Homotopy minimal periods for \(NR\)-solvmanifolds maps
type: Item
instance of: scholarly article
title: Homotopy minimal periods for \(NR\)-solvmanifolds maps
zbMATH Open document ID: 1057.55001
author: Jerzy Jezierski
author: Jarek Kȩdra
author: Wacław Marzantowicz
published in: Topology and its Applications
publication date: 2004-11-19T00:00:00Z
review text: A natural number \(m\) is a homotopy minimal period of a map \(f:X \to X\) if every map homotopic to \(f\) has periodic points of minimal period \(m\). The set of homotopy minimal periods of \(f\) is denoted \(HPer(f)\). Previous work of the first and third author [Math. Z. 239, No. 2, 381--414 (2002; Zbl 1167.55301)] determined \(HPer(f)\) for maps of compact nilmanifolds. The purpose of the present paper is to determine \(HPer(f)\) for maps of certain solvmanifolds. Let \(G\) be a simply-connected solvable Lie group. A compact solvmanifold is of the form \(S=G/ \Delta\) where \(\Delta\) is a uniform closed subgroup. If \(G\) is exponential, that is the exponential map from the Lie algebra of \(G\) to \(G\) is surjective, then \(S\) is called an exponential solvmanifold. These solvmanifolds can be characterized in terms of the eigenvalues of a certain finite set of matrices, namely, that none of the eigenvalues take values on the unit circle except perhaps 1. The \(NR\)-solvmanifolds of the title are a class that is more general than the exponential ones: the same matrices have no eigenvalues that are roots of unity except perhaps 1; thus \(NR\) stands for ``no roots'' [\textit{E. Keppelmann} and \textit{C. McCord}, Pac. J. Math. 170, No. 1, 143--159 (1995; Zbl 0856.55003)]. The main result of this paper is the determination of \(HPer(f)\) for maps of compact \(NR\)-solvmanifolds of dimension at least 4. They also determine \(HPer(f)\) for maps of a less general class of compact 3-dimensional solvmanifolds that are not nilmanifolds.
reviewed by: Robert F. Brown
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 2116782
zbMATH Keywords: Periodic point
zbMATH Keywords: Minimal period
zbMATH Keywords: Minimal homotopy period
zbMATH Keywords: Compact solvmanifold
zbMATH Keywords: Nielsen fixed point theory
MaRDI profile type: Publication
full work available at URL: j.topol.2004.02.018
OpenAlex ID: W2049471962
cites work: Minimal sets of periods for torus maps via Nielsen numbers
cites work: Discrete Uniform Subgroups of Solvable Lie Groups
cites work: Q3893024
cites work: Q3729722
cites work: Q5690485
cites work: Periodic points on tori
cites work: Q3279826
cites work: Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds. I
cites work: Model solvmanifolds for Lefschetz and Nielsen theories
cites work: Cancelling periodic points
cites work: Wecken's theorem for periodic points
cites work: Weak Wecken's theorem for periodic points in dimension 3
cites work: Homotopy minimal periods for nilmanifold maps
cites work: Homotopy minimal periods for maps of three-dimensional nilmanifolds.
cites work: Lectures on Nielsen fixed point theory
cites work: On the Least Number of Fixed Points
cites work: Estimation of the number of periodic orbits
cites work: Minimal sets of periods for torus maps
cites work: The Anosov theorem for exponential solvmanifolds
cites work: Q5791242
cites work: Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds
cites work: Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds. II
cites work: Factor spaces of solvable groups
cites work: On the cohomology of compact homogeneous spaces of nilpotent Lie groups
cites work: Q5669004
cites work: Q3244133
cites work: Q3803359
cites work: Q4847408
DOI: 10.1016/J.TOPOL.2004.02.018
Recommended article: Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds
Recommended article: Q3374444
Recommended article: Homotopy minimal periods for nilmanifold maps
Recommended article: Density of the homotopy minimal periods of maps on infra-solvmanifolds
Recommended article: Nielsen type numbers and homotopy minimal periods for maps on 3-solvmanifolds
Recommended article: Homotopy minimal periods for maps of three-dimensional solvmanifolds
Recommended article: Periodic points on nilmanifolds and solvmanifolds
Recommended article: Homotopy minimal period self-maps on flat manifolds with cyclic holonomies
Recommended article: Homotopy minimal periods for maps of three-dimensional nilmanifolds.
Recommended article: Q5694100
description: scientific article; zbMATH DE number 2116782