The coincidence Nielsen number on non-orientable manifolds
This paper presents the Nielsen Theory for coincidence of maps \((f,g): M\to N\) between closed manifolds of the same dimension without assuming that \(M\) and \(N\) are orientable. This is a quite desirable theory. In order to define \(N(f,g)\) the authors work with a pair of maps \((f,g)\) which are transversal. They define some sort of local index, called semi- index which is either a positive integer or an element of \(Z_ 2\), the cyclic group of order 2. Then they develop the Nielsen theory by defining \(N(f,g)\), and they show a type of Wecken theorem. As an application they compute \(N(f,g)\) for pairs of maps \((f,g): K\to K\) where \(K\) is the Klein bottle. I shall mention that the paper [\textit{E. Fadell} and \textit{S. Husseini}, Topology 20, 53-92 (1981; Zbl 0453.55002)] gives enough evidence that in the non-orientable case the local index should be something which lies in \(Z\) or \(Z_ 2\).
scientific article; zbMATH DE number 218633
scientific article