The Nielsen number as an isotopy invariant
Given a map \(f : M \to M\) of a compact \(n\)-manifold, every map homotopic to \(f\) has at least \(N(f)\) fixed points, where \(N(f)\) denotes the Nielsen number of \(f\). If \(n \geq 3\), then there exists a map homotopic to \(f\) with exactly \(N(f)\) fixed points, but that is not necessarily true if \(n = 2\). However, \textit{B. Jiang} and \textit{J. Guo} [Pac. J. Math. 160, 67-89 (1993; Zbl 0829.55001)] proved that if \(f : M \to M\) is a homeomorphism of a surface, then there is a homeomorphism isotopic to \(f\) with exactly \(N(f)\) fixed points. Thus it is a natural, and very attractive, problem to investigate whether this property of homeomorphisms under isotopy extends to higher-dimensional manifolds. The main result of this paper states that a homeomorphism \(f : M \to M\) of an \(n\)-manifold is isotopic to one with \(N(f)\) fixed points if \(n \geq 5\) and the boundary of \(M\) is empty. More generally, if \(f : M \to M\) is an embedding of an \(n\)-manifold, \(n \geq 5\), then there is an ambient isotopy of \(M\) carrying \(f\) to an embedding with exactly \(N(f)\) fixed points. The author shows that the fixed point set can be made finite, with each fixed point in a neighborhood, called an isotopy-standard ball, that is based on an explicitly described Euclidean model. The Nielsen number counts certain of the classes of fixed points under the relation that two fixed points are equivalent if there is a path between them that is homotopic, rel the endpoints, to its image under \(f\). Thus, the key of the proof is to be able to isotope \(M\) in such a way that any two equivalent fixed points can be replaced by a single fixed point. This is accomplished in the paper by means of extensive geometric constructions that occupy Section 3 and that are summarized as Proposition 3.5. The description of the necessary constructions is quite condensed and a more detailed presentation of this material would have made the argument more convincing. The methods of this paper do not extend to homeomorphisms of manifolds with nonempty boundary. Wang has obtained a corresponding result for homeomorphisms of hyperbolic 3-manifolds [\textit{S. Wang}, Nielsen number of self-homeomorphisms on 3-manifolds, preprint]. Nothing at all is known for the case \(n = 4\).
scientific article; zbMATH DE number 740456