Lifting classes for the fixed point theory of n-valued maps

Abstract

The theory of lifting classes and the Reidemeister number of single-valued maps of a finite polyhedron X is extended to n-valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicoténcatl. The liftings of an n-valued map f split into self-maps of the universal covering space of X that we call lift-factors. An equivalence relation is defined on the lift-factors of f and the number of equivalence classes is the Reidemeister number of f. The fixed point classes of f are the projections of the fixed point sets of the lift-factors and are the same as those of Schirmer. An equivalence relation is defined on the fundamental group of X such that the number of equivalence classes equals the Reidemeister number. We prove that if X is a smooth closed manifold of dimension at least three, then algebraically the orbit configuration space approach is the same as one utilizing the universal covering space. The Jiang subgroup is extended to n-valued maps as a subgroup of the group of covering transformations of the orbit configuration space and used to find conditions under which the Nielsen number of an n-valued map equals its Reidemeister number. If an n-valued map splits into n single-valued maps, then its n-valued Reidemeister number is the sum of their Reidemeister numbers.