Table of Contents:

  1. Introduction

  2. Kuiper's initial question

    1. Tight orientable surfaces

    2. Tight non-orientable surfaces
      1. A Tight projective plane with two handles

    3. There is no tight projective plane

    4. There is no tight Klein bottle

    5. Tightness and its consequences
      1. A Bound on the total absolute curvature integral
      2. Tightness and the convex hull
        1. Top sets and top cycles
      3. Tightness and polar height functions
      4. Tightness and the two-piece property
      5. Tightness and homology: the modern definition
      6. Tightness and polyhedral surfaces
        1. Examples of non-tight polyhedra
        2. Proof of the tightness lemma

  3. The smooth solution
    1. Differences between the polyhedral and the smooth cases

  4. The polyhedral solution
    1. Vertex stars of immersions
    2. Triple points of immersions
    3. Tightness for polyhedral surfaces
      1. Examples of non-tight polyhedra
      2. Proof of the tightness lemma
    4. Comparison to Kuiper's level sets
      1. Level 1: The initial circle
      2. Level 2: Pulling one side across
      3. Level 3: The initial self-intersection
      4. Level 4: Adding the triple point
      5. Level 5: The critical level
      6. Level 6: After the critical level
      7. Level 7: No more self-intersection
      8. Level 8: The final circle

  5. Pictures of the polyhedral solution
    1. Interactive pictures
      1. The complete model
      2. The central core
    2. Movies of the model
      1. The sequence of level sets
      2. Building the surface up from the bottom
      3. The central core rotating
    3. Still pictures of the model
    4. Comparison to Kuiper's level sets
    5. Templates for making the model

  6. Other related results

  7. Table of Figures

  8. Bibliography


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8/11/94 dpvc@geom.umn.edu -- The Geometry Center