The fundamental group of three-dimensional space with its axes removed

Let \(L\) be the union of the three coordinate axes. Since \(0\in L\), every point of \(X=\mathbb R^3\setminus L\) is nonzero. Radial projection gives a deformation retraction \[ X\longrightarrow S^2\setminus(L\cap S^2). \] The six points of \(L\cap S^2\) are \[ \pm e_1,\ \pm e_2,\ \pm e_3. \] Hence \(X\simeq S^2\setminus\{6\text{ points}\}\). Removing one of these points identifies the punctured sphere with \(\mathbb R^2\); removing the remaining five points gives \(\mathbb R^2\) minus five points, which deformation retracts onto a wedge of five circles. Therefore \[ \boxed{\pi_1(X)\cong F_5}. \]