Coincidence Nielsen numbers for covering maps for smooth manifolds (Q1044874)

Let \(f: X\to X\) be a self-map on a compact manifold (or more generally a compact polyhedron). The fundamental invariant in Nielsen fixed point theory is the Nielsen number \(N(f)\), which provides a lower bound for the number of fixed points of the maps in the homotopy class of \(f\). Consider a finite-sheeted regular covering \(p:\widetilde X \to X\) and assume that a self-map \(f\) has a lifting. J. Jezierski found out some relations among the Nielsen numbers of \(f\) and those of its lifts with respect to the covering \(p:\widetilde X \to X\). The author of this paper generalizes the idea of J. Jezierski to the case of coincidence points. Suppose that \(f,g:M\to N\) are two maps between closed smooth manifolds of same dimension. By using some combinations of the Nielsen coincidence numbers of the lifts of \(f\) and \(g\), two invariants \(N_L(f,g)\) and \(N_{ED}(f,g)\) are defined, satisfying \(N_L(f,g)+N_{ED}(f,g)=N(f,g)\), where \(N(f,g)\) is the Nielsen coincidence number of \(f\) and \(g\) given in [\textit{R. Dobreńko} and \textit{J. Jezierski}, Rocky Mt. J. Math. 23, No.~1, 67--85 (1993; Zbl 0787.55003)]. Some examples are given. Moreover, the author also obtains a product formula of semi-index. In fact, the relation of Nielsen coincidence numbers between finite-sheet coverings was treated in [\textit{S. W. Kim} and \textit{J. B. Lee}, Nagoya Math. J. 186, 69--93 (2007; Zbl 1129.55002)].