1. Introduction

A longstanding open problem in the study of tight surfaces centered around a question posed by Nicolaas Kuiper [1961] asking whether the surface with Euler characteristic —1 (a real projective plane with one handle) could be tightly immersed in three-space. Kuiper had established that all other surfaces admitted tight immersions in space except for the Klein bottle and the real projective plane, which do not. More than thirty years passed before Frangois Haab [1992] proved that, for smooth surfaces, no such immersion exists. In light of this result and the failure of the attempts to find a polyhedral counterexample, it seemed only a matter of time before a corresponding proof would be found for the polyhedral case as well. Surprisingly, a polyhedral tight immersion of this surface does exist, as shown recently in [Cervone 1994]. Although the smooth and polyhedral theories differ substantially for surfaces in high-dimensional spaces, they correspond quite closely in low dimensions; the case of the real projective plane with one handle is important in that it represents one of only a handful of low-dimensional examples where the theories differ in a significant way (see Section 3).