The minimizing of the Nielsen root classes
It is known that the root classes of two homotopic maps have a correspondence, they are homotopy related. Given a map \(f: X\to Y\) and an arbitrary root class \(C\), the authors define \(MR[f,C]\) to be the minimal cardinality among all root classes \(C'\), where \(C'\) is a root class of \(f\), and is \(H\)-related to \(C\) for a homotopy \(H\) between \(f'\) and \(f\). The root classes \(C\) with \(MR[f,C]>0\) are said to be the essential root classes. The sum \(\sum MR[f,C]\), where \(C\) ranges over all root classes of \(f\), is of course a lower bound for the number of roots. The authors give a sufficient and necessary condition that this lower bound can be realized for maps between surfaces. The condition is that the numbers of roots in all classes can be minimized at the same time. The authors also present maps without the above minimization from \(n\)-dimensional complexes to \(n\)-dimensional manifolds for all \(n\geq 3\).
scientific article; zbMATH DE number 2046458