$S^{1}$上$S^{1}$-丛上纤维映射的同伦极小周期
Homotopy minimal periods for fiber maps on circle bundles over circle
AI总结：
研究纤维丛上纤维映射的同伦极小周期，给出了基和纤维均为$S^{1}$的纤维丛上同伦极小周期的完整描述。
AI中文摘要：
给定纤维丛$Y \to M \stackrel{p}{\to} B$以及在$B$上的纤维映射$f: M \to M$，引入$f$的同伦极小周期$H_{B}Per(f)$的定义。在$M$是基和纤维为$S^{1}$的纤维丛的情形下，给出了$H_{S^{1}}Per(f)$的完整描述。
英文摘要：
Given a fiber bundle $Y\rightarrow M\xrightarrow{p}B$ and a fiber map $f: M\rightarrow M$ over $B$, we introduce the notion of fiberwise homotopy minimal periods, denoted by $H_BPer(f).$ This invariant records the periods that occur among representatives of the fiberwise homotopy class of $f.$ We investigate the case in which both the base and the fiber are circles. Up to isomorphism, the corresponding total spaces are the torus and the Klein bottle. Using Nielsen theory for fiber maps and Nielsen-type periodic numbers, we obtain a complete classification of $H_{S^1}Per(f)$ for fiber maps of the torus and the Klein bottle over $S^1$. In particular, we identify the exceptional cases in which some periods can be removed by fiberwise homotopy, including the case $H_{S^1}Per(f)=\mathbb{N}\setminus{2}.$
