Elements of Algebraic Topology

1 Homology Groups of a Simplicial Complex

1.1 Introduction

1.2 Simplices

1.3 Simplicial Complexes and Simplicial Maps

1.4 Abstract Simplicial Complexes

1.5 Review of Abelian Groups

1.6 Homology Groups

1.7 Homology Groups of Surfaces

1.8 Zero-Dimensional Homology

1.9 The Homology of a Cone

1.10 Relative Homology

1.11 ?Homology with Arbitrary Coefficients

1.12 ?The Computability of Homology Groups

1.13 Homomorphisms Induced by Simplicial Maps

1.14 Chain Complexes and Acyclic Carriers

2 Topological Invariance of the Homology Groups

2.1 Introduction

2.2 Simplicial Approximations

2.3 Barycentric Subdivision

2.4 The Simplicial Approximation Theorem

2.5 The Algebra of Subdivision

2.6 Topological Invariance of the Homology Groups

2.7 Homomorphisms Induced by Homotopic Maps

2.8 Review of Quotient Spaces

2.9 ?Application: Maps of Spheres

2.10 ?The Lefschetz Fixed Point Theorem

3 Relative Homology and the Eilenberg-Steenrod Axioms

3.1 Introduction

3.2 The Exact Homology Sequence

3.3 The Zig-Zag Lemma

3.4 The Mayer-Vietoris Sequence

3.5 The Eilenberg-Steenrod Axioms

3.6 The Axioms for Simplicial Theory

3.7 ?Categories and Functors

4 Singular Homology Theory

4.1 Introduction

4.2 The Singular Homology Groups

4.3 The Axioms for Singular Theory

4.4 Excisionin Singular Homology

4.5 ?Acyclic Models

4.6 Mayer-Vietoris Sequences

4.7 The Isomorphism Between Simplicial and Singular Homology

4.8 ?Application: Local Homology Groups and Manifolds

4.9 ?Application: The Jordan Curve Theorem

4.10 The Fundamental Group

4.11 More on Quotient Spaces

4.12 CW Complexes

4.13 The Homology of CW Complexes

4.14 ?Application: Projective Spaces and Lens Spaces

5 Cohomology

5.1 Introduction

5.2 The Hom Functor

5.3 Simplicial Cohomology Groups

5.4 Relative Cohomology

5.5 Cohomology Theory

5.6 The Cohomology of Free Chain Complexes

5.7 ?Chain Equivalences in Free Chain Complexes

5.8 The Cohomology of CW Complexes

5.9 Cup Products

5.10 Cohomology Rings of Surfaces

6 Homology with Coefficients

6.1 Introduction

6.2 Tensor Products

6.3 Homology with Arbitrary Coefficients

7 Homological Algebra

7.1 Introduction

7.2 The Ext Functor

7.3 The Universal Coefficient Theorem

7.4 Torsion Products

7.5 The Universal Coefficient Theorem for Homology

7.6 ?Other Universal Coefficient Theorems

7.7 Tensor Products of Chain Complexes

7.8 The Kuenneth Theorem

7.9 TheEilenberg-Zilber Theorem

7.10 ?The Kuenneth Theorem for Cohomolgy

7.11 ?Application: The Cohomology Ring of a Product Space

8 Duality in Manifolds

8.1 Introduction

8.2 The Join of Two Complexes

8.3 Homology Manifolds

8.4 The Dual Block Complex

8.5 Poincare Duality

8.6 Cap Products

8.7 A Second Proof of Poincare Duality

8.8 ?Application: Cohomology Rings of Manifolds

8.9 ?Application: Homotopy Classification of Lens Spaces

8.10 Lefschetz Duality

8.11 Alexander Duality

8.12 Natural Versions of Duality

8.13 Cech Cohomology

8.14 Alexander-Pontryagin Duality