Root problem for convenient maps (Q647567)

Given a map \(f: K\to Y\), the pre-image of \(f\) at a point \(y_0\in Y\) is called the root set of \(f\) at \(y_0\). Nielsen root theory provides a number, \(N(f, y_0)\), which is called the root number and turns out to be a lower bound for the number of roots at \(y_0\) for all maps homotopic to \(f\). The authors of the paper under review consider the cases where \(K\) is a \(2\)-dimensional complex and \(Y\) is a closed surface. The maps involved are so-called convenient ones, defined from \(\pi_2(Y)\) by using the language of cohomology. Since \(\pi_2(Y)\) is trivial if \(Y\neq S^2, {\mathbb R}P^2\), non-convenient maps must be those from \(K\) into either \(S^2\) or \({\mathbb R}P^2\). The main results of this paper are some sufficient and necessary conditions for such a map to be homotopic to a map having no root. It should be mentioned that in this situation \(N(f, y_0)\) is independent of the choice of \(y_0\). The vanishing of \(N(f, y_0)\) is a necessary condition for a map to be homotopic to a root free map. The sufficiency of this condition in a more general context was treated by \textit{D. Gonçalves} and \textit{P. Wong} [Proc. Am. Math. Soc. 133, No. 9, 2779--2782 (2005; Zbl 1071.55001)].