Elements of Algebraic Topology [Hardcover]

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 EilenbergSteenrod Axioms

3.1 Introduction

3.2 The Exact Homology Sequence

3.3 The Zig-Zag Lemma

3.4 The MayerVietoris Sequence

3.5 The EilenbergSteenrod 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 AlÃN