I need to compute the fundamental group of $\mathbb R^2 \setminus S^1$ .
First I notice this is a disconnected set, hence I will have to differentiate between two cases. If I choose the based point inside the bounded set ( the interior of $S^1$ ).
Clearly this is the trivial group since the open unit ball is convex.
On the other hand if I choose the based point $ x_{0} \in \mathbb R^2 $ with $ |x_{0}| >1$, I guess this fundamental group could be isomorphic to $\mathbb Z$ since the loops I can have are pretty similar as the ones I can get if I have the punctured plane which fundamental group is $\mathbb Z$, but I can't find a satisfactory argument for this latest assertion. Am I right so far?
The circle $S^1$ is a deformation retraction of the complement of the unit ball by $H_t(x)={1\over \|x\|}tx +(1-t)x$ so its fundamental group is $Z$.