Cacti, braids and complex polynomials.
Catégorification des coefficients de la matrice de la représentation de Burau
Nous catégorifions explicitement les coefficients de la matrice de la représentation de Burau en utilisant des méthodes géométriques élémentaires. Nous montrons que cette catégorification est fidèle dans le sens où elle détecte la tresse triviale.
Categorification of the virtual braid groups
We extend Rouquier’s categorification of the braid groups by complexes of Soergel bimodules to the virtual braid groups.
Chern classes of fibered products of surfaces.
Cohomology rings of Artin groups
In this paper integer cohomology rings of Artin groups associated with exceptional groups are determined. Computations have been carried out by using an effective method for calculation of cup product in cellular cohomology which we introduce here. Actually, our method works in general for any finite regular complex with identifications, the regular complex being geometrically realized by a compact orientable manifold, possibly with boundary.
Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie
A bipolynomial is a holomorphic mapping of a sphere onto a sphere such that some point on the target sphere has exactly two preimages. The topological invariants of spaces of bipolynomials without multiple roots are connected with characteristic classes of rational functions with two poles and generalized braid groups associated to extended affine Weyl groups of the serie . We prove that the cohomology rings of the spaces of bipolynomials of bidegree stabilize as tends to infinity and that...
Commensurability of graph products.
Computation of centralizers in Braid groups and Garside groups.
We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conjugacy problem given by the authors in [9], are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups [2]. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.
Coxeter group actions on the complement of hyperplanes and special involutions
We consider both standard and twisted actions of a (real) Coxeter group on the complement to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in and give explicit formulae which describe both actions on the total cohomology in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group , the Weyl groups...