Wecken property for periodic points on the Klein bottle
Let \(f: X\to X\) be a self map on a compact polyhedron. Two homotopy invariants \(NP_n(f)\) and \(NF_n(f)\) (or written as \(N\Phi_n(f)\)) were defined for each natural number \(n\), which are respectively lower bounds for the number \(\sharp(\text{Fix}(g^n)-\bigcup_{k<n}\text{Fix}(g^n))\) and \(\sharp \text{Fix}(g^n)\) for any map \(g\) in the homotopy class of \(f\). When \(n=1\), these two invariants turn out to be the classical Nielsen number \(N(f)\). In general, Nielsen fixed point theory focuses on lower bounds for the numbers of fixed points or periodic points. As a pioneer work, Wecken proved that \(N(f)\) can be realized by some map \(f'\) homotopic to \(f\) provided the space \(X\) is a manifold with dimension greater than \(2\). For this reason, a lower bound in Nielsen fixed point theory is said to be Wecken if it can be realized by the number of fixed points or periodic points of some map. The first author of this paper proved that \(NF_n(f)\) is Wecken if \(f\) is a self map on a compact manifold with dimension greater than \(3\) [Topology 42, No. 5, 1101--1124 (2003; Zbl 1026.55001)]. In this paper, the authors show that \(NF_n(f)\) is Wecken for all self maps on the Klein bottle. As a corollary, the homotopy minimal periods of all self maps on the Klein bottle are obtained. The homotopy minimal period \(Hper(f)\) of a self map is a subset of the natural numbers. A natural number \(m\) lies in \(Hper(f)\) if and only if every map \(g\) in the homotopy class of \(f\) has a periodic point of least period \(m\). This result confirms the statement of \textit{J. Llibre} about \(Hper(f)\) for maps on the Klein bottle [Pac. J. Math. 157, No. 1, 87--93 (1993; Zbl 0832.55003)], where \(Hper(f)\) is said originally to be ``set of period'', written as \(Mper(f)\). This result is also included in a work of \textit{J. Y. Kim, S. S. Kim} and the reviewer [J. Korean Math. Soc. 45, No. 3, 883--902 (2008; Zbl 1147.55004)]. The homotopy classification of self maps on the Klein bottle was given by Halpern in his manuscript. The computations of \(NP_n(f)\) and \(NF_n(f)\) for self maps on the Klein bottle were done in [Kim, Kim and the reviewer (loc. cit.)].
scientific article; zbMATH DE number 5578029