an alternative method to deduce relations in A 0 D . Indeed, a direct consequence of the following proposition is that M D D B 0 t B C t B is the polarized basis arising from a Gysin decomposition in the sense of [9].
where @ is the multiplication with ı 1ˇ1n 1 . It can be described on skyline diagrams by the operation of replacing each column corresponding to ı m k with the diagram corresponding to ı mC1 1ˇı m k 1 .
Proof By a general fact, the connecting homomorphism @ is the multiplication with the Euler class e of the double covering. In the case n D 1, this covering is isomorphic to the universal double covering S 1 ! P 1 .R/, and its Euler class is ı 1 . For bigger n, the Euler class is ı 1ˇ1n 1 because it is the only class in the right degree that restricts to ı 1 .
tr ı D 0 because we are working modulo 2. Therefore the transfer of a neutral gathered block is 0. If b D bl ;t is a charged gathered block, then the restriction of tr.b/ must be b C Ã.b/, and the multiplication with ı 1ˇ1n 1 must kill tr.b/. These two conditions force tr.b/ D b l;t . Since tr preserves the transfer productˇ, the formula for a general Hopf monomial follows.

Restriction to elementary abelian subgroups
We recall here some theorems from Swenson's thesis [18], which constitute the formal framework in which we will calculate the cohomology of W B n and W D n . We will then exploit these theorems to determine the restriction of our generators in A B and A D to elementary abelian 2-subgroups. This yields the restriction of all the cohomology of the groups W B n and W D n to maximal elementary abelian subgroups, because the structural morphisms of our almost-Hopf rings behave in a predictable way: cup products and coproducts are preserved by such restriction, while the relation with transfer product is determined via double cosets formulas, as stated in Adem and Milgram's book [1, Section II.6].

Quillen's theorem for finite reflection groups
The relevance of these restriction maps is encompassed by a result of Quillen [15; 16], which we state here. Let G be a finite group and F a family of subgroups. Let Â g W H .KI F p / ! H .gKg 1 I F p / be the conjugation homomorphism. Define Alternatively, we can consider F as a category in which Hom.K; K 0 / D fg j g 1 Kg Â K 0 g: Then F n is the inverse limit of the functor H n from F into the category of F p -vector spaces. In other words, F consists of collections of cohomology classes of groups in F that are stable under restrictions and conjugation by elements of G. Observe that F D L n F n has a natural ring structure.
Theorem 6.1 [15, Theorem 6.2, page 564] Let G be a finite group. Let F be as before. The map q G W H .GI F p / ! F given by q G .f / D ff j K g K is a well-defined ring homomorphism. Moreover, if F is the family of elementary abelian p-subgroups, then the kernel and cokernel of q G are nilpotent.
Hence elementary abelian p-subgroups give much information on the F p -cohomology of a group. In the case of a finite reflection group, an even stronger property holds.
Theorem 6.2 [18, Theorem 11, page 2] If G is a finite reflection group and F is the family of elementary abelian p-subgroups of G, then q G is an isomorphism.

Restriction from A B
For the reasons explained in the previous subsection, Swenson has calculated the elementary abelian 2-subgroups of W B n . Before stating his result, we need to recall the structure of elementary abelian 2-groups of the symmetric group † n on n objects. The relevant calculations are reviewed in [1]. † n admits a transitive elementary abelian 2-subgroup if and only if n D 2 k . In this case, all these subgroups are conjugated in † n to the image V k of the homomorphism k W F k 2 ,! † 2 k given by the regular action of F k 2 on itself. More generally, a maximal elementary abelian 2-subgroup of † n is conjugated to a direct product Hence, conjugacy classes of maximal elementary abelian 2-subgroups in † n are parametrized by partitions of n such that every element of is an integral power of 2 and the multiplicity of 1 D 2 0 in is at most 1.
To further simplify notation, we borrow from Swenson's thesis the following definition.
Definition 6.3 [18] Let n 2 N. We say that a partition of n is admissible if it consists only of parts that are integral powers of 2.
