Homotopy minimal periods for \(NR\)-solvmanifolds maps
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.
scientific article; zbMATH DE number 2116782
scientific article