We shall be concerned with the
behavior of a mapping π from one oriented compact surface-with-boundary to
another, which may fail to be a covering projection in one of two ways. Firstly, π
need not be a local homeomorphism, although its interior singularities will be
of a restricted type, called branch points. Secondly, boundary points may
be mapped into the interior, although we shall assume the restriction of π
to the boundary is injective. We shall show that π must then be a local
homeomorphism except on a finite set. Moreover, we shall analyze the behavior of π
near the boundary in sufficient detail to derive a formula relating Euler
characteristics of the domain and of the image, with multiplicities, to the total
order of branching of π. These results may be used to study ramification and
ramified branch points of parametric minimal surfaces of general topological
type.
