×

The Nielsen number as an isotopy invariant. (English) Zbl 0839.55002

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, B. Jiang and 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 [S. Wang, Nielsen number of self-homeomorphisms on 3-manifolds, preprint]. Nothing at all is known for the case \(n = 4\).

MSC:

55M20 Fixed points and coincidences in algebraic topology
57N37 Isotopy and pseudo-isotopy

Citations:

Zbl 0829.55001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brown, R. F., The Lefschetz Fixed Point Theorem (1971), Scott, Foresman: Scott, Foresman Chicago, IL · Zbl 0216.19601
[2] Edwards, R. D.; Kirby, R. C., Deformations of spaces of embeddings, Ann. of Math., 93, 63-88 (1971) · Zbl 0214.50303
[3] Ivanov, N. V., Nielsen numbers of maps of surfaces, J. Soviet Math., 26, 1636-1641 (1984) · Zbl 0544.55001
[4] Jiang, B. J., On the least number of fixed points, Amer. J. Math., 102, 673-749 (1980)
[5] Jiang, B. J., Lectures on Nielsen Fixed Point Theory, (Contemporary Mathematics, 14 (1983), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0512.55003
[6] Jiang, B. J., Fixed points and braids, Invent. Math., 75, 69-74 (1984) · Zbl 0565.55005
[7] Jiang, B. J., Fixed points and braids II, Math. Ann., 272, 249-256 (1985) · Zbl 0617.55001
[8] Jiang, B. J.; Guo, J., Fixed points of surface diffeomorphisms, Pacific J. Math., 160, 67-89 (1993) · Zbl 0829.55001
[9] Kelly, M. R., Minimizing the number of fixed points for self-maps of compact surfaces, Pacific J. Math., 126, 81-123 (1987) · Zbl 0571.55003
[10] Kelly, M. R., Isotopic homeomorphisms and Nielsen fixed point theory, Rocky Mountain J. Math., 24, 2, 563-578 (1994) · Zbl 0822.55002
[11] Kirby, R. C.; Siebenmann, L. C., Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, (Annals of Mathematical Studies, 88 (1977), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0361.57004
[12] Rourke, C. P.; Sanderson, B. J., Introduction to Piecewise-Linear Topology (1972), Springer: Springer Berlin · Zbl 0254.57010
[13] Rushing, T. B., Topological Embeddings, (Pure and Applied Mathematics, 52 (1973), Academic Press: Academic Press New York) · Zbl 0176.22001
[14] Schirmer, H., A relative Nielsen number, Pacific J. Math., 122, 459-473 (1986) · Zbl 0553.55001
[15] Wecken, F., Fixpunktklassen III, Math. Ann., 118, 544-577 (1942) · JFM 68.0504.02
[16] Zhang, X. G., The least number of fixed points can be arbitrarily larger than the Nielsen number, Acta Sci. Natur. Univ. Pekinensis, 3, 15-25 (1986) · Zbl 0615.55005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.