$0 < a_1 < a_2 < \dots < a_r$, and we choose $y = \widetilde{p(x)}$ to be $y = (a_1^{\frac{1}{d}},\dots,
a_r^{\frac{1}{d}})$. The above loop thus lifts under $r$ to the path $(a_1 e^{2 \ii \pi t/d},a_2,\dots,a_r)$ in $\mathcal{M}^{\#}(d,r)$. By definition of $\psi$, the class of this path $\sigma = \psi^{-1}(\xi_0^2)$. Similarly, we can determine $\psi^{-1}(\xi_i)$ when $i \geq 1$ : the image of $\xi_i$ under $p$ is a path in $\mathcal{M}^{\#}(r)$ homotopic to

On the open cone described in the picture, the map $z \mapsto z^d$ is a positive homeomorphism, and this enables one to lift this path to

meaning that $\tau_i = \psi^{-1}(\xi_i)$ is the class of this path, from $(a_1^{1/d}, \dots, a_i^{1/d}, a_{i+1}^{1/d}, \dots, a_r^{1/d})$ to $(a_1^{1/d}, \dots, a_{i+1}^{1/d}, a_{i}^{1/d}, \dots, a_r^{1/d})$. We recall that ${{\mathrm{B}}}(de,e,r)$ is defined by $\pi_1( \mathcal{M}^{\#}(de,r)/G(de,e,r))$ when $d > 1$. If moreover $e=1$, then ${{\mathrm{B}}}(d,1,r)$ is generated by $\sigma,
\tau_1,\dots,\tau_{r-1}$ ; in general, it is generated by $\sigma^e,
\tau_1,\dots,\tau_{r-1}$. Now, the morphism $\tilde{\psi} : {{\mathrm{B}}}(d,1,r) \into
{{\mathrm{Br}}}(r+1) = \pi_1(\mathcal{M}(r+1)/\mathfrak{S}_{r+1})$ commutes with the natural morphisms $$\xymatrix{ {{\mathrm{B}}}(d,1,r) \ar[rr] \ar[d] & & {{\mathrm{Br}}}(r+1) \ar[d] \\
G(d,1,r) \ar[rr] \ar[dr] & & \mathfrak{S}_{r+1} \\ 
 & \mathfrak{S}_r \ar[ur] & }$$ Letting as in [@BMR] $\xi'_1 = \xi_0^2 \xi_1 