Minimizing roots of maps into the two-sphere

Abstract

This article is a study of the root theory for maps from two-dimensional CW-complexes into the 2-sphere. Given such a map $f:K\rightarrow S^2$ we define two integers $\zeta(f)$ and $\zeta(K,d_f)$, which are upper bounds for the minimal number of roots of $f$, denote be $\mu(f)$. The number $\zeta(f)$ is only defined when $f$ is a cellular map and $\zeta(K,d_f)$ is defined when $K$ is homotopy equivalent to the 2-sphere. When these two numbers are defined, we have the inequality $\mu(f)\leq\zeta(K,d_f)\leq\zeta(f)$, where $d_f$ is the so-called homological degree of $f$. We use these results to present two very interesting examples of maps from 2-complexes homotopy equivalent to the sphere into the sphere.