Minimizing the Nielsen root classes for maps between CW-complexes and manifolds of the same dimension ⩾3

Given a map f : K → M , where K is a CW-complex and M a manifold, both of the same dimension n ⩾ 3 , and a Nielsen root class, there is a number associated to this root class, which is the minimum number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We discuss the following question: Is there a map g homotopic to f in which all classes have cardinality equal to the minimal number? We show that the question has a positive answer if f is homotopic to a map that has a Nielsen class with minimum number of points contained in the interiors of n-cells. In the particular case where K is a simplicial complex, we give a sufficient condition on K so that the question has a positive answer.