# Finding fundamental groups of two spaces.

What are the fundamental groups of

1. $$\Bbb R^3 \setminus \{\rm { circle} \}$$.
2. $$\Bbb R^3 \setminus \{\rm {two\ disjoint\ circles} \}$$.

• The stereographic projection of any circle on the sphere not passing through the north pole would be a circle on the complex plane which is homeomorphic to $\Bbb R^2$. Hence $\Bbb R^3 \setminus \{\rm {circle\ not\ passing\ through\ the\ origin } \}$ is homeomorphic to $\Bbb R^2 \setminus \{\rm {circle} \}$. Hence they have isomorphic fundamental groups. – math maniac. Nov 28 '18 at 7:53
• $\pi_1 (\Bbb R^3 \setminus \{\rm {circle} \}) \simeq \mathbb Z$. – math maniac. Nov 28 '18 at 10:18