basis $\mathcal{X}_n$. A differential $\partial_n : C_n \to C_{n-1}$ is defined recursively through two auxiliary $\Z$-module homomorphisms $s_n : C_n \to C_{n+1}$ and $r_n : C_n \to C_n$. Let $[\alpha,A]$ be a $(n+1)$-cell, with $\alpha \in \mathcal{X}$ and $A$ a $n$-cell. We let $\alpha_{/A}$ denote the unique element in $M$ such that $(\alpha_{/A}) lcm(A) = lcm(\alpha,A)$. The defining equations for $\partial$ and $r$ are the following ones. $$\partial_{n+1}[\alpha,A] = \alpha_{/A} [A] - r_n(\alpha_{/A} [A]), \ \ r_{n+1} = s_n \circ
\partial_{n+1}, \ \ r_0(m[\emptyset]) = [\emptyset].$$ In order to define $s_n$, we say that $x [A]$ for $x \in M$ and $A$ a $n$-cell is *irreducible* if $x = 1$ and $A = \emptyset$, or if $\alpha = \md(x \lcm(A))$ coincides with the first coefficient in $A$. In that case, we let $s_n(x [A]) = 0$, and otherwise $$s_n(x[A]) = y[\alpha,A] + s_n(y r_n(\alpha_{/A} [A]))$$ with $x = y \alpha_{/A}$.

The Charney-Meyer-Wittlesey complex
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Let again $G$ denote the group of fractions of a Garside monoid $M$, with Garside element $\Delta$. Let $\mathcal{D}$ denote the set of simple elements in $M$, namely the (finite) set of proper divisors of $\Delta$. We let $\mathcal{D}_n$ denote the set of $n$-tuples $[\mu_1|\dots|\mu_n]$ such 