Closed orientable genus $g$ has $\chi=2-2g$; closed nonorientable genus $k$ has $\chi=2-k$.Suggest a slogan
Let \(g\geq 0\) and \(k\geq 1\). The closed orientable surface \(\Sigma_g\) of genus \(g\) and the closed nonorientable surface \(N_k\) of genus \(k\) have Euler characteristics \[ \chi(\Sigma_g) = 2-2g, \qquad \chi(N_k) = 2-k \] [@Hat02]. Removing the interiors of \(b\) disjoint closed discs from a closed surface lowers \(\chi\) by \(b\).
What is the Euler characteristic in terms of genus?