Homotopy minimal periods for fiber maps on circle bundles over circle
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 $H_{S^1}Per(f)=\mathbb{N}\setminus{2}.$