Wecken property for periodic points on the Klein bottle

Abstract

Suppose $f\colon M\to M$ on a compact manifold. Let $m$ be a natural number. One of the most important questions in the topological theory of periodic points is whether the Nielsen-Jiang periodic number $NF_m(f)$ is a sharp lower bound on $\# {\rm Fix}(g^m)$ over all $g\sim f$. This question has a positive answer if ${\rm dim}\, M\geq 3$ but in general a negative answer for self maps of compact surfaces. However, we show the answer to be positive when $M=\mathbb K$ is the Klein bottle. As a consequence, we reconfirm a result of Llibre and compute the set ${\rm HPer} (f)$ of homotopy minimal periods on the Klein bottle.