# Fundamental group of $\mathbb{C^*}/G$

Calculate the fundamental group of $$\mathbb{C^*/G}$$ where $$G=\{\phi^n:n\in \mathbb{Z}\}$$ is the group of the homeomorphisms s.t. $$\phi (z)=2z$$ and $$\mathbb{C^*}=\mathbb{C}-\{0\}$$

Any hints on how to attack this exercise? thanks

• could you be a bit more explicit about the definition of $G$? – Pink Panther Mar 10 at 21:51
• I cannot that is how the exercise is presented to me, however i made some mistakes it should be $\mathbb{C^*}=\mathbb{C}-\{0\}$ – Alfdav Mar 10 at 21:55

Hint Construct a homeomorphism $$\Phi : \Bbb C^* \stackrel{\cong}{\to} \Bbb S^1 \times \Bbb R_+$$ through which $$G$$ factors into a (continuous) action on the second factor, or more precisely, for which there is an action $$\ast$$ of $$G$$ on $$\Bbb R_+$$ such that $$\Phi(g \cdot z) = (\Phi_1(z), g \ast \Phi_2(z))$$ for all $$g \in G$$. Then, $$\pi_1(\Bbb C^* / G) \cong \pi_1(\Bbb S^1 \times (\Bbb R_+ / G)) \cong \pi_1(\Bbb S^1) \times \pi_1(\Bbb R_+ / G),$$ so it remains only to understand the simpler quotient space $$\Bbb R_+ / G$$.