The main results about elementary abelian 2-subgroups in W B n is the following: We can calculate the restriction of our generating classes k;n and ı n to these abelian subgroups. The calculation for k;n has been essentially carried out by Giusti, Salvatore and Sinha [7]. We state here the result.
otherwise: Proposition 6.6 Let n 0. Let D .2 k 1 ; : : : ; 2 k r / be an admissible partition. The restriction of ı n to the cohomology of the maximal elementary abelian 2-subgroup A is equal to N r i D1 f 2 k i . Moreover, ı n is the unique class in H n .W B n I F 2 / that has this property for every .
Proof We observe that the restrictions of a cohomology class to A for all the partitions of n determine its restriction to every elementary abelian 2-subgroup (not necessarily maximal). Hence, by Theorem 6.2, a class that satisfies the condition in the statement for every is necessarily unique.
Let U n D R n be the reflection representation of W B n . Recall that, if n D 2 k and D .2 k /, then A D V k C k , where C k D hti is a cyclic group of order 2, the center of W B n , and V k D hv 1 ; : : : ; v k i Ä † 2 k is the subgroup defined above. H .A I F 2 / is polynomial on degree 1 elements x; y 1 ; : : : ; y k , the linear duals to t; v 1 ; : : : ; v k , respectively. Given a 2 A n f1g, let " a , sgn a , and Rhai be the 1-dimensional trivial representation, the signum representation, and the regular representation of hai Š F 2 , respectively. We first observe that, since t acts on U n as the multiplication by 1, U n j A Š sgn t˝U n j V k . Moreover, the inclusion of V k in † 2 k is given by the regular representation; hence where U S;i is equal to sgn v i if i 2 S , and to " v i if i … S . Thus, with the notation used before in this document, the Stiefel-Whitney class of U n j A is Y S Âf1;:::;ng Its n-dimensional part is exactly f 2 k . Hence, the thesis for D .2 k / follows from the naturality of the characteristic classes and Proposition 5.3 In the case of a general admissible partition D .2 k 1 ; : : : ; 2 k r /, the proposition follows from the fact that A Š To complete the calculation of the restriction morphisms from A B to maximal elementary abelian 2-subgroups, we need to describe how such maps behave with the structural morphisms of A B . Restrictions preserve cup products, and, regarding the coproduct, there is nothing to say because every maximal elementary abelian subgroup of W B n W B m is itself a maximal elementary abelian subgroup of W B nCm . The only nontrivial behavior occurs with the transfer product. We describe it in the following proposition. where the sum runs over all partitions f1; : : : ; rg D I t J of f1; : : : ; rg into two subsets such that P i 2I 2 k i D n (and , consequently, P j 2J 2 k j D m), and is the obvious permutation of tensor factors.
Proof We begin by assuming that r D 1; thus D .2 k / for some k and n C m D 2 k . Then, since A acts transitively on f1; : : : Given that A is abelian, the classically known property stated in [1, Proposition 5.6, page 69] implies that the transfer map is identically zero. Eilenberg's double coset formula then guarantees that the composition of the restriction with the transfer product In the general case, the restriction of xˇy to this subgroup factors through the r-fold coproduct. By the calculations above, addends in this coproduct for which a factor is a nontrivial transfer product restrict to 0. Sinceˇand form a bialgebra structure on A B , the other addends have the desired form.

Restriction from A D and proof of relations
We can adapt the argument to calculate the restriction to elementary abelian subgroups of generators also in the D n case. First, we state the analog of Proposition 6.4. Recall that a partition of n is admissible if and only if it consists of parts that are powers of 2.
x j : Conversely, if m 1 D m 2 D 0, then the W B nconjugacy class of A contains exactly two W D n -conjugacy classes of elementary abelian 2-subgroups.
We now determine the restriction of our generators to the elementary abelian subgroups. Proposition 6.9 Let n D 2 k m, for some k; m 1. Let be an admissible partition of n. Let m 1 and m 2 be the multiplicities of 1 and 2 in . Then with the convention that 1 W Br D 0 when r < 0.
