# Compute the fundamental group of the complement of the three coordinate axes in $\mathbb{R}^3$, giving explicit generators.

This is problem 10-5 from John Lee's ITM.

Compute the fundamental group of the complement of the three coordinate axes in $$\mathbb{R}^3$$, giving explicit generators.

I figured out that the space $$X$$ is homotopic to the sphere minus six points. And further since the sphere minus six points is homeomorphic to the plane minus five points, which is homotopic to the bouquet of $$5$$ circles, $$X$$ has fundamental group isomorphic to $$\mathbb{Z}^{*5}$$. Hence it is generated by $$5$$ loops, each corresponding to a circle around a distinct half-axis. However, I cannot see or imagine how the $$6th$$ loop, that is the loop around the one left half-axis can be generated by a free product of $$5$$ other generators. I would greatly appreciate any help with understanding the explicit generator for this group.