The boundary-Wecken classification of surfaces

Abstract

Let $X$ be a compact $2$-manifold with nonempty boundary $\partial X$ and let $f:\left(X,\partial X\right)\to \left(X,\partial X\right)$ be a boundary-preserving map. Denote by $M{F}_{\partial }\left[f\right]$ the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to $f$. The relative Nielsen number $N_{\partial }\left(f\right)$ is the sum of the number of essential fixed point classes of the restriction $\bar{f}:\partial X\to \partial X$ and the number of essential fixed point classes of $f$ that do not contain essential fixed point classes of $\bar{f}$. We prove that if $X$ is the Möbius band with one (open) disc removed, then $M{F}_{\partial }\left[f\right]-N_{\partial }\left(f\right)\le 1$ for all maps $f:\left(X,\partial X\right)\to \left(X,\partial X\right)$. This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If $X$ is the disc, annulus or Möbius band, then $X$ is boundary-Wecken, that is, $M{F}_{\partial }\left[f\right]=N_{\partial }\left(f\right)$ for all boundary-preserving maps. If $X$ is the disc with two discs removed or the Möbius band with one disc removed, then $X$ is not boundary-Wecken, but $M{F}_{\partial }\left[f\right]-N_{\partial }\left(f\right)\le 1$. All other surfaces are totally non-boundary-Wecken, that is, given an integer $k\ge 1$, there is a map $f_k:\left(X,\partial X\right)\to \left(X,\partial X\right)$ such that $M{F}_{\partial }\left[f_k\right]-N_{\partial }\left(f_k\right)\ge k$.