complex from an implementation by Jean Michel in CHEVIE.

However, for $G_{31}$, $G_{33}$ and $G_{34}$, the complexes obtained are too large to be dealt with completely through usual computers and software. The one missing for $G_{31}$ and $G_{33}$ are the middle homology $H_2(B,\Z)$ for $G_{31}$ and $H_3(B,\Z)$ for $G_{33}$. For $G_{33}$ the Dehornoy-Lafont complex for $G_{33}$ is however computable in reasonable time, and its small size enables to compute the whole homology by standard methods. For $G_{31}$, for which there is so far no construction analogous to the Dehornoy-Lafont complex, we used the following method for computing $H_2(B,\Z)$.

We first get $H_2(B,\Q) = 0$ by computing the second Betti number from the lattice. Indeed, recall from [@OT] (cor. 6.17, p.223) that the Betti numbers of $X/W$ can be in principle computed from the lattice of the arrangement. Precisely, the second Betti number of $X/W$ is given by $\sum_{Z \in T_2} |\mathcal{H}_Z/W_Z| -1$ where $T_i$ is a system of representatives modulo $W$ of codimension $i$ subspaces in the arrangement lattice ; for $Z$ such a subspace, $\mathcal{H}_Z = \{ H \in \mathcal{A} \ | \ 
H \supset Z \}$, $W_Z = \{ w \in W \ | \ w(Z) = Z 