Kuiper's Initial Question:

When Kuiper initiated the study of tight immersions, he showed [K1, K2] the following:

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

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 [More]. 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.

[Right] Tightness and its consequences
[Right] The smooth result
[Left] Introduction

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10/14/94 dpvc@geom.umn.edu -- The Geometry Center