Stabilizers of fixed point classes and Nielsen numbers of \(n\)-valued maps (Q1704369)

Let \(X\) be a connected finite polyhedron, \(\widetilde{X}\) a connected space, and \(p : \widetilde{X} \to X\) a finite cover of degree \(n\). For a continuous map \(f : X \to X\), the \(n\)-valued lift of \(f\) is defined as the multimap \(\phi_{p,f} : \widetilde{X} \multimap \widetilde{X}\), \(\phi_{p,f} (\widetilde{x}) = p^{-1}(fp(\widetilde{x})).\) By using the notion of a stabilizer of a fixed point class of \(f\), introduced by \textit{B. Jiang} et al. [Algebr. Geom. Topol. 11, No. 4, 2297--2318 (2011; Zbl 1232.55006)] the authors study relations between the Nielsen numbers \(N(\phi_{p,f})\) and \(N(f).\) In particular, they refine the computations of the Nielsen numbers for \(n\)-valued lifts of maps of graphs with positive Lefschetz numbers and of maps of spaces with abelian fundamental group.