Algebraic Topology

Note

This project is under active development.

Copyright 2001 by Allen Hatcher

Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved.

Electronic copy

Contents:

• Preface
  • Standard Notations
• Chapter 0. Some Underlying Geometric Notations
  • Homotopy and Homotopy Type
  • Cell complexes
  • Operations on Spaces
  • Two Criteria for Homotopy Equivalence
    • Collapsing Subspaces
    • Attaching Spaces
  • The Homotopy Extension Property
  • Exercises
• Chapter 1. The Fundamental Group
  • The Idea of the Fundamental Group
    • 1.1. Basic Constructions
    • 1.2. Van Kampen’s Theorem
    • 1.3. Covering Spaces
    • Additional Topics
• Chapter 2. Homology
  • The Idea of Homology
    • 2.1. Simplicial and Singular Homology
    • 2.2. Computations and Applications
    • 2.3. The Formal Viewpoint
    • Additional Topics
• 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. Poincaré 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

Indices and tables

• Index

• Module Index

• Search Page