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Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint n-dimensional open discs in $S^n$. Then collapsing $K$ to a point we find that $S^n/K\simeq \bigvee _{i=1}^d S^n$. Identifying (choosing a homemomorphism) each of the sphere in the disjoint union with $S^n$ (with the approriate orientation), the composition of the two maps $$ S^n\rightarrow \bigvee _{i=1}^d S^n \rightarrow S^n, $$ gives us a map $\phi_d:S^{n}\rightarrow S^n$ of degree $d$.

In general, if $f:S^n\rightarrow S^n$ is a map of degree $d$ and $x\in S^{n}$ is such that the fiber $f^{-1}(x)$ is finite, then one has from excision theorem that $$ \sum_{y\in f^{-1}(x)} deg_f(y)=d. $$

Q1: How would you construct a (continuous) map $f:S^n\rightarrow S^n$ of degree $d$ such that on a dense subset of points $X\subseteq S^{n}$ one has that for every $x\in X$ that the fiber $f^{-1}(x)$ is infinite?

Q2: Having a map $f$ as in Q1 and a point $x\in X$, is it possible to take some kind natural average sum over the local degrees of the elements of $f^{-1}(x)$ in such a way that the sum converges to $d$ (you may assume that $S^n$ is endowed with a metric if you think it helps) ?

added: Note that if one constructs a map $f:S^n\rightarrow S^n$ of degree one with infinite fibers (on a dense set) then by post-composing with a map of degree $d$ we obtain a map of degree $d$ which satisfies all the conditions.

edited Oct 29 at 22:22

asked Oct 28 at 17:50

Hugo Chapdelaine
3,337●3●18

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Think about the following example $f\colon \mathbb S^1\to \mathbb S^1$ defined as $$f(x)=x+\sum_n \tfrac1n\sin (n^2\cdot x) \pmod{2\cdot\pi}.$$ Note that the preimage $f^{-1}(y)$ for any $y$ is a Cantor set. – Anton Petrunin Oct 28 at 19:37

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In the actual Q1 (in contrast to the title), you only asked that some point $x$ have an infinite fiber. The example you gave at the beginning of the question already does that. The set $K$ that you collapse to a point is one of the fibers of your map. – Andreas Blass Oct 29 at 0:35

Yes you are right, I had in mind something different. – Hugo Chapdelaine Oct 29 at 14:57

@Anton, I have 2 simple questions: what is the degree of your map $f$ (and how to compute it) and how do you compute a fiber $f^{-1}(y)$? – Hugo Chapdelaine Oct 29 at 15:11

Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

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