---
author:
- 'Filippo Callegaro & Ivan Marin'
bibliography:
- 'biblio.bib'
date: 'November 18, 2010'
title: '[**Homology computations for complex braid groups**]{}'
---

Scuola Normale Superiore\
Piazza dei Cavalieri, 7\
56126 Pisa\
Italy\
—\
Institut de Mathématiques de Jussieu\
Université Paris 7\
175 rue du Chevaleret\
75013 Paris\
France

[**Abstract.**]{} Complex 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.

[**MSC 2010 : 20J06, 20F36, 20F55**]{}

Complexes from Garside theory
=============================

We recall a few homological constructions from the theory of Garside monoids and groups. Recall that a Garside group $G$ is the group of fractions of a Garside monoid $M$, where Garside means that $M$ satisfy several conditions for which we refer to [@dehpar]. In particular, $M$ admits (left) lcm’s, and contains a special element, called the Garside element. We denote $\mathcal{X}$ the set of atoms in $M$, assumed to 