Up to sign, the Chern class of the $T(N_{m_{0}})$-fibre bundle $q_{1}:Y_{\Sigma}-q_{1}^{-1}(\Delta^{\prime})\rightarrow Y_{\Sigma}/T(N_{m_{0}})-\Delta^{\prime}$ is the 1-chain $c_{1}$ on $\Delta^{\prime}\cup\{\infty\}$ described as follows. Choose an orientation on the plane containing $P$. View $\Delta^{\prime}$ as contained in this same plane as the 1-skeleton of the dual cell complex of the triangulation of $P$ determining $\Sigma$. Each oriented edge $E$ of $\Delta^{\prime}$ intersects a unique edge $\langle\tau_{i},\tau_{j}\rangle$ in the triangulation of $P$, oriented so that $E$ and $\langle\tau_{i},\tau_{j}\rangle$ intersect positively with respect to the chosen orientation on $P$. The 1-chain $c_{1}$ assigns to the oriented edge $E$ the element $\tau_{j}-\tau_{i}$ of $N_{m_{0}}$. (See Figure 3.4) This $1$-chain has the property that all coefficients of edges are primitive in $N_{m_{0}}$, and the three coefficients associated to edges with a common vertex span $N_{m_{0}}$.

Figure $3.4$