= Euler characteristic of a paired-polygon surface cover
{title2=$\chi(M)=|T|(1-s+\sum_jq_j^{-1})$}
For a paired $2s$-gon with $r$ base <vertex cycles of a paired polygon>, there are one face, $s$ edges and $r$ vertices on the base, so $\chi(M_0)=1-s+r$. The full finite-group construction has $|T|$ faces, $s|T|$ edges and $\sum_j|T|/q_j$ vertices, giving the displayed ordinary <Euler characteristic>. If all lifted angles are $2\pi$, the <orbifold Euler characteristic> formula gives equivalently $\chi(M)=|T|\chi_{\rm orb}(M_0)$.
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