Let $ X = \mathbb R^3 \setminus A$, where $A$ is a circle. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all – I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is path connected on which to use van Kampen.
Any help would be appreciated. Thanks
Best Answer
I am not sure whether there is a nicer choice but this is how I think about it. Intuitively the fundamental group should be $\mathbb Z$ - a path may jump through the hoop a couple of times or not. I choose the open sets to model this somewhat. One open set is the interiour of a filled torus with the circle lying on the surface. The other set is the whole of $\mathbb R^3$ with the closed disk (bounded by the circle) removed. Then the first set contracts to a circle, the second set contracts to a sphere and the intersection is contractible.
Edit: To make the sets more precise: $$U=\mathbb R^3-D^2\simeq S^2$$ such that $$A=\partial D^2\subseteq D^2$$ and $$V=int(S^1\times D^2)\simeq S^1$$ such that $$A=\ast\times \partial D^2\subseteq S^1\times D^2.$$ Then $$U\cap V=int(S^1\times D^2-\ast\times D^2)\cong int(I\times D^2)\simeq\ast$$