Endomorphisms of degree d on a sphere with infinite fibers on a dense subset - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:24:03Z http://mathoverflow.net/feeds/question/110925 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110925/endomorphisms-of-degree-d-on-a-sphere-with-infinite-fibers-on-a-dense-subset Endomorphisms of degree d on a sphere with infinite fibers on a dense subset Hugo Chapdelaine 2012-10-28T17:50:07Z 2012-10-29T22:22:08Z <p>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 $d$. </p> <p>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 <strong>finite</strong>, then one has from excision theorem that $$\sum_{y\in f^{-1}(x)} deg_f(y)=d.$$</p> <p>Q1: How would you construct a (continuous) map $f:S^n\rightarrow S^n$ of degree $d$ such that on a <strong>dense subset of points</strong> $X\subseteq S^{n}$ one has that for every $x\in X$ that the fiber $f^{-1}(x)$ is infinite?</p> <p>Q2: Having a map $f$ as in Q1 and a point $x\in X$, is it possible to take some kind <strong>natural</strong> 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) ?</p> <p><strong>added</strong>: 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. </p>