Trying to prove that Möbius band can't retract onto the boundary circle, I got an idea that I must show the below thing.
If $\alpha\in\pi_1 (\partial M)$ is a generator, its image $i^*(\alpha) \in \pi_ 1 (M)$ under the inclusion $i:\partial M\to M$ is the square of an element of $\pi_1 (M)$.
I can visualize such phenomenon in my head, but can't formulate an argument using explicit homotopies and paths. Could anyone suggest me how to rigorously prove the above fact?
