Can you help me to compute the fundamental group of $\mathbb{R}^3\setminus X$, where $X = \mathbb{S}^1\cup \{(0,0,z)\in\mathbb{R}^3\mid z\in\mathbb{R}\}\ ?$
I'm sorry if I repeated the question and I'm sorry if the question it's not well written and ... thanks in advance ! I'd say that the fundamental group is isomorphic to $F_2$, but I'm not even sure if the intuition is right or not.