Fundamental Group of $\mathbb{R}^3\setminus S$ where $S$ is the unit circle in the $x-y$ plane.

Question

I have been asked to find the fundamental group of $\mathbb{R}^3\setminus S$ where S is the unit circle in the $x-y$ plane. I have started to use Seifret-Van Kampen and so I constructed the spaces: $$U=\mathbb{R^3}\setminus\{(x,y,0)\in\mathbb{R^3}:x^2+y^2\leq1\}$$ $$V=\{(x,y,z)\in\mathbb{R^3}:x^2+y^2<1,|z|<1\}$$ That is, the real 3-plane with the unit disk removed, and the open cylinder. Both of these spaces are simply connected, but I'm confused as to how to find $\pi_1(U\cap V)$ as it is two disjoint cylinders and so I can't use Seifret-Van Kampen as they have an empty intersection.

Answer 1

Here's a solution that does not uses S-V theorem directly.

First, the space is homotopic to the 2-dimensional sphere where two poles are identified (try to imagine this), which is again homotopic to the wedge sum of $S^2$ and $S^1$. Hence S-V theorem gives $\pi_1(X) \simeq \pi_1(S^1) * \pi_1(S^2) \simeq \mathbb{Z}$.