Proof If has more than 1 element and is different from .1; 1; : : : ; 1/, then the restriction to y A or y A s 0 factors through the coproduct. Thus, by applying the coproduct formulas of Proposition 5.12, we can inductively reduce to these two cases.
We begin by assuming that D .2 n / has only one element, and we prove the first statement. If k 1 and n 2, the restriction of k ;2 l to A (n D k C l) must be N W D 2 n .A /-invariant. Hence, for degree reasons, it can be 0 or d 2 n 2 l . Since i C . C k;2 l / D k;2 l (resp. i C . k;2 l / D 0) by Proposition 5.27, its restriction to A \ † 2 n must be the Dickson invariant of degree 2 n 2 l (resp. 0). This forces C k;2 l j A D d 2 n 2 l D k;2 l j A (resp. k;2 l j A D 0). By essentially the same argument, considering i instead of i C , we determine the restrictions to A s 0 , proving the first point.
As in A B , the behavior of the restriction to maximal elementary abelian 2-subgroups with the cup product and coproduct is straightforward. We describe the relation between such restriction maps and the transfer product in the following proposition, which is the counterpart of Proposition 6.7. where the sum runs over all partitions f1; : : : ; rg D I t J of f1; : : : ; rg into two subsets such that P i 2I 2 k i D n (and , consequently, P j 2J 2 k j D m) and at least one between I and J does not contains any l 2 f1; : : : ; rg such that k l D 0, and where where I; J; I;J are as above, and c s 0 W H . y A I F 2 / ! H . y A s 0 I F 2 / is induced by the conjugation with s 0 .
Proof We cannot repeat the proof of Proposition 6.7 because in A 0 D the transfer product and the coproduct do not form a bialgebra. Therefore, we argue by considering Eilenberg's double coset formula associated with the two subgroups W D n W D m and y A of W D nCm . We preliminarily fix some notation. Let P be the partition of the set f1; : : : ; n C mg given by P D f1; : : : ; 2 k 1 g; f2 k 1 C 1; : : : ; 2 k 1 C 2 k 2 g; : : : ; r 1 X lD1 2 k l C 1; : : : ; n C m : Moreover, let P j; D˚P j 1 lD1 2 k l C 1; : : : ; Assume that 1 … . A set of representatives for W D nCm =.W D n W D m / is the set Sh.n; m/ f1; tg, where Sh.n:m/ Â † nCm ,! W D nCm is the set of .n; m/-shuffles, and t D s 0 s 0 2 W B n W B m . Note that y A Â . t " /.W D n W D m /. t " / 1 if and only if .f1; : : : ; ng/ is a union of parts of P . Since y A is abelian, these provide the only nonzero terms in the summation of the double coset formula. Moreover, by inspecting the image of f1; : : : ; ng Â f1; : : : ; n C mg under the signed permutation action of W D nCm Â W B nCm , we see that if t " and 0 t " 0 are two coset representatives satisfying this condition, then  The case of y A s 0 where 1; 2 … is done similarly. If 1 2 , the same argument holds, but if there exists i 2 I and j 2 J such that k i D k j D 0, then, interpreting the elements of W D nCm as signed permutations, .p i ; p i /.p j ; p j / belongs to y A but not to . I t " /.W D n W D m /. I t " / 1 , where P i; D fp i g and P j; D fp j g. Thus, we need to restrict the summation only to partitions f1; : : : ; rg D I t J in which all the occurrences of 1 in belong to the same part.
This result provides a way to detect the charge of a Hopf monomial via restriction to maximal elementary abelian 2-subgroups. We first fix preliminary notation. Proof Every positively charged gathered block b restricts to an element of H C A . Nontrivial computations arise only if b D .ı 0 2mW0 / r . C 1;m / s with r 0 and s > 0 and A D A .2/ m . In this case, with the notation of Theorem 6.8, we observe that where z i D x i C y i . Thus h 2 k m 2 H C .2/ m . Let 2 k be the biggest power of 2 smaller than s. Then h s 2 k m is a sum of pure tensors of the form w 1˝ ˝w m , where w i is a monomial in x i and z i with total degree smaller than 2 k . Therefore We see the corresponding statement for negatively charged gathered blocks by noting that conjugation with s 0 exchanges H C A and H A s 0 .
