I know that $\mathbb R^3-\text{circle}$ has a fundamental group equal to $\mathbb Z$ and I was wondering if some things could be said about $\mathbb R^n-\text{circle}$ for $n>3$. In particular, I'd like to show that it is simply connected.
Do you know if it is true and how to prove it ?
Thanks !
P.S: I have tried to show that it strong retracts to a simpler space but all my attempts failed.
For $n>3,\;\pi_1(\mathbb{R}^n-S^1)\cong0$ . Intuitively because $n$ is at least 4 ( and 1+1<4) we know that any two loops can be homotoped to be disjoint therefore, nullhomotopic loops will stay null homotopic after removing $S^1.$
$\textbf{Idea:}$ Van-Kampen's theorem can be used to show that the inclusion of $\mathbb{R}^n-K\hookrightarrow S^n-K$ induces an isomorphism on $\pi_1.$ Then it follows from $S^n-S^m\simeq S^{n-m-1}$ if $n-m>2$ the fundamental group remains trivial. (Shown here: Fundamental group of $S^n \setminus S^m$)