fundamental group of complex Projective Line minus points

Question

I want to compute the fundamental group of $$\mathbb{P}^1(\mathbb{C})\setminus \{a,b,c\} $$ Since $\mathbb{P}^1(\mathbb{C})$ is homeomorphic to $\mathbb{S}^2$, this is a punctured sphere and thus i think the fundamental group should be a finitely generated group by 2 generators. I wanted to ask, if someones knows how I can formally compute this fundamental group (like with Seifert-van-Kampens thrm.)

Answer 1

It's the free group on two generators. $S^2$ minus three points is homeomorphic to $\Bbb R^2$ minus two points. Divide $\Bbb R^2$ in two open half spaces that intersect in a stripe, in such fashion that each half-space contains exactly one of the two points you are removing. $\Bbb R^2\setminus\{b,c\}$ is therefore union of two connected open sets with fundamental group $\Bbb Z$ and simply connected intersection. By Seifert-Van Kampen the fundamental group is isomorphic to $\Bbb Z*\Bbb Z$.