# Table of Contents:

**Introduction**

**Kuiper's initial question**

- Tight orientable surfaces

- Tight non-orientable surfaces
- A Tight projective plane
with two handles

- There is no tight projective plane

- There is no tight Klein bottle

- Tightness and its consequences
- A Bound on the total absolute
curvature integral
- Tightness and the convex hull
- Top sets and top cycles

- Tightness and polar height functions
- Tightness and the two-piece property
- Tightness and homology: the
modern definition
- Tightness and polyhedral surfaces
- Examples of non-tight polyhedra
- Proof of the tightness lemma

**The smooth solution**
- Differences between the
polyhedral and the smooth cases

**The polyhedral solution**
- Vertex stars of immersions
- Triple points of immersions
- Tightness for polyhedral surfaces
- Examples of non-tight polyhedra
- Proof of the tightness lemma

- Comparison to Kuiper's level sets
- 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 self-intersection
- Level 8: The final circle

**Pictures of the polyhedral solution**
- Interactive pictures
- The complete model
- The central core

- Movies of the model
- The sequence of level sets
- Building the surface up
from the bottom
- The central core rotating

- Still pictures of the model
- Comparison to Kuiper's level sets
- Templates for making the model

**Other related results**

**Table of Figures**

**Bibliography**

*Main entry point*

* 8/11/94 dpvc@geom.umn.edu -- *

*The Geometry Center*