Decomposing a surface $S$ with a simple closed curve $\Gamma$

Question

In class we learned about how the Euler characteristic changes when we take a connected sum of surfaces $M_1$ and $M_2$: $$\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - 2,$$

and it made me wonder how the Euler characteristic would change if we decomposed $S$ by cutting along a simple closed curve (a non-intersecting curve that ends where it begins).

I have two questions:

Suppose we have a surface $S$ with Euler characteristic $\chi(S)$ and a simple closed curve $\Gamma$ on $S$. If we decompose $S$ into two surfaces $S_1$ and $S_2$ by cutting along $\Gamma$, how do the Euler characteristics, $\chi(S_1), \chi(S_2)$ relate to $\chi(S)$?

What if $\Gamma$ decomposes $S$ into one surface $S_0$ with Euler characteristic $\chi(S_0)$? How does $\chi(S_0)$ relate to $\chi(S)$?

Answer 1

Imagine that you have a cell decomposition of $S$ in which the circle $\Gamma$ is a subcomplex, and so $\Gamma$ consists of $k$ zero dimensional cells and $k$ one dimensional cells for some integer $k \ge 1$. Now cut along $\Gamma$ to get $S_1 \cup S_2$. Count cells:

• ($\#$0-cells of $S) + k = (\#$0-cells of $S_1$) + ($\#$0-cells 0f $S_2)$
• ($\#$1-cells of $S) + k = (\#$1-cells of $S_1$) + ($\#$1-cells 0f $S_2)$
• ($\#$2-cells of $S) = (\#$2-cells of $S_1$) + ($\#$2-cells of $S_2$)

Take the first line, minus the second line, plus the third line, and you get

• $\chi(S) = \chi(S_1) + \chi(S_2)$

ADDED: For your second question, a similar logic prevails:

• ($\#$0-cells of $S) + k = (\#$0-cells of $S_0$)
• ($\#$1-cells of $S) + k = (\#$1-cells of $S_0$)
• ($\#$2-cells of $S) = (\#$2-cells of $S_0$)

and so

• $\chi(S) = \chi(S_0)$