On approximability by embeddings of cycles in the plane

Abstract

We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximable by embeddings if and only if for each i=0,...,s the i-th derivative of the map f (defined by Minc) neither contains transversal self-intersections nor is the standard winding of degree greater than 1. We deduce from the Minc result the completeness of the van Kampen obstruction to approximability by embeddings of piecewise linear maps of a segment to the plane. We also generalize these criteria to simplicial maps of a graph without vertices of degree >3 to a circle.