Let $L_i$ ($i=1,2,3$) be three different lines passing through $0 \in \mathbb C^2$, and $D = \bigcup_i L_i$. What is $\pi_1(\mathbb C ^2 \setminus D)$ ?
Remark : I think if the three lines intersects each other at different points, then we have $\pi_1(\mathbb C ^2 \setminus D) = \mathbb Z^3$. Also the group we are looking for is not abelian.
What I tried : this space is a fiber bundle over $\mathbb C \setminus \{ 0,1\}$ with fiber $\mathbb C^*$. Alternatively, it's a fiber bundle over $\mathbb C^*$ with fiber $\mathbb C \setminus \{0,1\}$. We get a short exact sequence $\{e\} \to \mathbb Z \to \pi_1(\mathbb C^2 \setminus D) \to \mathbb Z \times \mathbb Z \to \{e\}$. But I don't know if it's split or not.
Actually, it's also possible to retract this space on $S^3$ minus three circles, but it's unclear how to use it. Also Van Kampen doesn't seems to work here.