Immersions of Surfaces:

A map f :M -> R^3 of a polyhedral surface into space is an embedding if it is a one-to-one map. It is an immersion if it is locally one-to-one. (For smooth surfaces, there are some additional requirements that guarantee a well-defined tangent plane at each point of M.) A non-orientable surface can not be embedded into three-space, but every surface can be immersed there.

For polyhedral surfaces, which are mapped linearly into space, there is an easy characterization of immersions in terms of the map's behavior at the vertices:

Lemma: A polyhedral map f :M -> R^3 is an immersion if, and only if, the star of every vertex of M is embedded by f .

To see this, first note that if some vertex star is not embedded, then the map is not an immersion (since the map is linear every neighborhood of the vertex looks essentially like its star). On the other hand, if every vertex star is embedded, than in particular, no edge degenerates to a point, and no face degenerates to a line segment or point. So (since the mapping is linear) the interiors of edges and faces are embedded, and the mapping is an immersion.

Kuiper's initial question

The polyhedral solution

Introduction

8/10/94 dpvc@geom.umn.edu -- The Geometry Center