Introduction

In 1960, Nicolaas Kuiper [K1] showed that every surface can be tightly immersed in three-space except for the real projective plane and the Klein bottle, for which no such immersion exists, and the real projective plane with one handle, for which he could find neither a tight example nor a proof that one does not exist. The status of this last surface went undetermined for 30 years until in 1992, François Haab [H1] proved that there is no smooth tight immersion into three-space of the projective plane with one handle. Haab's proof is valid only for smooth surfaces, but it, together with the fact that no polyhedral example had been found in the preceding 30 years, strongly suggested that the same would be true of polyhedral surfaces as well.

Surprisingly, this is not the case. A tight polyhedral immersion of the real projective plane with one handle exists, and it is the topic of this presentation.

Kuiper's original question

The smooth solution

The polyhedral solution

Pictures of the polyhedral solution

Other related results

Main entry point

7/19/94 dpvc@geom.umn.edu -- The Geometry Center