Comparison to Kuiper's Level Sets:
In Kuiper's original paper
he drew the following sequence of
to describe an immersion of the
real projective plane
The figure has three
(the minimum possible), indicated by the arrows. Kuiper used this
picture to show how to build a tight model of a smooth projective
plane with two handles
but was not able to produce a tight version with only one handle.
The core of the tight polyhedral model presented here
follows Kuiper's original drawings quite closely. Starting with this
initial projective plane, the top and bottom are cut off and a tube
added around the outside of the object to complete the tight immersion
of the projective plane with one handle.
The following pages describe the relationship between Kuiper's
drawings and the polyhedral
in more detail:
Level 1: The initial circle.
Level 2: Pulling one side across.
Level 3: The initial self-intersection.
Level 4: Adding the triple point.
Level 5: The critical level.
Level 6: After the critical level.
Level 7: No more intersection.
Level 8: The final circle.
The polyhedral solution
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