The Nielsen Number of a Fibre Map

Abstract

For X a compact metric ANR and f: X-) X a map, N(f), the Nielsen number of f, is a lower bound for the number of fixed points of every map homotopic to f. Let 2 = (E, p, B) be a fibre space. A fibre (-preserving) map f: E E induces maps f: B B and fb: p-'(b) , p-1(b) for all b e B. We will prove that, under appropriate hypotheses, N(f) N(f) * N(fb) for all fb. Thus a class of fibre maps may be added to the set of maps whose Nielsen number can be effectively computed. Furthermore, since the Nielsen number is, under an additional hypothesis, a sharp lower bound in the sense that there exists g: E d E homotopic to f with exactly N(f ) fixed points, we also obtain new results on the existence of fixed point free maps homotopic to a given map. Section 1 consists of basic definitions and a precise statement of the conditions under which N(f ) = N(f) . N(fb). Applications are presented in ? 2. The rest of the paper is devoted to a proof of the theorem.