# Nice underestimated elementary topology problem

There is a nice elementary topology problem (proposition) that is often
missing from the introductory books on the topic.

PROBLEM. Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a continuous
self-map of the circle of degree $\deg(\varphi)=d$. Then $\varphi$ has at
least $|d-1|$ fixed points. (For example, if $\varphi$ is an orientation
reversing homeomorphism, then it has at least 2 fixed points – the ‘two monks
walking in opposite directions’ problem.)

Its place should be just after the notion of degree, fundamental group etc. In
my opinion, it is a very good exercise, as it combines basic notions, such as
degree, fixed point, $\pi_{1}(\mathbb{S}^{1})$ and has useful applications.
Paradoxically, I don’t see it in the appropriate place (“degree”, “fundamental
group”), but in the more heavy advanced context of Nielsen theory. Nielsen
theory, in its turn, is often missing from elementary topology books. The
available to me ones are dealing with almost one and the same list of problems
(nice, indeed), but this one seems not to be present there.

So my question is: Does anyone know a good elementary proof of this
problem/proposition (without referring to advanced things such as Nielsen index
theory or so). Any references are welcome as well. Thanks in advance.

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I thought I’d better make my comment into an answer:

Lift $\varphi$ to a continuous map $f: \mathbb{R} \to \mathbb{R}$ and look at its graph. The condition on the degree forces $f(x+1) = f(x) + d$ for all $x \in \mathbb{R}$. But this implies that the graph over $[0,1)$ must intersect at least $d-1$ among the graphs $y = x + k$ with $k \in \mathbb{Z}$ by the mean value theorem (to be specific, the ones with $k$ between $[f(0), f(0)+d)$ of which there are at worst $d-1$). I let you flesh out the details, but that’s what I call completely elementary.