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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.

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  • $\begingroup$ Are you asking about this specific problem or for more general advice? $\endgroup$ – Alex Becker Apr 12 '14 at 5:18
  • $\begingroup$ Both! I do have a "sort of" solution that I did turn in, but I feel like I don't really understand my own work. Maybe it's an issue with not understanding what the question is asking..which is typical so far for this class. $\endgroup$ – DaveNine Apr 12 '14 at 5:22
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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 $\varphi_k(z)=z^k$?

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  • $\begingroup$ Hatcher kind of proves the fundemental group of $S^1$ by kind of using this winding thing, although I'm not sure why I would have thought that we would want to find a map that wraps around the circle $k$ times. The solution makes sense, but I just have a hard time getting to a point where I have an objective. Is the isomorphism you describe due to the fundemental group being isomporphic to $\mathbb{Z}$? Is the induced homomorphism here $f_{k}$?I guess it's just written a little differently than I had it $\endgroup$ – DaveNine Apr 12 '14 at 5:46
  • $\begingroup$ I apologize for asking so many questions, I really would like to understand this! $\endgroup$ – DaveNine Apr 12 '14 at 5:46
  • $\begingroup$ @DaveNine "Is the isomorphism you describe due to the fundemental group being isomporphic to Z?" Not sure what you're asking. When you say that the fundamental group is isomorphic to Z, this means there is some isomorphism $\pi_1\to\mathbb Z$. So I think the answer to your question is "yes". "Is the induced homomorphism here $f_k$?" No, $f_k$ is a curve on $S^1$, and its homotopy class maps to $k$ under the isomorphism $\pi_1(S^1)\to \mathbb Z$. The map $\varphi_k$ is such that $f_k=\varphi_k f$, so the induced homomorphism ${\varphi_k}_*$ sends the homotopy class of $f$ to that of $f_k$. $\endgroup$ – Alex Becker Apr 12 '14 at 5:50

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