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 Xicot\'encatl. 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 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.Comment: near complete rewrite from previous versio
