Euler characteristic of a connected sum of surfaces.

Question

How do I see that the Euler characteristic of a connected sum of surfaces $S_1$ and $S_2$ is given by$$\chi(S_1 \# S_2) = \chi(S_1) + \chi(S_2) - 2?$$

Answer 1

To define the connected sum of $S_1$ and $S_2$, consider a triangulation $T_1$ of $S_1$ and $T_2$ of $S_1$, remove a triangle $t_1\in T_1, t_2\in T_2$ and glue along the boundaries of $t_1$ and $t_2$. You obtain a triangulation of $S_1\# S_2$ induced by $T_1$ and $T_2$.

If $s_i$ is the number of vertices of $T_i$, $a_i$ the number of edges of $T_i$ and $n_i$ the number of triangles of $T_i, i=1,2$;

The number of vertices of this triangulation is $s_1+s_2-3$ since you identify one vertex of $t_1$ with one vertex of $t_2$,

The number of edges of this triangulation is $a_1+a_2-3$ since you identify one edge of $t_1$ with one edge of $t_2$,

The number of triangles is $n_1+n_2-2$ since you remove $t_1$ and $t_2$.

We deduce that: $\chi(S_1 \# S_2)=(s_1+s_2-3)-(a_1+a_2-3)+(n_1+n_2-2)=\chi(S_1)+\chi(S_2)-2$.