fundamental group of $\mathbb{C^*}/\{e,a\}$

Question

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$.

My thoughts are that $\mathbb{C^*}$ is a $G$-space with $G=\{e,a\}$, so if I know what $\mathbb{C^*}/\{e,a\}$ is isomorphic to then I'll be able to figure out the fundamental group for this quotient space. But so far, we only know the fundamental group of simply-connected spaces, the circle, and products of spaces, so I guess it has to be isomorphic to one of these spaces.

Can someone please help out?

Answer 1

The quotient is homoemorphic to $\{z \in \mathbb{C}|z \ne 0, \Re(z) \ge 0\}$, which is contractible.