I am reading the proof of $\mathbb{Z}\approx\pi_1(S^1)$ from Hatcher and didn't understand the last paragraph in the picture (the homomorphism part):
Isn't $\tau_m\tilde{\omega}_n:I\to \Bbb{R}$ and $\tau_m\tilde{\omega}_n(s)=ns+m$? So what is the image of $\tau_m\tilde{\omega}_n$ under $p$? It must be something like $\omega_?(s)=(\cos 2\pi (ns+m), \sin 2\pi (ns+m))$. But this loop doesn't look like $\omega_n$. Could you give me more detail? What part did I misunderstand?