The boundary-Wecken classification of surfaces

Let X be a compact 2-manifold with nonempty boundary ∂X and let f : (X, ∂X) → (X, ∂X) be a boundary-preserving map. Denote by MF∂ [f ] 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∂(f) is the sum of the number of essential fixed point classes of the restriction f̄ : ∂X → ∂X and the number of essential fixed point classes of f that do not contain essential fixed point classes of f̄ . We prove that if X is the Möbius band with one (open) disc removed, then MF∂ [f ] − N∂(f) ≤ 1 for all maps f : (X, ∂X) → (X, ∂X). 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, MF∂ [f ] = N∂(f) 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 MF∂ [f ]−N∂(f) ≤ 1. All other surfaces are totally non-boundary-Wecken, that is, given an integer k ≥ 1, there is a map fk: (X, ∂X) → (X, ∂X) such that MF∂ [fk]− N∂(fk) ≥ k . AMS Classification 55M20; 54H25, 57N05