Algebraic Topology

Description

In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. Cover Table of Contents Preface 0. Some Underlying Geometric Notions Homotopy and Homotopy Type Cell Complexes Operations on Spaces Two Criteria for Homotopy Equivalence The Homotopy Extension Property 1. The Fundamental Group 1.1. Basic Constructions Paths and Homotopy The Fundamental Group of the Circle Induced Homomorphism 1.2. Van Kampen's Theorem Free Products of Groups The van Kanpen Theorem Applications to Cell Complexes 1.3. Covering Spaces Lifting Properties The Classification of Covering Spaces Deck Transformations and Group Actions 1.A. Graphs and Free Groups 1.B. K(G,1) Spaces and Graphs of Groups 2. Homology 2.1. Simplicial and Singular Homology Delta-Complexes Simplicial Homology Singular Homology Homotopy Invariance Exact Sequence 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 2.A. Homology and Fundamental Group 2.B. Classical Approximations 2.C. Simplicial Approximation 3. Cohomology 3.1. Cohomology Groups The Universal Coefficient Theorem Cohomology of Spaces 3.2. Cup Product The Cohomology Ring A Kunneth Formula Spaces with Polynomial Cohomology 3.3. Poincare Duality Orientations and Homology The Duality Theorem Connection with Cup Product Other Forms of Duality 3.A. Universal Coefficients for Homology 3.B. The General Kunneth 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 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 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