1,720,995 research outputs found
Salvetti complex, spectral sequences and cohomology of Artin groups
Get PDFThe aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented
The homology of the Milnor fiber for classical braid groups
No full textIn this paper we compute the homology of the braid groups, with coefficients in the module Z[q(+/- 1)] given by the ring of Laurent polynomials with integer coefficients and where the action of the braid group is defined by mapping each generator of the standard presentation to multiplication by -q.
The homology thus computed is isomorphic to the homology with constant coefficients of the Milnor fiber of the discriminantal singularity
Homology of the family of hyperelliptic curves
Get PDFHomology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus g with one boundary component, that are double coverings of the disk ramified over n = 2g+1 points. The main part of such homology is described by the homology of the braid group with coefficients in a symplectic representation, namely the braid group Brn acts on the first homology group of a genus g surface via Dehn twists. Our computations show that such groups have only 2-torsion. We also investigate stabilization properties and provide Poincaré series, for both unstable and stable homology
Families of superelliptic curves, complex braid groups and generalized Dehn twists
Get PDFWe consider the universal family End of superelliptic curves: each curve Σnd in the family is a d-fold covering of the unit disk, totally ramified over aset P of n distinct points; Σnd↪End→Cn is a fiber bundle, where Cn is the configuration space of n distinct points. We find that End is the classifying space for the complex braid group of type B(d, d, n) and we compute a big part of the integral homology of End, including a complete calculation of the stable groups over finite fields by means of Poincaré series. The computation of the main part of the above homology reduces to the computation of the homology of the classical braid group with coefficients in the first homology group of Σnd, endowed with the monodromy action. While giving a geometric description of such monodromy of the above bundle, we introduce generalized 1d-twists, associated to each standard generator of the braid group, which reduce to standard Dehn twists for d = 2
Homology computations for complex braid groups
Get PDFComplex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincaré polynomial with coefficients in a finite field for one large series of such groups, and compute the second integral cohomology group for all of them. As a consequence we get non-isomorphism results for these groups
On the cohomology of Artin groups in local systems and the associated Milnor fiber
No full textLet W be a finite irreducible Coxeter group and let X-W be the classifying space for G(W), the associated Artin group. If A is a commutative unitary ring, we consider the two local systems L-q and L'(q) over X-W, respectively over the modules A[q, q(-1)] and A[[q, q(-1)]] ,given by sending each standard generator of G(W) into the automorphism given by the multiplication by q. We show that H*(X-W, L'(q)) = H*(+1) (X-W, L-q) and we generalize this relation to a particular class of algebraic complexes. We remark that H*(X-W, L'(q)) is equal to the cohomology with trivial coefficients A of the Milnor fiber of the discriminant bundle of the associated reflection group
The integer cohomology algebra of toric arrangements
Get PDFWe compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain: – a “combinatorial” version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements,– a uniqueness result for realizations of arithmetic matroids with at least one basis of multiplicity 1
- …
