Let $m
Consider $\mathbb{R}^m$ as a subspace of $\mathbb{R}^n$ via $\mathbb{R}^m\times \{(0,0,...0)\}$.
Any suggestions on how to compute $\pi_1(\mathbb{R}^n\backslash\mathbb{R}^m)$?
I have no idea how to tackle this in the general case, and for the computation of fundamental groups, Van Kampen is the only real method at my disposal so far.