## 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.

### Appendix

Topology of Cell Complexes. The Compact-Open Topology.

**Bibliography**

**Index**