Lefschetz indices for \(n\)-valued maps
In order to study Nielsen fixed point theory, two main ingredients are necessary: a) Nielsen classes; b) a local index. The same type of theory can be studied in more general situations, like coincidence, parametrized fixed point theory, the equivariant setting, \(n\)-multivalued maps, etc., which require suitable extensions of the two ingredients referred to above. The first ingredient usually admits a rather straightforward extension for the new situation. The second one, the definition of an index, is much more settled and tricky. In principle it is not clear where this new type of local index should lie. The present work studies Nielsen Theory in the very general situation of \(n\)-multivalued maps in several contexts like the equivariant and parametrized one. Nielsen classes are introduced as well as suitable indices, which in the latter case are elements of a stable homotopy group, using the author's works [J. Fixed Point Theory Appl. 2, No. 2, 171--193 (2007; Zbl 1142.37020)] and [ibid. 7, No. 1, 1--32 (2010; Zbl 1205.55002)]. The index is denoted by \(^{G}\omega_0(A)\), where \(A\) is the suitable homotopy fixed-point set. A trace formula is derived for the nonequivariant case. In the equivariant case it is shown that the local index satisfies the classical properties: localization, additivity, homotopy invariance, multiplicativity and that the \(G\)-equivariant index is determined by the family of nonequivariant indices of the restrictions of the given map \(f\) to the fixed set \(X^H\) as \(H\) runs over the family of closed subgroups of \(G\) with Weil group finite. Then the \(G\)-parametrized case is studied and a generalization of the fixed point transfer of Dold is obtained in the context of multivalued maps. The author also proves many other results extending known results of the classical case. These are too long and technical to be stated here. The paper is very well organized and requires a solid knowledge of the homotopy theory involved which deals mainly with the indices. Enough details are given and in general the subtle points are highlighted.
scientific article; zbMATH DE number 6485523
scientific article