Exercise 14, § 5

Let P be a plane of $\mathbf{R}^3$, let C be a circle contained in P and let D be a line of $\mathbf{R}^3$. Let $U =\mathbf{R}^3-(C\cup D)$ and let $a$ be a point of U.

(a) Prove that U is arcwise connected.

(b) Suppose that D does not meet the convex hull of C. Prove that $\pi_1(U, a)$ is a free group with two generators.

(c) Suppose that D does not meet C but that it meets the convex hull of C. Prove that $\pi_1(U, a)$ is isomorphic to $\mathbf{Z}^2$.

(d) Suppose that D meets C at a unique point. Prove that $\pi_1(U, a)$ is a free group with two generators.

(e) Suppose that D meets C at two points. Prove that $\pi_1(U, a)$ is a free group with three generators.