I know that the fundamental group of the Möbius strip $M$ is $\pi_1(M)=\mathbb{Z}$ because it retracts onto a circle. However, I am trying to show this using Van Kampen's theorem.
As usual I would take a disk inside the Möbius band as an open set $U$ and the complement of a smaller disk as $V$. Then $\pi_1(U)=0$ and $\pi_1(U\cap V)=\langle \varepsilon\mid\rangle = \mathbb{Z}$. The puncture Möbius strip retracts to a circle given by the boundary, so $\pi_1(V)=\langle \alpha\mid\rangle = \mathbb{Z}$.
When I compute $i_{2*}:\pi_1(U\cap V)\to\pi_1(V)$ I get that $\varepsilon\mapsto \alpha^2$, because when we go around in the polygon representation we pass through the boundary twice in the same direction.
However, this would implye that $\pi_1(M)=\langle \alpha\mid \alpha^2\rangle = \mathbb{Z}/2\mathbb{Z}$.
What I am doing wrong? I guess my computation of $i_{2*}$ is wrong somehow, but I don't see why.