- Every orientable surface admits a tight immersion into three-space.
- Every non-orientable surface with Euler characteristic strictly less than -1 admits a tight immersion into three-space.
- The real projective plane, which has Euler characteristic 1, can not be tightly immersed in three-space.
- The Klein bottle, which has Euler characteristic 0, can not be tightly immersed in three-space.

He was primarily concerned with
smooth
surfaces, but it is not hard to modify his constructions to generate
polyhedral
examples for (1) and (2), and his proof of (3) goes through for the
polyhedral case as well. The proof for (4) needs some modification
for the polyhedral case, but Kuiper provided this in
[**K3**].

The only surface that is not covered above is the surface
with
Euler characteristic
-1, a non-orientable surface formed by
adding a handle to the real projective plane
.
Kuiper could not find an example of a tight immersion of this surface
(though his attempts to find one did help him to generate an example
of the real projective plane with two handles
[**K2**]).

Kuiper conjectured that the real projective plane with one handle could not be tightly immersed in space, though he was not able to prove it. This became a well-known open question in the field, and many attempts were made to prove his conjecture, or to find a counterexample.

The solution in the smooth case was obtained in 1992 by
François Haab
[**H1**], who proved it by looking at
projections into the plane of immersions in space. He uses a counting
argument concerning the number of
fold curves in a projection of a
tight immersion and the number of different types of
saddle points
on those curves to arrive at a contradiction.

* 10/14/94 dpvc@geom.umn.edu -- *

*The Geometry Center*