This problem was asked in a previous homework set:
Show that every homomorphism $\pi_1(S^1)\rightarrow \pi_1(S^1)$ can be realized as the induced homomorphism $\varphi_*$ of a map $\varphi:S^1 \rightarrow S^1$.
Hatcher defines an induced homomorphisms as $\varphi_*:\pi_1(X,x_0) \rightarrow \pi_1(Y,y_0)$ by composing loops $f:[0,1]\rightarrow X$ based at $x_0$ with $\varphi$. So that $\varphi_*[f] = [\varphi f]$
We had already proven, after a couple days, that $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$.
I'm not sure how to use this fact, and even after staring at definitions, I'm not exactly sure about what I'm supposed to do. I just feel like I'm getting lost in the notation. Where do I start? Are there any better resources than using this book? It seems so dense. Our instructor does help with things, but still eventually tells us we need to just sit down and "work it out", which isn't working out for me with this class.
Note that the homomorphisms of $\mathbb Z$ are all of the form $n\mapsto kn$ for some $k\in\mathbb Z$, and are completely determined by the fact that $1$ maps to $k$. You want to show that each of these can be realized as as the induced homomorphism of a map $\varphi_k:S^1\to S^1$.
Intuitively, the isomorphism $\pi_1(S^1)\to \mathbb Z$ counts how many times a curve winds around the circle. Thus the curve $f(t)=e^{2\pi i t}$ corresponds to $1$, and we want to find a map which makes this curve wind around $k$ times. The simplest curve corresponding to $k$ is $f_k(t)=e^{2\pi i kt}$, so let's find some $\varphi_k$ such that $f_k=\varphi_k\circ f$. How about $\varphi_k(z)=z^k$?