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.
