A note on the set of periods for Klein bottle maps
By using the Nielsen fixed point theory we characterize the set of periods attained by continuous self-maps on the Klein bottle belonging to a given homotopy class. The main result can be stated as follows (here \(2\mathbb{N}\) denotes the set of all even natural numbers): Theorem. Let \(f\) be a Klein bottle map and let \(u\) and \(v\) be its associated integers. (1) Suppose \(|u|>1\). Then \[ \begin{alignedat}{2} \text{Per} (f)&= \mathbb{N} && \text{if }|v|\neq 1, \qquad \qquad \text{and}\\ \text{Per} (f) &\supset \mathbb{N} \smallsetminus 2\mathbb{N} \qquad &&\text{if }v= - 1.\end{alignedat} \] Furthermore, the last result cannot be improved without additional hypothesis on \(f\). (2) Suppose \(|u|\leq 1\). Then \[ \begin{alignedat}{2} \text{Per} (f)&= \mathbb{N} &&\text{ if } v\not\in \{-2, -1, 0,1\}, \\ \text{Per} (f) &\supset N\smallsetminus \{2\} \qquad &&\text{ if } v=-2, \\ \text{Per} (f) &\supset \{1\} &&\text{ if } v\in \{-1, 0\}.\end{alignedat} \] Furthermore, these last two results cannot be improved without additional hypotheses on \(f\). (3) If \(v=1\) then the set \(\text{Per} (f)\) can be empty. The proof is based on the results of \textit{B. Halpern} [Periodic points on the Klein bottle (Preprint)]\ on the Nielsen numbers of Klein bottle maps. Unfortunately this nice paper of Halpern remains unpublished. As we shall see the Nielsen numbers of the Klein bottle map \(f\) only depend on its associated integers. Then, since the Nielsen numbers are homotopy invariants, the characterization of the set of periods given in the theorem works for all Klein bottle maps homotopic to \(f\).
periods attained by continuous self-maps on the Klein bottle
scientific article; zbMATH DE number 508853