Question 1: Find the Fundamental group of the complement to three infinite straight lines that have no common points in $\mathbb{R^3}$
$\mathbb{R^3}$" />Question 2 Compute the fundamental group of the complement of $S^1 \cup Z \subset \mathbb{R^3}$ where $S^1$ is the unit circle in the $xy$ plane and $Z$ is the vertical axis, see picture.
I know that the fundamental group of $(X, x_0)$ denoted $\pi_1(X, x_0)$ is the set of homotopy classes of loops in $(X, x_0)$. If $\lambda$ is a loop in $(X, x_0)$, we write $[\lambda]$ for the homotopy class $\lambda$ that is:
$[\lambda]=\{\mu | \lambda \simeq \mu \}$
$\lambda$ and $\mu$ are homotopic in $ (X, x_0)$ if there is a map $H$ such that:
$H:[0,1] \times [0,1] \rightarrow X$ satisfying:
- $H(s,0)=\lambda(s) \forall s \in I$
- $H(s,1)=\mu(s) \forall s \in I$
- $H(0,t)=H(1,t)=x_0 \forall t \in I$
I think van Kampen's theorem could be used in question 2, as we are dealing with union of two space. The theorem says $(X, x_0)$ a pointed space, $X= A \cup B$ where $A \cap B$ is path connected and contains the base point $x_0$. Then: $\pi_1(X, x_0)=\pi_1(A, x_0) *_{\pi_1(A \cap B, X_0)} \pi_1(B, x_0)$. But the problem is we are dealingn with the complement of the union. How would this change things?
Many thanks for your help, I am really struggling with these types of questions