Fixed points of boundary-preserving maps of punctured discs

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 Δn denote the two-dimensional disc with n open discs removed. Brown and Sanderson have shown that Δn is boundary-Wecken for n = 0, 1 and almost boundary-Wecken for n = 2, with a bound of 1. A map f: (Δn, ∂Δn) → (Δn, ∂Δn) is called boundary inessential if f is null homotopic on each boundary component. We show, for n⩾2: f: (Δn, ∂Δn) → (Δn, ∂Δn) is boundary inessential, then f is boundary-Wecken. f f: (Δn, ∂Δn) → (Δn, ∂Δn) is a map such that the image of the boundary of Δn intersects each boundary component, then f is boundary-Wecken. There exists a class of maps f: (Δn, ∂Δn) → (Δn, ∂Δn), called maps with one essential component, such that MF∂[f] − N∂(f) ⩽ 1. ticular, for n = 2 the results (i)-(iii) imply that Δ2 is almost boundary-Wecken. We give an example of a family of maps f[m]: (Δn, ∂Δn) → (Δn, ∂Δn) for n⩾3 which have two essential components, and such that N∂(f[m]) = 1 but MF∂[f[m]] = 4m + 1, which implies Δn is totally non-boundary-Wecken for n⩾3.