Let be a compact -manifold with nonempty boundary and let be a boundary-preserving map. Denote by the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to . The relative Nielsen number is the sum of the number of essential fixed point classes of the restriction and the number of essential fixed point classes of that do not contain essential fixed point classes of . We prove that if is the Möbius band with one (open) disc removed, then for all maps . This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If is the disc, annulus or Möbius band, then is boundary-Wecken, that is, for all boundary-preserving maps. If is the disc with two discs removed or the Möbius band with one disc removed, then is not boundary-Wecken, but . All other surfaces are totally non-boundary-Wecken, that is, given an integer , there is a map such that .
"The boundary-Wecken classification of surfaces." Algebr. Geom. Topol. 4 (1) 49 - 71, 2004. https://doi.org/10.2140/agt.2004.4.49