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I am trying to do the following exercise from Hirsch :

Let $f:S^1 \rightarrow \mathbb{R}$ be a $C^1$ continuous map and $y$ a regular value . Then $|f^{-1}(y)|=2n$, where $n\in \mathbb{N}.$

Now I was able to prove that we have $|f^{-1}(y)|<\infty$, since this will be a $C^0$ manifold and compact. Now for the even number of points part I have no idea. Maybe this as something to do with antipodal points but I can't seem to get anything to work . I was also thought about using the Borsuk-Ulam theorem but again I don't seem to find anywhere to use it . Any tip or hint is appreciated , just wanted something to get me thinking about an approach . Thanks in advance.

New edit : I have been thinking about this more and I belive it will be to the fact that we have a point of maximum and minimum and the existece of a $z$ such that $f(z)=f(-z)$ and the fact that the functions is continuous , this facts together will forces to pass trough $f^{-1}(q)$ an even number of times, and we know that $x$ is not the point of maximum and minimum since it's derivative is surjective. Still haven't made it very rigorous but I think this is the way.

If there is another way to prove this I would be happy to hear it.

Someone
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1 Answers1

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Maybe say $f^{-1}(y)=\{ x_1,\cdots,x_r \}$. Suppose $x_k=e^{2i\pi \theta_k}$ $0 \leqslant \theta_1 < \cdots < \theta_r <1$, that is we order the $x_k$'s on the circle.

Locally at each $x_k$, $f$ is a bijection $$ \varphi_k:(\theta_k-\alpha,\theta_k+\alpha^\prime) \to (y-\epsilon,y+\epsilon^\prime), $$ because by assumption $f$ is regular at $x_k$, so is a local diffeomorphism. $\varphi_k$ is either increasing or decresing. Say $\varphi_1$ is increasing.

By the Mean value theorem, $\varphi_2$ must be decreasing (otherwise there would be another preimage of $y$ between $x_1$ and $x_2$). Similarly $\varphi_3$ must be increasing, etc. $\varphi_r$ must be decreasing since $\varphi_1$ is increasing : $r$ must be even.