This book is intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory (including cup products and the duality theorems of Poincare and Alexander). It consists of material from the first five chapters of the author's earlier book Algebraic Topology: An Introduction (GTM 56) together with almost all of his book Singular Homology Theory (GTM 70). This material from the two earlier books has been revised, corrected, and brought up to date. There is enough here for a full-year course.

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

10.The Existence Theorem for Covering Spaces References

CHAPTER Ⅵ

Background and Motivation for Homology Theory

1.Introduction

2.Summary of Some of the Basic Properties of Homology Theory

3.Some Examples of Problems which Motivated the Development of Homology Theory in the Nineteenth Century References

CHAPTER Ⅶ

Definitions and Basic Properties of Homology Theory

1.Introduction

2.Definition of Cubical Singular Homology Groups

3.The Homomorphism Induced by a Continuous Map

4.The Homotopy Property of the Induced Homomorphisms

5.The Exact Homology Sequence of a Pair

6.The Main Properties of Relative Homology Groups

7.The Subdivision of Singular Cubes and the Proof of Theorem 6.4

CHAPTER Ⅷ

Determination of the Homology Groups of Certain Spaces:

Applications and Further Properties of Homology Theory

1.Introduction

2.Homology Groups of Cells and Spheres——Applications

3.Homology of Finite Graphs

4.Homology of Compact Surfaces

5.The Mayer-Vietoris Exact Sequence

6.The Jordan-Brouwer Separation Theorem and lnvariance of Domain

7.The Relation between the Fundamental Group and the First Homology Group

References

CHAPTER Ⅸ

Homology of CW-Complexes

1.Introduction

2.Adjoining Cells to a Space

3.CW-Complexes

4.The Homology Groups of a CW-Complex

5.Incidence Numbers and Orientations of Cells

6.Regular CW-Complexes

7.Determination of Incidence Numbers for a Regular Cell Complex

8.Homology Groups of a Pseudomanifold References

CHAPTER Ⅹ

Homology with Arbitrary Coefficient Groups

1.Introduction

2.Chain Complexes

3.Definition and Basic Properties of Homology with Arbitrary Coefficients

4.Intuitive Geometric Picture of a Cycle with Coefficients in G

5.Coefficient Homomorphisms and Coefficient Exact Sequences

6.The Universal Coefficient Theorem

7.Further Properties of Homology with Arbitrary Coefficients References

CHAPTER Ⅺ

The Homology of Product Spaces

1.Introduction

2.The Product of CW-Complexes and the Tensor Product of Chain Complexes

3.The Singular Chain Complex of a Product Space

4.The Homology of the Tensor Product of Chain Complexes (The Kiinneth Theorem)

5.Proof of the Eilenberg-Zilber Theorem

6.Formulas for the Homology Groups of Product Spaces References

CHAPTER Ⅻ

Cohomology Theory

CHAPTER ⅩⅢ

Products in Homology and Cohomology

CHAPTER ⅩⅣ

Duality Theorems for the Homology of Manifolds

CHAPTER ⅩⅤ

Cup Products in Projective Spaces and Applications of Cup Products

APPENDIX A

A Proof of De Rham's Theorem

APPENDIX B

Permutation Groups or Transformation Groups

Index