Pseudo-immersions of oriented surfaces with one boundary component into the plane

Abstract

Let M be a compact connected oriented surface with exactly one boundary component. A pseudo-immersion of M into the plane is a smooth map such that it has only fold singularities and is an orientation-preserving immersion near the boundary of M. For a pseudo-immersion we study the relation between the winding number of the boundary curve and the number of the singular set components.