Nice underestimated elementary topology problem

There is a wonderful primary geography trouble (suggestion) that is usually missing out on from the initial publications on the subject.

TROUBLE. Allow $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a continual self - map of the circle of level $\deg(\varphi)=d$. After that $\varphi$ contends the very least $|d-1|$ dealt with factors. (For instance, if $\varphi$ is an alignment turning around homeomorphism, after that it contends the very least 2 dealt with factors - the '2 monks strolling in contrary instructions' trouble.)

Its area needs to be following the idea of level, basic team etc In my point of view, it is a great workout, as it incorporates standard ideas, such as level, dealt with factor, $\pi_{1}(\mathbb{S}^{1})$ and also has valuable applications. Paradoxically, I do not see it in the ideal area ("degree", "fundamental group"), yet in the extra hefty innovative context of Nielsen concept. Nielsen concept, in its turn, is usually missing out on from primary geography publications. The readily available to me ones are managing virtually identical checklist of troubles (wonderful, without a doubt), yet this set appears not to be existing there.

So my inquiry is: Does any person recognize an excellent primary evidence of this problem/proposition (without describing innovative points such as Nielsen index concept approximately). Any kind of referrals rate too. Many thanks beforehand.

2022-06-07 14:32:40
Source Share
Answers: 1

I assumed I would certainly much better make my comment right into a solution:

Lift $\varphi$ to a continual map $f: \mathbb{R} \to \mathbb{R}$ and also consider its chart. The problem on the level pressures $f(x+1) = f(x) + d$ for all $x \in \mathbb{R}$. Yet this indicates that the chart over $[0,1)$ have to converge at the very least $d-1$ amongst the charts $y = x + k$ with $k \in \mathbb{Z}$ by the mean value theory (to be details, the ones with $k$ in between $[f(0), f(0)+d)$ of which there go to worst $d-1$). I allow you expand the information, yet that is what I call entirely primary.

2022-06-07 14:56:53