Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$.
I am trying to use the Van Kampen theorem but I do not know how to apply it correctly, how do I find $U, V$ in such a way that $X=U\cup V$ and $U,V, U\cap V$ is connected by paths? Or could you calculate this fundamental group by finding a deformation retraction whose fundamental group is known? Thank you very much.
The map $H:X\times [0,1]\to X$ given by the equation $$ H(x,t)=(1-t)x + t \frac{x}{\| x \|}$$ shows that $Y:=\mathbb{S}^2-\{(1,0,0),(0,1,0),(0,0,1),(-1,0,0),(0,-1,0),(0,0,-1)\}$ is a deformation retract of $X.$ Moreover, via the stereographic projection, one can show that there is a homeomorphism of $Y$ onto $\mathbb{R}^2$ with 5 points removed. Using Seifert-van Kampen, one can also show that the latter has fundamental group a free group with 5 generators.