Fixed points of boundary-preserving maps of punctured discs

Abstract

Let M be a compact connected manifold with nonempty boundary and let f: ( M, ∂M) → ( M, ∂M) be a boundary-preserving map. We denote the relative Nielsen number of f, in the sense of Schirmer, by N ∂ ( f) and write MF ∂ [ f] for the minimum number of fixed points of all maps homotopic to f as maps of pairs. Let Δ n denote the two-dimensional disc with n open discs removed. Brown and Sanderson have shown that Δ n is boundary-Wecken for n = 0, 1 and almost boundary-Wecken for n = 2, with a bound of 1. A map f: ( Δ n , ∂Δ n ) → ( Δ n , ∂Δ n ) is called boundary inessential if f is null homotopic on each boundary component. We show, for n⩾2: (i) If f: ( Δ n , ∂Δ n ) → ( Δ n , ∂Δ n ) is boundary inessential, then f is boundary-Wecken. (ii) If f: ( Δ n , ∂Δ n ) → ( Δ n , ∂Δ n ) is a map such that the image of the boundary of Δ n intersects each boundary component, then f is boundary-Wecken. (iii) There exists a class of maps f: ( Δ n , ∂Δ n ) → ( Δ n , ∂Δ n ), called maps with one essential component, such that MF ∂ [ f] − N ∂ ( f) ⩽ 1. In particular, for n = 2 the results (i)-(iii) imply that Δ 2 is almost boundary-Wecken. We give an example of a family of maps f [ m] : ( Δ n , ∂Δ n ) → ( Δ n , ∂Δ n ) for n⩾3 which have two essential components, and such that N ∂ ( f [ m] ) = 1 but MF ∂ [ f [ m] ] = 4 m + 1, which implies Δ n is totally non-boundary-Wecken for n⩾3.