Product formulae for Nielsen numbers of fibre maps
The relationship between fixed point theory and fiber spaces was established in 1967 [the reviewer, Ann. Math., II. Ser. 85, 483-493 (1967; Zbl 0149.203)]. During the next 15 years, a substantial literature explored the nature of that relationship. Then, \textit{Chen-ye You} [Pac. J. Math. 100, 217-241 (1982; Zbl 0512.55004)] unified and extended virtually all the previous results and began a new generation of research. In the present paper, the author re-examines You's results and achieves a notable clarification and simplification of this material. His approach consists of separating the algebraic from the geometric aspects of the subject. Furthermore, the algebra is unified by deriving most of the results from a single exact sequence that had been developed for a different purpose but is very well suited to the present context. The paper also contains a number of new results. The following is a sample of one of them. Let \(p: E\to B\) be a fiber space and \(f: E\to E\) a fiber preserving map inducing a self-map \(\bar f\) of the base and, by restriction, a self-map \(f_ b\) of a fiber over a fixed point of \(\bar f.\) Let K be the kernel of the homomorphism of fundamental groups induced from the inclusion of the fiber into E. Let N denote the Nielsen number and \(N_ K\) the K-Nielsen number in the sense of \textit{D. McCord} [ibid. 66, 195-203 (1976; Zbl 0342.55005)]. The author proves that if the endomorphism of the fundamental group of B induced by \(\bar f\) is nilpotent (that is, a power is constant), then \(N(f)=N_ K(f_ b)N(\bar f)\).
scientific article; zbMATH DE number 3911282