Filippo Callegaro

The K(p,1) problem for the affine Artin group of type Bn and its cohomology

In this paper we prove that the complement to the affine complex arrangement of type \widetilde{B}_n is a K(\pi, 1) space. We also compute the cohomology of the affine Artin group G of type \widetilde{B}_n with coefficients over several interesting local systems. In particular, we consider the module Q[q^{\pm 1}, t^{\pm 1}], where the first n-standard generators of G act by (-q)-multiplication while the last generator acts by (-t)-multiplication. Such representation generalizes the analog 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor fibre. The cohomology of G with trivial coefficients is derived from the previous one.

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type An, B n and A n with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B n with coefficients over the module Q[q ±1 ,t ±1 ]. Here the first n - 1 standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiros lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type An as well as the cohomology of the classical braid group Br n with coefficients in the n-dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(π, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

An Explicit Description of Coxeter Homology Complexes

Rains (2010) computes the integral homology of real De Concini-Procesi models of subspace arrangements, using some homology complexes whose main ingredients are nested sets and building sets of subspaces. We think that it is useful to provide various different descriptions of these complexes, since they encode relevant information about the homotopy type of the models and there still are interesting open questions about ℤ-bases of the homology modulo its torsion (see the work by Rains (2010)). In this paper we focus on the case of the Coxeter arrangements: we give an explicit and elementary description, in terms of the combinatorics of the Coxeter groups, of the cells and of the boundary maps of these complexes.

Cohomology of Artin groups of type tilde{A}_n, B_n and applications

We consider two natural embeddings between Artin groups: the group G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} of type A_n. The cohomologies of these groups are related, by standard results, in a precise way. By using techniques developed in previous papers, we give precise formulas (sketching the proofs) for the cohomology of G_{B_n} with coefficients over the module Q[q^{+-1},t^{+-1}], where the action is (-q)-multiplication for the standard generators associated to the first n-1 nodes of the Dynkin diagram, while is (-t)-multiplication for the generator associated to the last node. As a corollary we obtain the rational cohomology for G_{tilde{A}_n} as well as the cohomology of Br_{n+1} with coefficients in the (n+1)-dimensional representation obtained by Tong, Yang and Ma. We stress the topological significance, recalling some constructions of explicit finite CW-complexes for orbit spaces of Artin groups. In case of groups of infinite type, we indicate the (few) variations to be done with respect to the finite type case. For affine groups, some of these orbit spaces are known to be K(pi,1) spaces (in particular, for type tilde{A}_n). We point out that the above cohomology of G_{B_n} gives (as a module over the monodromy operator) the rational cohomology of the fibre (analog to a Milnor fibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.

The S n +1 Action on Spherical Models and Supermaximal Models of Tipe A n −1

In this paper we recall the construction of the De Concini–Procesi wonderful models of the braid arrangement: these models, in the case of the braid arrangement of type An−1, are equipped with a natural S n action, but only the minimal model admits an ‘hidden’ symmetry, i.e. an action of Sn+1 that comes from its moduli space interpretation. We show that this hidden action can be lifted to the face poset of the corresponding minimal real spherical model and we compute the number of its orbits. Then we provide a spherical version of the construction of the supermaximal model (see Callegaro, Gaiffi, On models of the braid arrangement and their hidden symmetries. Int. Math. Res. Not. (published online 2015). doi: 10.1093/imrn/rnv009), i.e. the smallest model that can be projected onto the maximal model and again admits the extended Sn+1 action.