Let f : X → X denote a self
map of a compact ANR and let N(f) denote the Nielsen number of f which measures
the number of essential fixed points of f. Then it is well-known that f ∼ g : X → X
implies N(f) = N(g). Suppose Y is another ANR and g : Y → Y is a map such
that for a homotopy equivalence h : X → Y , we have hf ∼ gh. Then Jiang
(1964) proved that in these more general circumstances, Nf = N(g), in
the special case when π1(X) is finite. This paper contains a proof of the
result without this restriction and applies it to give a technique for extending
results in the Nielsen theory of fiber-preserving maps from locally trivial fiber
bundles in the category of polyhedra to Hurewicz fibrations in the ANR
category.
