exercise about fundamental group

Question

I have to calculate fundamental group of $R^3\setminus$(2 parallel line and one transversal line). The line are represented in the figure 1. Can you help me?

Answer 1

This arrangement is isotopic to $\mathbb R^3 \setminus L$ where $L$ is the union of three lines through the origin. We may intersect this with a unit $3$-ball and consider the resulting space (which is a deformation retract.)

Using a radial projection from the origin, it is also clear that $(\mathbb R^3 \setminus L) \cap D^3$ retracts onto $S^2 \setminus \{6 \mathrm{\,points}\}$.

This is homeomorphic to the disk with $5$ points removed, which retracts onto the wedge of $5$ circles, and by the the Van-Kampen theorem, its fundamental group is $F_5$, the free group on $5$-generators.