Intereting Posts

Modular Arithmetic with Powers and Large Numbers
Solve the initial value problem $u_{xx}+2u_{xy}-3u_{yy}=0,\ u(x,0)=\sin{x},\ u_{y}(x,0)=x$
How one can obtain roots at the negative even integers of the Zeta function?
Is there a 'Mathematics wiki' analogous to 'String theory wiki'?
Coupon Collector Problem – expected number of draws for some coupon to be drawn twice
How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?
Requesting abstract algebra book recommendations
How to solve an definite integral of floor valute function?
Discontinuity points of a Distribution function
Proof of Reeb's theorem without using Morse Lemma
Why is the Laplacian important in Riemannian geometry?
Good introductory book for Markov processes
Showing that $\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$
idempotents acting as local identities
Is my understanding of product sigma algebra (or topology) correct?

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.

- Principal Bundles, Chern Classes, and Abelian Instantons
- Do finite products commute with colimits in the category of spaces?
- Homotopy functions
- Surface of genus $g$ does not retract to circle (Hatcher exercise)
- Homology Group of Quotient Space
- Fiber bundle with null-homotopic fiber inclusion

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.

- Why are all k-cells convex?
- Build a topological manifold starting from a set.
- Closed unit interval is connected proof
- Does proper map $f$ take discrete sets to discrete sets?
- Why not just study the consequences of Hausdorff axiom? What do statements like, “The arbitrary union of open sets is open,” gain us?
- Raising a partial function to the power of an ordinal
- What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?
- Whatever Happened to Nearness Spaces?
- Unnecessary property in definition of topological space
- How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?

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.

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- Iterated Limits