Let $V$ be a finite real vector space, $W$ a subspace. What is the fundamental group of $V-W$?

Question

I am not sure is this the same with $\Bbb R^n-\Bbb R^m$. Even in this case I am not sure about how to compute the fundamental group. When $n$ and $m$ are less than $3$, everything thing seems OK. But what about $n$ and $m$ larger?

Answer 1

Every finite-dimensional real vector space is linearly homeomorphic to some $\Bbb R^n$, so we do not lose generality in solving the problem for $\Bbb R^n \setminus \Bbb R^m$. Normalization gives that $\Bbb R^n \setminus \Bbb R^m$ is homotopically equivalent to $\Bbb S^{n-m-1}$, so: $$\pi_1(\Bbb R^n \setminus \Bbb R^m) = \begin{cases} \Bbb Z, & \text{if } m = n-2,\\ \{0\}, & \text{otherwise.}\end{cases}$$