# fundamental group of the complement of lines in C^2

Here is an interesting problem, which I could not solve and would appreciate any comment in solving it;

Assume that we have omitted infinitely many lines from $\mathbb{C}P^2$ to obtain $\mathbb{C}^2$ and now, consider the following lines in which $(z,w)$ is the coordinate of $\mathbb{C}^2$: $z=0, z=1, w=0, w=1, z=w.$

What is the fundamental group of the complement of these lines in $\mathbb{C}^2$?

What you have is an arrangement of hyperplanes in $\mathbb C^2$ obtained by complexifying a real arrangement. There is a simple way to describe the fundamental group of its complement, due to Randell. You can find the details and references in the beautiful book Arrangement of hyperplanes by Peter Orlik and Hiroaki Terao, in section 5.3.