Preface
Notation and Terminology
CHAPTER Ⅰ Two-Dimensional Manifolds
1.Introduction
2.Definition and Examples of n-Manifolds
3.Orientable vs.Nonorientable Manifolds
4.Examples of Compact, Connected 2-Manifolds
5.Statement of the Classification Theorem for Compact Surfaces
6.Triangulations of Compact Surfaces
7.Proof of Theorem 5.1
8.The Euler Characteristic of a Surface
References
CHAPTER Ⅱ The Fundamental Group
1.Introduction
2.Basic Notation and Terminology
3.Definition of the Fundamental Group of a Space
4.The Effect of a Continuous Mapping on the Fundamental Group
5.The Fundamental Group of a Circle IS Infinite Cyclic
6.Application: The Brouwer Fixed-Point Theorem in Dimension 2
7.The Fundamental Group of a Product Space
8.Homotopy Type and Homotopy Equivalence of Spaces
References
CHAPTER Ⅲ Free Groups and Free Products of Groups
1.Introduction
2.The Weak Product of Abelian Groups
3.Free Abelian Groups
4.Free Products of Groups
5.Free Groups
6.The Presentation of Groups by Generators and Relations
7.Universal Mapping Problems
References
CHAPTER Ⅳ Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces.Applications
1.Introduction
2.Statement and Proof of the Theorem of Seifert and Van Kampen
3.First Application of Theorem 2.1
4.Second Application of Theorem 2.1
5.Structure of the Fundamental Group of a Compact Surface
6.Application to Knot Theory
7.Proof of Lemma 2.4
References
CHAPTER Ⅴ Covering Spaces
1.Introduction
2.Definition and Some Examples of Covering Spaces
3.Lifting of Paths to a Covering Space
4.The Fundamental Group of a Covering Space
5.Lifting of Arbitrary Maps to a Covering Space
6.Homomorphisms and Automorphisms of Covering Spaces
7.The Action of the Group π(X,x) on the Set p-1 (x)
8.Regular Covering Spaces and Quotient Spaces
9.Application: The Borsuk-Ulam Theorem for the 2-Sphere
