An equality for 2-sided surfaces with a finite number of wild points

Abstract

Let S be a 2-sided surface in a 3-manifold that is wild from one side U at just m points. It is shown that the minimal genus possible for all members of a sequence of surfaces in U converging to S (where these surfaces each separate the same point from S in U ∪ S U \cup S ) is equal to the sum of the genus of S and a certain multiple of the sum of m special topological invariants associated with the wild points. In this equality, the sum of these invariants is multiplied by just one of the numbers 0, 1, or 2, dependent upon the genus and orientability class of S and the value of m. As an application, an upper bound is given for the number of nonpiercing points that a 2-sided surface has with respect to one side.