[Home ] [Title List ] [Tables of Contents and Reviews] [Ordering Information]

Topology of Surfaces

by A. Gramain

A beautiful introduction to the concepts and methods of differential topology. Originally published in 1974 by Presses Univeritaires de France. Translated into English with the assistance of the author.

Table of Contents

1. Foreword to the French edition
   
2. Foreword to the English edition
   
3. Chapter I: -- The Fundamental Group
   
   1. Arcwise connectedness
      
   2. Paths: equivalence, products
      
   3. Loops, the fundamental group
      
   4. Basic examples: circle, torus
      
4. Chapter II: -- The Van Kampen Theorem
   
   1. The fundamental group of the sphere
      
   2. Rudiments of group theory
      
   3. The Van Kampen theorem
      
   4. Basic applications
      
   5. Gluing applications
      
   6. Compact surfaces
      
   7. Non-orientable surfaces
      
5. Chapter III: -- Differentiable Functions and Manifolds
   
   1. Differentiable mappings
      
   2. Inverse function theorem
      
   3. Differentiable manifolds
      
   4. Regular values of a differentiable function
      
   5. Critical points
      
6. Chapter IV: -- Morse Functions on Surfaces
   
   1. Morse functions on a compact surface
      
   2. Vector fields and one-parameter groups of diffeomorphisms
      
   3. Regular values of a Morse function
      
   4. Jumping over a critical value
      
   5. Modification of a Morse function in a canonical neighborhood
      
7. Chapter V: -- The Classification of Surfaces
   
   1. The classification of curves
      
   2. Preliminaries to the classification of surfaces
      
   3. Beginning of the proof
      
   4. Orientable and non-orientable surfaces
      
   5. A special case
      
   6. The general case
      
   7. Remarks on the gluing of homeomorphisms
      
8. Chapter VI: -- Knots
   
   1. Definitions
      
   2. The group of a knot
      
9. Chapter VII: -- Surfaces in Euclidean Space R^3
   
   1. Separation of R^3 by a compact connected manifold
      
   2. Orientable surfaces can be embedded in R^3
      
   3. Non-orientable surfaces can be embedded in R^4 but not R^3
      
10. Index
    

[Home ] [Title List ] [Tables of Contents and Reviews] [Ordering Information]