Use Van Kampen theorem to show that two sets are not homeomorphic.

Question

Let $S^1$ be the unit circle, $L_1$ a line passing inside $S^1$ and $L_2$ another line passing outside $S^1$. By using Van Kampen theorem, show that $U=\mathbb{R}^3\setminus\{L_1\cup S^1\}$ and $V=\mathbb{R}^3\setminus\{L_2\cup S^1\}$ are not homeomorphic.

I know that $U\cong T=S^1\times S^1$, then $\pi_1(U)=\mathbb{Z}\times \mathbb{Z}$, but how can I compute $\pi_1(V)$?

I will appreciate any hint.

Thank you.

Answer 1

You can split $V$ into two open half spaces (whose intersection is contractible) one of which contains the missing circle and the other containing the missing line.

Both have fundamental group isomorphic to $\mathbb Z$, while the intersection has $\pi_1 = 0$. Therefore, by Van Kampen : $$\pi_1(\mathbb R^3 \backslash \{\mathbb S^1 \cup L_2\}) \simeq \mathbb Z *\mathbb Z\not\simeq \mathbb Z^2$$