Piecewise linear immersions

Abstract

0. Introduction. An immersion f of one space X in another Y is a continuous map which is locally an embedding; that is, for any x ∈ X there is a neighbourhood N(x) of x such that f|N(x) is an embedding. Thus the image of an immersion may intersect itself, but it can contain no worse singularities. For example, the well-known model of the Klein Bottle is the image of an immersion into ordinary 3-space; the locus of points of self intersection is a circle, where two distinct circles in the Klein Bottle have their images. This paper obtains a sufficient condition for a map of PL manifolds to be deformable into a PL immersion.