3, 5 \}$ we have $H_2(B,\Z_p) = H_2(P,\Z_p)^W = (H_2(P,\Z)^W) \otimes \Z_p$. But $0=H_2(B,\Q) = H_2(P,\Q)^W = H_2(P,\Z)^W \otimes \Q$ whence $H_2(P,\Z)^W = 0$ and $H_2(B,\Z_p) = 0$. Now $H_1(B,\Z) = \Z$ is torsion-free, hence $H_2(B,\Z_n) \simeq
H_2(B,\Z) \otimes \Z_n$ for any $n$ by the universal coefficients theorem. Since $H_2(B,\Z)$ is a $\Z$-module of finite type this yields $H_2(B,\Z) = \Z_6$ and completes the computation for $G_{31}$.

Embeddings between Artin-like monoids {#sectembed}
-------------------------------------

We end this section by proving a few lemmas concerning submonoids, which will be helpful in computing differentials in concrete cases.

We consider Garside monoids with set of generators $S$ and endowed with a length function, namely a monoid morphism $\ell : M \to \N = \Z_{\geq 0}$ such that $\ell(x) = 0 \Leftrightarrow
x = 1$ and $\ell(s) = 1$ for all $s \in S$. We consider the divisibility relation on the left (that is $U | V$ means $\exists m \ V = U m$) and recall that such a monoid admit lcm’s (on the left).

Let $M,N$ be two such monoids, and $\varphi : M \to N$ a monoid morphism such that

1.  $\forall s \in S \ | \ \varphi(s) \neq 1$

2.  