Compute the fundamental group of $\mathbb{C} \setminus \{0\}$

Question

Compute the fundamental group of $\mathbb{C} \setminus \{0\}$.

Can you give me a hint on how to approach these kind of exercises?

Answer 1

The loops in $\mathbb C\setminus\{0\}$ are completely characterised up to homotopy by how many times they wind around $0$, which can be any integer. Furthermore, adding two loops with winding numbers $a$ and $b$ produces a loop with winding number $a+b$. Therefore the fundamental group of $\mathbb C\setminus\{0\}$ is $\mathbb Z$.

Answer 2

$\Bbb C\setminus\{0\}$ has the circle $S^1$ as a deformation retract. Thus $\pi_1(\Bbb C\setminus\{0\})=\pi_1(S^1)=\Bbb Z$.