This article is a study of the root theory for maps from two-dimensional CW-complexes into the 2- sphere. Given such a map ?: K -> S~2 we define two integers ξ (?) and ξ(K,d?), which are upper bounds for the minimal number of roots of ?, denote be μ(?). The number ζ(?) is only defined when ? is a cellular map and ζ(K, df) is defined when K is homotopy equivalent to the 2-sphere. When these two numbers are defined, we have the inequality μ(?) ≤ ζ(K, df) < ((f), where df is the so-called homological degree of ?. We use these results to present two very interesting examples of maps from 2-complexes homotopy equivalent to the sphere into the sphere.
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