Fundamental Group of Klein Bottle?

Question

Let $C^{*}=C \setminus \{ 0 \}$. What is the fundamental group of $C^{*}/H$, here $H=\{\psi^n;n \in \mathbb{Z}\}$ with $\psi(z) = 2 \bar{z}$?

Answer 1

Letting $\gamma(t) = e^{2 \pi i t}$ be a loop (with base point $1+0i$) representing the generator for $\pi_1(C^*)$, you get the presentation $$\pi_1(C^*/H) = \langle \gamma,\psi \,\, | \,\, \psi \gamma \psi^{-1} = \gamma^{-1}\rangle $$