Orbifold Euler characteristic (${\chi }_{\mathrm{orb}}$)

For a compact topological surface with cone-point orders ${\alpha }_1,\dots ,{\alpha }_k$, the orbifold Euler characteristic is ${\chi }_{\mathrm{orb}}=\chi (\text{underlying surface})-{\sum }_i(1-1/{\alpha }_i)$. An orbifold covering multiplies this quantity by its degree.