how to calculate the fundamental group using intitutive ideas? I am trying to calculate the fundamental group of $\mathbb{R}^3- \{ x\text{-axis}\cup y\text{-axis}\cup z\text{-axis}\}$.
Idea: I think we can show it deformation retracts on 2-sphere minus 4 points.

Since sphere delete one point is homeomorphic to $R^2$, so $S^2$\ $\{ X-axis, Y-axis,Z-axis\}$ is homotopy equivalent to unit disk $D^2$\ $\{ 5 points\}$, so fundamental group will be free product $\mathbb Z*\mathbb Z* \mathbb Z* \mathbb Z* \mathbb Z$
