Table of Contents
Outline
Tight Surfaces
Definition of Tightness
Tightness and Height Functions
Construction Theorem
A Tight Example
Top Cycles
Historical Note
Polyhedral Tightness
Some Polyhedral Examples
Classification of Surfaces
Adding Handles Tightly
Making Surfaces Tight
Kuiper's Projective Plane
Kuiper's Results
No Tight Projective Plane
No Tight Klein Bottle
The Remaining Surface
Compace Eulcidean Space Forms
Definition of CES
Geometric Meaning
How many CES are there?
Classification of CES
Surfaces in CES
Some Tori in CES
Tight Surfaces in CES
Tightness in CES
Tightness and Curvature
Tightness and Euler Characteristic
Height Functions in CES
Tightness by Euler Characteristic
Tight Spheres
No Tight Projective Planes
Tight Tori
Adding Tight Handles
Tight Tori with Handles
Tight Klein Bottles
Flat Klein Bottles with Handles
Klein Bottles with Handles
Projective Planes with Handles
Projective Planes with Handles (II)
Otto's Conjectures
Connected Sums
Brehm's Projective Plane
Replacing a Saddle
Smoothable Projective Plane with Two Handles
What Surface is This?
The Surface Revealed
Computing the Euler Characteristic
A Tight Projective Plane with a Handle
Trying the Other CES
Final Results
Summary
Conclusion