# Fundamental group of complement space

Let $$I=[-1,1],$$ and $$J=\partial I=\{-1,1\},$$ and consider $$A=(I\times I\times J)\cup(I\times J\times I)$$ ($$A$$ is a cube without two parallel faces). The fundamental group $$\pi_1(C)$$ is $$\mathbb{Z}$$ since $$A$$ deformation retracts to the square $$\{0\}\times I\times I.$$ I want to calculate $$\pi_1(\mathbb{R}^3-A).$$ Since I usually have problem with these kind of problems, I chose this one to make some questions. Are there some usual techniques to solve these problems: in particular based on van Kampen or deformation retracts? Are there some links between $$\pi_1(X)$$ and $$\pi_1(Y-X)?$$

Let $$S^2$$ be a sphere of radius $$2$$ centred at the origin, and let $$L$$ be the straight line segment connecting the points $$(2, 0, 0)$$ and $$(-2, 0, 0)$$. The space $$R^3 - A$$ deformation retracts onto $$S^2 \cup L$$, whose fundamental group is $$\mathbb Z$$.
Hatcher, page 46, example 1.23 contains pictures of these kinds of deformation retractions for a number of examples, including $$\mathbb R^3 - S^1$$, which is very similar to the present example.