Algebraic Topology

Table of Contents

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. Fundamental Group and Covering Spaces

1. Basic Constructions.
Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms.
2. Van Kampen's Theorem
Free Products of Groups. The van Kampen Theorem. Applications to Cell Complexes.
3. Covering Spaces
Lifting Properties. The Classification of Covering Spaces. Deck Transformations and Group Actions.
4. Additional Topics
Graphs and Free Groups. K(G,1) Spaces and Graphs of Groups.

Chapter 2. Homology

1. Simplicial and Singular Homology
Delta-Complexes. Simplicial Homology. Singular Homology. Homotopy Invariance. Exact Sequences and Excision. The Equivalence of Simplicial and Singular Homology.
2. Computations and Applications
Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with Coefficients.
3. The Formal Viewpoint
Axioms for Homology. Categories and Functors.
4. Additional Topics
Homology and Fundamental Group. Classical Applications. Simplicial Approximation.

Chapter 3. Cohomology

1. Cohomology Groups
The Universal Coefficient Theorem. Cohomology of Spaces.
2. Cup Product
The Cohomology Ring. A Kunneth Formula. Spaces with Polynomial Cohomology.
3. Poincare Duality
Orientations and Homology. The Duality Theorem. Cup Product and Duality. Other Forms of Duality.
4. Additional Topics
The Universal Coefficient Theorem for Homology. The General Kunneth Formula. H-Spaces and Hopf Algebras. The Cohomology of SO(n). Bockstein Homomorphisms. Limits. More About Ext. Transfer Homomorphisms. Local Coefficients.

Chapter 4. Homotopy Theory

1. Homotopy Groups
Definitions and Basic Constructions. Whitehead's Theorem. Cellular Approximation. CW Approximation.
2. Elementary Methods of Calculation
Excision for Homotopy Groups. The Hurewicz Theorem. Fiber Bundles. Stable Homotopy Groups.
3. Connections with Cohomology
The Homotopy Construction of Cohomology. Fibrations. Postnikov Towers. Obstruction Theory.
4. Additional Topics
Basepoints and Homotopy. The Hopf Invariant. Minimal Cell Structures. Cohomology of Fiber Bundles. The Brown Representability Theorem. Spectra and Homology Theories. Gluing Constructions. Eckmann-Hilton Duality. Stable Splittings of Spaces. The Loopspace of a Suspension. Symmetric Products and the Dold-Thom Theorem. Steenrod Squares and Powers.


Topology of Cell Complexes. The Compact-Open Topology.