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I am reading "Compact Riemann Surfaces" by Jurgen Rost. In an exercise before the chapter covering spaces, he wants the reader to compute the fundamental group of a $\mathbb S^1$ in a direct way. He wants us to show the maps $t \mapsto e(nt)$,$t \mapsto e(mt)$ are not homotopic to each other if $n \neq m$. Here is what I tried:

Suppose $F$ is a homotopy between the before mentioned maps. My idea in the case m=2, n=1 was to look at $$\alpha = \text{inf}\lbrace \beta \in [0,1] : \text{there are} \space 0\le t\lt z\le 1 \space \text{such that} \space \\ \qquad F(\beta,t)=p_0=F(\beta,z) \space \text{and length}\bigl(F(\beta,.)\vert_{\lbrack t,z\rbrack} \bigr) \geq 2\pi \rbrace $$ Using some compactness argument we would get a convergent sequence $(\beta_n)_n$ together with $(t_n)_n$, $(z_n)_n$. Then we set $t$ and $z$ to be the limit of $(t_n)_n$ and $(z_n)_n$. Thus we get that $\alpha$ is contained in the set. My idea was now to see what happens when I approach $\alpha$ from below, but I got stuck. I know this is a very elementary approach, but something like this should work, because this is chapter 1 of the book and the introduction was very gentle. Any help is greatly appreciated. I know a question like this was asked earlier here Calculate the fundamental group of $\pi_1(S^1)$, but the solutions seemed to be quite sophisticated.

Dude1662
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    This is probably a duplicate. The based homotopy of those loops must lift to some homotopy of paths in $\Bbb R$. The endpoints must project to the same basepoint, so, must be integers. Continuous integer-valued paths are constant, so we find the endpoint integers cannot change. So if they are homotopic, $n=m$ is forced – FShrike Jul 28 '23 at 16:38
  • What exactly is the domain of definition of $F$? Since it is a homotopy, I would have expected something like $F : [0,1] \times \mathbb S^1 \to \mathbb S^1$, but "$\beta \in \mathbb R$" throws me off. I would have expected instead $\beta \in [0,1]$, but in that case clearly $\alpha=0$. – Lee Mosher Jul 28 '23 at 17:04
  • Presumable, $e(t)=\exp(2\pi i t)$ defined on $[0,1]?$ – Thomas Andrews Jul 28 '23 at 17:08
  • e(t) is defined in this way. – Dude1662 Jul 28 '23 at 17:16
  • Lee thank you for noticing that. Yes in the general case where m not equal to n i would have to look at a different set. I changed the comment in a way, so that the idea is more clear. – Dude1662 Jul 28 '23 at 17:20
  • Fshrike thanks a lot for answering the question. I am assuming after the lifting to some homotopy of paths, the integer value of the endpoint tells me to which e(nt) my path is homotopic. Why? – Dude1662 Jul 28 '23 at 17:25
  • A complex analytic approach is to show that the winding number of a curve is a homotopy invariant (continuous maps to $\mathbb{Z}$, $\mathbb{Z}$ is discrete). – yoyo Jul 28 '23 at 18:10
  • Jost, Jürgen Jost, not Rost – Roman Hric Jul 28 '23 at 20:19

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