Product formulae for Nielsen numbers of fibre maps
type: Item
instance of: scholarly article
title: Product formulae for Nielsen numbers of fibre maps
zbMATH Open document ID: 0571.55002
author: Philip R. Heath
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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 \(N(f)=N_ K(f_ b)N(\bar f)\).
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 55R05
zbMATH DE Number: 3911282
zbMATH Keywords: fixed point theory
zbMATH Keywords: fiber preserving map
zbMATH Keywords: self-map
zbMATH Keywords: homomorphism of fundamental groups
zbMATH Keywords: Nielsen number
reviewed by: Robert F. Brown
MaRDI profile type: Publication
full work available at URL: pjm.1985.117.267
OpenAlex ID: W2001733937
DOI: 10.2140/PJM.1985.117.267
description: scientific article; zbMATH DE number 3911282