For a vector subspace $W$ of $\mathbb R^n$, where $n \ge 4$ and $\dim W\le n-3$, is $\mathbb R^n\setminus W $ simply connected?

Question

Let $W$ be a vector subspace of $\mathbb R^n$, where $n \ge 4$ and $\dim W\le n-3$. Then is $\mathbb R^n\setminus W $ simply connected ?

I can only see that it is path connected.

Please help.

Answer 1

Let $U$ be a complement (so $\Bbb R^n=W\oplus U$). Any Loop in $\Bbb R^n\setminus W$ can be moved to $U\setminus \{0\}$ and contracted there because $\dim U\ge 3$.