Let $M$ be a connected discrete surface. From the last post we know $\textrm{dim Im }d_1=\#F-1$ and we already knew $\textrm{dim Ker }d_0=1$. From $\textrm{Im }d_0 \subset \textrm{Ker }d_1$ we see that the first Betti number $\beta_1(M) := \textrm{dim Ker … Continue reading Euler characteristic and genus