Given a model 2-complex K P of a group presentation P, we associate to it an integer matrix ΔP and we prove that a cellular map f: K P → S 2 is root free (is not strongly surjective) if and only if the diophantine linear system ΔP Y = $$ \overrightarrow {deg} $$(f) has an integer solution, here $$ \overrightarrow {deg} $$(f)is the so-called vector-degree of f

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Given a continuous map "Equation missing" from a 2-dimensional CW complex into a closed surface, the Nielsen root number "Equation missing" and the minimal number of roots "Equation missing" of "Equation missing" satisfy "Equation missing". But, there is a number "Equation missing" associated to each Nielsen root class of "Equation missing" and an important problem is to know when "Equation missing". In addition to investigate this problem, we determine a relationship between "Equation missing" and "Equation missing", when "Equation missing" is a lifting of "Equation missing" through a covering space, and we find a connection between this problems, with which we answer several questions related to them when the range of the maps is the projective plane.

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