Algebraic Topology

"This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology." Table of Contents Preface Standard Notations Chapter 0. Some Underlying Geometric Notions Homotopy and Homotopy Type Cell Complexes Operations on Spaces Two Criteria for Homotopy Equivalence The Homotopy Extension Property Chapter 1. The Fundamental Group 1.1. Basic Constructions Paths and Homotopy The Fundamental Group of the Circle Induced Homomorphisms 1.2. Van Kampen’s Theorem Free Products of Groups The van Kampen Theorem Applications to Cell Complexes 1.3. Covering Spaces Lifting Properties The Classification of Covering Spaces Deck Transformations and Group Actions Additional Topics 1.A. Graphs and Free Groups 1.B. K(G,1) Spaces and Graphs of Groups Chapter 2. Homology 2.1. Simplicial and Singular Homology ∆ Complexes Simplicial Homology Singular Homology Homotopy Invariance Exact Sequences and Excision The Equivalence of Simplicial and Singular Homology 2.2. Computations and Applications Degree Cellular Homology Mayer-Vietoris Sequences Homology with Coefficients 2.3. The Formal Viewpoint Axioms for Homology Categories and Functors Additional Topics 2.A. Homology and Fundamental Group 2.B. Classical Applications 2.C. Simplicial Approximation Chapter 3. Cohomology 3.1. Cohomology Groups The Universal Coefficient Theorem Cohomology of Spaces 3.2. Cup Product The Cohomology Ring A Künneth Formula Spaces with Polynomial Cohomology 3.3. Poincare Duality Orientations and Homology The Duality Theorem Connection with Cup Product Other Forms of Duality Additional Topics 3.A. Universal Coefficients for Homology 3.B. The General Künneth Formula 3.C. H–Spaces and Hopf Algebras 3.D. The Cohomology of SO(n) 3.E. Bockstein Homomorphisms 3.F. Limits and Ext 3.G. Transfer Homomorphisms 3.H. Local Coefficients Chapter 4. Homotopy Theory 4.1. Homotopy Groups Definitions and Basic Constructions Whitehead’s Theorem Cellular Approximation CW Approximation 4.2. Elementary Methods of Calculation Excision for Homotopy Groups The Hurewicz Theorem Fiber Bundles Stable Homotopy Groups 4.3. Connections with Cohomology The Homotopy Construction of Cohomology Fibrations Postnikov Towers Obstruction Theory Additional Topics 4.A. Basepoints and Homotopy 4.B. The Hopf Invariant 4.C. Minimal Cell Structures 4.D. Cohomology of Fiber Bundles 4.E. The Brown Representability Theorem 4.F. Spectra and Homology Theories 4.G. Gluing Constructions 4.H. Eckmann-Hilton Duality 4.I. Stable Splittings of Spaces 4.J. The Loopspace of a Suspension 4.K. The Dold-Thom Theorem 4.L. Steenrod Squares and Powers Appendix Topology of Cell Complexes The Compact-Open Topology The Homotopy Extension Property Simplicial CW Structures Bibliography Index