that each $\mu_i$ as well as the product $\mu_1 \dots \mu_n$ lie in $\mathcal{D}$. The differential from the free $\Z M$-modules $\Z M \mathcal{D}_n$ to $\Z M \mathcal{D}_{n-1}$ is given by $$\partial_n [\mu_1|\dots|\mu_n] = \mu_1 [\mu_2|\dots|\mu_n]
+ \sum_{i=1}^{n-1} (-1)^i [\mu_1,\dots,\mu_i \mu_{i+1},\dots,\mu_n]
+ (-1)^n [\mu_1|\dots|\mu_{n-1}]$$ This complex in general has larger cells than the previous one. Its main advantage for us is that the definition of the differential is simpler, and does not involve many recursion levels anymore.

Application to the exceptional groups
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When $W$ is well-generated, meaning that it can be generated by $n$ reflections, where $n$ denotes the rank of $W$, then $B$ is the group fractions of (usually) several Garside monoids that generalize the Birman-Ko-Lee monoid of the usual braid groups. These monoids have been introduced by D. Bessis in [@BESSISKPI1] and call there dual braid monoids. They are determined by the choice of a so-called Coxeter element $c$. Such an element is regular, meaning that it admits only one eigenvalue different from 1 with the corresponding eigenvector outside the reflection hyperplanes. A Coxeter element is a regular element with eigenvalue $\exp(2 \ii \pi/h)$, where $h$ denotes the (generalized) Coxeter number for $W$, namely its highest degree 