In general, a positively (resp. negatively) charged Hopf monomial x is a transfer product of gathered blocks, all positively charged (resp. all positively charged except one). Consequently, Proposition 6.10 yields the statement for x.
We can finally complete our relations for A 0 D by providing the proofs of the two leftover propositions of Section 5.2.
Proof of Proposition 5.14 Let b be a positively charged gathered block in A D and x 2 A D . From Lemma 5.10 and the definition of 0 we deduce that .b/ D 0 .b/ C .Ã˝Ã/ 0 .b/, and that .Ã.b// D .id˝Ã C Ã˝id/ 0 .b/. During this proof, we assume, by convention, that xj A D 0 when x 2 H .W D n I F 2 / and is not an admissible partition of n. Let D .2 k 1 ; : : : ; 2 k r / and 0 D .2 h 1 ; : : : ; 2 h s / be admissible partitions of some integers. From Proposition 6.10, we deduce that OE.ˇ˝ˇ/.id˝ ˝id/. 0˝ /.b˝x/j y In these equalities we used the identities of Lemma 5.10 to perform the substitutions .Ã˝Ã/.x/ D .x/ and .id˝Ã/.x/ D .Ã˝id/.x/ D .Ã.x//; t 0 is assumed to be .2 k 1 ; : : : ; 2 k r ; 2 h 1 ; : : : ; 2 h s /; I D I \ f1; : : : ; rg and J D J \ f1; : : : ; rg, while I 0 and J 0 are I \ fr C 1; : : : ; r C sg and J \ fr C 1; : : : ; r C sg suitably shifted. The sum should be over all I , J , I 0 and J 0 such that at least one between I and J does not contain an l such that k l D 0 and at least one between I 0 and J 0 does not contain an l such that h l D 0. However, since the restriction of positively charged gathered blocks is zero on elementary abelian 2-subgroups corresponding to admissible partitions containing 1, we can restrict the sum only to the terms for which k i ¤ 0 for all i 2 I and h i ¤ 0 for all i 0 2 I 0 . This condition is equivalent to I not containing 1, and we can, once again, restrict the last sum only to these terms and get the last equality.
Proof of Proposition 5.13 Using Proposition 6.9, the newly proved Proposition 5.14, Proposition 6.10, and the fact that cup products commute with restrictions, we check that the desired identity hold when restricted to maximal elementary abelian 2-subgroups. Then Theorem 6.2 yields the relations in A D .

Proofs of the main theorems
We devote this section to the proofs of the presentation theorems for A B and A D . They will be proved by comparing restrictions to elementary abelian 2-subgroups and exploiting Theorem 6.2. We will separate two technical lemmas from the proofs for the sake of clarity of exposition.
We first provide a proof for our structure theorem for A B . Proof From Swenson's description of A , stated as in Proposition 6.4, we can identify A .2 k / with the image of the diagonal embedding id d W † 2 V k 1 ! † 2 oV k 1 ! W B 2 k . Its intersection with the product A .2 k 1 ;2 k 1 / D V k 1 V k 1 is identified with the subgroup V k 1 Â † 2 V k 1 , embedded diagonally in W B 2 k .
The restriction to this subgroup maps f 2 k to .f 2 k 1 / 2 , d 2 k 2 l to .d 2 k 1 2 l 1 / 2 if l > 0, and d 2 k 1 to 0. This is known, but we sketch a proof for completeness. If we chose bases fx; y 1 ; : : : ; y k g of H 1 .A .2 k / I F 2 / and fx; y 1 ; : : : ; y k 1 g of H 1 .A .2 k 1 / I F 2 / as in Section 6.2, the restriction is given by x 7 ! x, y i 7 ! y i if 1 Ä i < k and y k 7 ! 0. The polynomial F k .t/ D Q v2H 1 .V k IF 2 / .t C v/ in H .V k I F 2 /OEt restricts to .F k 1 .t // 2 . Since f 2 k D F k .x/, we deduce the formula for f 2 k . The identities for d 2 k 2 l are obtained from this by using the classical identity F k .t/ D P k i D0 t 2 i d 2 k 2 i .
