Introduction
In 1960, Nicolaas Kuiper
[K1]
showed that every surface can be
tightly
immersed
in threespace 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 threespace 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