Minimizing roots of maps into the two-sphere - Fenille, Marcio Colombo (MATHEMATICA SCANDINAVICA, 2012)
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→ S^2 we define two integers ζ(f) and ζ(K,d_f), which are upper bounds for the minimal number of roots of f, denote be μ(f). The number ζ(f) is only defined when f is a cellular map and ζ(K,d_f) is defined when K is homotopy equivalent to the 2-sphere. When these two numbers are defined, we have the inequality μ(f)≤ζ(K,d_f)≤ζ(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.