Proof of Theorem 5.9 Let A 0 B be the Hopf ring generated by k;m and ı m with the desired relations. Since the relations mentioned above hold in A B , there exists an obvious morphism ' W A 0 B ! A B .
We need to fix a total ordering Ä on the set P n of admissible partitions of n such that, for all ; 0 2 P n , 0 > if 0 is a refinement of . In other words, Ä extends the partial ordering given by refinement. Let b be a nontrivial gathered block in A B . There exist unique nonnegative integers n and m such that b D Q n i D1 a i i;2 n i m ı a 0 2 n m with a n ¤ 0. We consider the partition of 2 n m b D .2 n ; : : : ; 2 n /. Given x D b 1ˇ ˇb r 2 M B , let x D F r i D1 b i . As a consequence of Propositions 6.5, 6.6 and 6.7, xj A ¤ 0 implies that x > . Explicitly, if b D Q n i D1 a i i;2 n i m ı a 0 2 n m , Ã˝m : For any x D b 1ˇ ˇb r 2 M B , xj A n is the symmetrization of N r i D1 b i j A b i . Given a partition , let M be the set of elements x 2 M such that x D .
We first prove that ' is injective. We proceed by contradiction, and we assume that there exists a nontrivial sum P i x i of elements of M B that is 0 when restricted to every elementary abelian 2-subgroup. Let be maximal among the set of partitions f x i g i . Since, by the explicit calculation above, the restrictions of the elements of M to A are linearly independent, this gives a contradiction.
To prove surjectivity, it is sufficient, by Theorem 6.2, to prove that an element˛of the Quillen group F W Bn can be written as the image via q W Bn of a linear combination of elements of M B . Note that such an˛is determined by its values˛ on the maximal abelian 2-subgroups A . Let N ˛D maxf 2 P n j˛ ¤ 0g with respect to the chosen linear ordering. We write N ˛D .2 k 1 ; : : : ; 2 k r /. We proceed by induction on N ˛. We must have a i;j;k j ¤ 0 for all i and j . Otherwise, we can define a partition 0 obtained from N ˛b y substituting 2 k j with two parts both equal to 2 k j 1 and observe that, by Lemma 7.1, we must have˛N ˛j A N ˛\ A 0 ¤ 0. Thus˛ 0 ¤ 0 and this would contradict the maximality of N ˛. By our calculations above, since˛N ˛m ust be invariant by permutations of tensor factors, this condition guarantees the existence of an element x in the linear span of M N ˛s uch that xj A N ˛D˛N ˛. This reduces the statement tǫ 0 D˛C q W Bn .x/ for which, by construction, N ˛0 < N ˛, and completes the induction argument.
We now focus on the presentation of A D .  Proof Note that, due to Theorem 6.8 and Proposition 6.9, the Hopf monomials in M 2 \ H .W D 2 I F 2 / restrict to linearly independent elements in H . y A .2/ I F 2 /. Therefore, to prove the linear independence claim for m > 1, it is enough to check that the restrictions of the elements of M 2 \ H .W D 2m I F 2 / to H .W m D 2 I F 2 / (which is a component .2/ m of the coproduct) are linearly independent. Let F be the weight filtration on A 0 D provided by Definition 5.20. It is enough to prove that this set is linearly independent when working in the associated graded spaces gr F .A 0 D / and gr F .H .W m D 2 I F 2 //. In this setting, the image of a gathered block b C l;t 2 M 2 (resp. b l;t 2 M 2 ) under gr F . .2/ m / is P " 1 ;:::;" l N m iD1 b " i 1;t , where the sum is over all l-tuples ." 1 ; : : : ; " l / with " i 2 fC; g and the cardinality of the set fi W 1 Ä i Ä l; " i D g is even (resp. odd). Combining this with Proposition 5.14, we check the claim directly. By Propositions 6.9 and 6.10, every element of M 2 restricts to 0 on y A whenever 1 2 . Therefore, it is contained in the kernel of 2;1 . We now prove the opposite inclusion. With the notation of Theorem 6.8, we write / is the ideal generated by h m and d˝m 1 D 2;1 . C 1;m C 1;m /j A .2/ m . Finally, the generators belong to the image of M 2 , the linear subspace generated by M 2 is a -subalgebra by our formulas in A 0 D , and restriction maps preserve cup products.
