Algebraic Topology

Description

Cover Title page Preface Contents Introduction 1. Set Theory 2. General Topology 3. Group Theory 4. Modules 5. Euclidean Spaces Chapter One: Homotopy and the Fundamental Group 1. Categories 2. Functors 3. Homotopy 4. Retraction and Deformation 5. H spaces 6. Suspension 7. The Fundamental Groupoid 8. The Fundamental Group Exercises Chapter Two: Covering Spaces and Fibrations 1. Covering Projections 2. The Homotopy Lifting Property 3. Relations with the Fundamental Group 4. The Lifting Problem 5. The Classification of Covering Spaces 6. Covering Transformations 7. Fiber Bundles 8. Fibrations Exercises Chapter Three: Polyhedra 1. Simplicial Complexes 2. Linearity in Simplicial Complexes 3. Subdivision 4. Simplicial Approximation 5. Contiguity Classes 6. The Edge-Path Groupoid 7. Graphs 8. Examples and Applications Exercises Chapter Four: Homology 1. Chain Complexes 2. Chain Homotopy 3. The Homology of Simplicial Complexes 4. Singular Homology 5. Exactness 6. Mayer-Vietoris Sequences 7. Some Applications of Homology 8. Axiomatic Characterization of Homology Exercises Chapter Five: Products 1. Homology with Coefficients 2. The Universal-Coefficient Theorem for Homology 3. The Künneth Formula 4. Cohomology 5. The Universal-Coefficient Theorem for Cohomology 6. Cup and Cap Products 7. Homology of Fiber Bundles 8. The Cohomology Algebra 9. The Steenrod Squaring Operations Exercises Chapter Six: General Cohomology Theory and Duality 1. The Slant Product 2. Duality in Topological Manifolds 3. The Fundamental Class of a Manifold 4. The Alexander Cohomology Theory 5. The Homotopy Axiom for the Alexander Theory 6. Tautness and Continuity 7. Presheaves 8. Fine Presheaves 9. Applications of the Cohomology of Presheaves 10. Characteristic Classes Exercises Chapter Seven: Homotopy Theory 1. Exact Sequences of Sets of Homotopy Classes 2. Higher Homotopy Groups 3. Change of Base Points 4. The Hurewicz Homomorphism 5. The Hurewicz Isomorphism Ttheorem 6. CW complexes 7. Homotopy Functors 8. Weak Homotopy Type Exercises Chapter Eight: Obstruction Theory 1. Eilenberg-MacLane Spaces 2. Principal Fibrations 3. Moore-Postnikov Factorizations 4. Obstruction Theory 5. The Suspension Map Exercises Chapter Nine: Spectral Sequences and Homotopy Groups of Spheres 1. Spectral Sequences 2. The Spectral Sequence of a Fibration 3. Applications of the Homology Spectral Sequence 4. Multiplicative Properties of Spectral Sequences 5. Applications of the Cohomology Spectral Sequence 6. Serre Classes of Abelian Groups 7. Homotopy Groups of Spheres Exercises Index