Group homomorphism on unit circle For $n\in \mathbb{Z}$, define the map $f_n:S^1\to S^1$ as $f_n(z)= z^n$, where the unit circle $S^1$ is observed as the subspace $\{z\in\mathbb{C}|\ |z|=1\}$.
How would one compute the induced group homomorphism $(f_n)_*\pi_1(S^1,1)\to \pi_1(S^1,1)$ with the identification of $\pi_1(S^1,1)$ with $\mathbb{Z}$?

You would start out with an explicit description of the isomorphism between $\mathbb Z$ and $\pi_1(S^1, 1)$, i.e. you would write down a loop $\alpha$ representing the class that corresponds to $1\in\mathbb Z$. Then you would see what happens to that class under $(f_n)_\ast$, i.e. you would look at $f_n\!\circ\!\alpha$. You would try to express this loop in terms of $\alpha$. You might try to see if drawing images for $n=-1,0,1,2,3$ helps.
