A Tight Polyhedral Immersion in Three-space of the Real Projective Plane with One Handle

In 1960, Nicolaas Kuiper 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. It was not until more than 30 years later, in 1992, that François Haab 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, which we demonstrate in this paper.