Let X be a compact
2-manifold with nonempty boundary ∂X. Given a boundary-preserving map
f : (X,∂X) → (X,∂X), let MF∂[f] denote the minimum number of fixed points
of all boundary-preserving maps homotopic to f as maps of pairs and let
N∂(f) be the relative Nielsen number of f in the sense of Schirmer [S]. Call
X boundary-Wecken, bW, if MF∂[f] = N∂(f) for all boundary-preserving
maps of X, almost bW if MF∂[f] − N∂(f) is bounded for all such f, and
totally non-bW otherwise. We show that if the euler characteristic of X
is non-negative, then X is bW. On the other hand, except for a relatively
small number of cases, we demonstrate that the 2-manifolds of negative euler
characteristic are totally non-bW. For one of the remaining cases, the pants surfaceP, we use techniques of transversality theory to examine the fixed point
behavior of boundary-preserving maps of P, and show that P is almost
bW.
