Exercise 12, § 5

a) Let D be a right of $\mathbf{R}^3$ and let $U =\mathbf{R}^3-$ D. Prove that U is arcwise connected and that for every point $a\in U$, the group $\pi_1(U, a)$ is isomorphic to $\mathbf{Z}$.

b) Let P be a plane of $\mathbf{R}^3$, let C be a circle contained in P and let $V =\mathbf{R}^3-$ C. Prove that V is arcwise connected and that, for every point $a\in V$, the group $\pi_1(V, a)$ is isomorphic to $\mathbf{Z}$. (Apply the theorem of van Kampen to the closed covering of $\mathbf{R}^3$ formed by the two half-spaces delimited by P; one can also note that the spaces U and V are homeomorphic.)