Nielsen numbers and fixed points of mappings of bouquets of circles

Abstract

In this note we construct maps of a bouquet of circles into itself, for which the familiar lower bound for the number of fixed points in terms of the Nielsen number is extremely ineffective. Precisely, we prove the following theorem: if n ⩾ 1 and if f: S1vS1→-S1vS1 is a map such that the homomorphism of the fundamental group induced by it carries the canonical generators α and β, respectively, into 1 and (αgbα−1gb−1)nαβα−1, then the Nielsen number of the map f is equal to 0, and any map homotopic to f has not less than 2n−1 fixed points.