Given a map f : X → X of a
compact ANR and any finite connected regular covering p :X→ X to which f
admits lifts, then one can compute a certain homotopy invariant NH(f) if the
Lefschetz numbers of the lifts and the relation of the lifts to the covering
transformations are known. H = p#π1(X). Every map homotopic to f has at least
NH(f) fixed points. If X is a finite polyhedron, then NH(f) ≦ N(f), the Nielsen
number. The smaller invariant is easier to compute by virtue of its smallness, but it is
adequate to discern for example homeomorphisms, h, of manifolds in all dimensions
with L(h) = 0 and N(h) ≧ 2.
