Fundamental group of surface of genus $g$

Question

Suppose we have a compact surface of genus $g$.

How to calculate its fundamental group ?

Answer 1

You should look it up in a standard book on algebraic topology (see Lee, or see Massey). The standard way this is done is by using regular polygons. You can represent your compact surface as a certain quotient space by the edges of a polygon. Once you have that representation the rest gets easier. Massey does it in Chapter 1.

Answer 2

If the surface is orientable, then we have the formula :

$$\pi_1(S)=\langle\;a_1,b_1,...,a_g,b_g\;:\;\;[a_1,b_1]\cdot\ldots\cdot [a_g,b_g]=1\;\rangle$$

(check theorem 6.3. in the above paper)