Fig. 7.

Example of the labeling induced by formula (11). We assume that $f(\tau )=\tilde{\sigma }$ where $\sigma$ is a simplex of $X^{\mathit{sc}}$ with three different vertices $V_0{V}_1{V}_2$. In this example, the tree connects $R-V_0-V_1$ as well as $R-V_0-V_2$ and the edge $V_1{V}_2$ is not in the tree. On the right, we give the induced labeling of vertices of ${\mathrm{Esd}}_k(\tau )$ which determines a simplicial map to $X^{\mathit{sc}}$. The bottom and left faces of $\sigma$ are in the tree, hence the bottom and left extended faces in ${\mathrm{Esd}}_k(\tau )$ are all mapped into R. The right face of $\sigma$ is the edge $V_1{V}_2$ that is not in the tree: the corresponding right extended face in ${\mathrm{Esd}}_k(\tau )$ is mapped to a loop $R-V_0-V_1-V_2-V_0-R$, where $V_1{V}_2$ is the only part that is not in the tree. The bold edges are sent to the edge $V_1{V}_2$. The most interior simplex in ${\mathrm{Esd}}_k(\tau )$ is highlighted and is the only one mapped to $\sigma$