Fundamental group of $\Bbb R^3$ minus a plane passing through the origin.

Question

I have solved a problem from Hatcher's book which can be found here Fundamental group of $\mathbb{R}^3$ \ finite number of lines passing through origin.

So, motivated by this problem, I was thinking about problem like computation of fundamental group of $\Bbb R^3-\{ax+by+cz=0\}$ And generalization of this problem as like I mentioned above.

Can you please give me any idea how to solve this problem?

Thanks in advance.

Answer 1

$\Bbb R^3 \setminus P$ where $P$ is a plane is just a disjoint sum of two copies of $\Bbb R^3$. If $p$ is the base point for the fundamental group, it's in one of these copies so the fundamental group is just the one for that copy (as a loop only lies inside that copy for connectedness reasons) and thus trival (as the space is contractible), i.e. $\{0\}$.