What are the fundamental groups of

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

Please help me in this regard. Thank you very much.

  • 1
    $\begingroup$ 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. $\endgroup$ – math maniac. Nov 28 '18 at 7:53
  • $\begingroup$ $\pi_1 (\Bbb R^3 \setminus \{\rm {circle} \}) \simeq \mathbb Z$. $\endgroup$ – math maniac. Nov 28 '18 at 10:18

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