The effective scale (effsc) of a gathered block in the cohomology of W B n (resp. W D n ) is the least l such that n=2 l is an integer and its restriction to W n=2 l B 2 l (resp. W n=2 l D 2 l ) is nonzero, and the effective scale of a Hopf monomial x D b 1ˇ ˇb r as min r i D1 effsc.b i /. A full-width monomial is a Hopf monomial in A B (resp. A D ) of which no constituent block is of the form 1 W Bn (resp. 1 W Dn ). Sq i .ı 2 n / is the sum of all the full-width monomials x 2 M B of degree 2 n C i with ht.x/ Ä 2 and effsc.x/ 1 such that a generator of the form ı k appears in every constituent gathered block of x.
Proof The calculation for Sq i . k;2 n / is an obvious consequence of [7,Theorem 8.3,page 191]. Regarding Sq i .ı 2 n /, since ı 2 n is the top-dimensional Stiefel-Whitney class of the reflection representation U 2 n by Proposition 5.3, by Wu's formula Sq i .ı 2 n / D w i .U 2 n /ı 2 n . Defining, by convention, k;0 D 1, let u i D X j 0 ;:::;j n 0; P n 1 rD1 2 r j r Cj n Cj 0 D2 n P n 1 rD1 .2 r 1/j r Cj n Di n 1 K rD1 r;j rˇı r nˇ1 W B j 0

:
We computed the restriction of w i .U 2 n / to the maximal elementary abelian subgroups A in the proof of Proposition 6.6. It coincides with the restriction of u i by our previous calculations based on Proposition 6.5. Thus, Sq i .ı 2 n / D w i .U 2 n /ı 2 n D u i ı 2 n ; and this class is exactly the sum of all the desired Hopf monomials x.
Regarding the calculation of the Steenrod squares on the generators of A D , We observe that the calculation for Sq i .ı nWm / is implicit in Theorem 8.2 since ı nWm D .ı nˇ1m / and commute with Steenrod operations. Thus we only need to consider generators of the form k ;n . Theorem 8.3 Let k; n 1 and i 0. Then, in A D , Sq i . C k;n / (resp. Sq i . k;n /) is the sum of all the full-width monomials x 2 B C (resp. x 2 B ) of degree 2 nCk 2 n C i with ht.x/ Ä 2 and effsc.x/ l in which generators of the form ı nWm do not appear.
Proof We recall that Definition 6.11 provides, for all maximal elementary abelian 2-subgroup A Â W D 2n , subspaces H C A and H A of the cohomology of A. A direct calculation shows that Sq i .h n / 2 H C .2/ n . Since restrictions preserve the Steenrod squares, Sq i . C 1;n / is mapped to an element of H C A for all maximal elementary abelian 2-subgroups A Â W D 2n and all choices of i and n. Similarly, the restriction of Sq i . 1;n / to every such subgroup A lies in H A . Let x C (resp. x ) be the sum of all the positively (resp. negatively) charged Hopf monomials considered in the statement. By Proposition 6.12, the restriction of x C (resp. x ) belongs to H C A (resp. H A ).
Moreover Sq i . C k;n / C Sq i . k;n / D .Sq i . k;n //. Consequently, Theorem 8.2 implies that Sq i . C k;n / C Sq i . k;n / D x C C x . Since H C A \ H A D 0 for all A, the two facts above guarantee that Sq i . C k;n / D x C and Sq i . k;n / D x .