We were taught in class that $\pi(S^1)=\mathbb{Z}$
I am a bit confused as to why this is true. The motivation behind this was that if we have $\alpha_1$ moving around the circle once in a clockwise direction, and $\alpha_3$ moving around three times then $\alpha_1 \cdot \alpha_3=\alpha_4$.
My question is why is $\alpha_1$ not homotopic to $\alpha_3$?
It seems a that if we just moved around the cirlce at $\frac{1}{3}$ the speed we could easily "deform" $\alpha_3$ in to $\alpha_1$.