A Concise Course in Algebraic Topology

Introduction
Chapter 1. The fundamental group and some of its applications
1. What is algebraic topology?
2. The fundamental group
3. Dependence on the basepoint
4. Homotopy invariance
5. Calculations: π₁(ℝ) = 0 and π₁(S¹) = ℤ
6. The Brouwer fixed point theorem
7. The fundamental theorem of algebra
Chapter 2. Categorical language and the van Kampen theorem
1. Categories
2. Functors
3. Natural transformations
4. Homotopy categories and homotopy equivalences
5. The fundamental groupoid
6. Limits and colimits
7. The van Kampen theorem
8. Examples of the van Kampen theorem
Chapter 3. Covering spaces
1. The definition of covering spaces
2. The unique path lifting property
3. Coverings of groupoids
4. Group actions and orbit categories
5. The classification of coverings of groupoids
6. The construction of coverings of groupoids
7. The classification of coverings of spaces
8. The construction of coverings of spaces
Chapter 4. Graphs
1. The definition of graphs
2. Edge paths and trees
3. The homotopy types of graphs
4. Covers of graphs and Euler characteristics
5. Applications to groups
Chapter 5. Compactly generated spaces
1. The definition of compactly generated spaces
2. The category of compactly generated spaces
Chapter 6. Cofibrations
1. The definition of cofibrations
2. Mapping cylinders and cofibrations
3. Replacing maps by cofibrations
4. A criterion for a map to be a cofibration
5. Cofiber homotopy equivalence
Chapter 7. Fibrations
1. The definition of fibrations
2. Path lifting functions and fibrations
3. Replacing maps by fibrations
4. A criterion for a map to be a fibration
5. Fiber homotopy equivalence
6. Change of fiber
Chapter 8. Based cofiber and fiber sequences
1. Based homotopy classes of maps
2. Cones, suspensions, paths, loops
3. Based cofibrations
4. Cofiber sequences
5. Based fibrations
6. Fiber sequences
7. Connections between cofiber and fiber sequences
Chapter 9. Higher homotopy groups
1. The definition of homotopy groups
2. Long exact sequences associated to pairs
3. Long exact sequences associated to fibrations
4. A few calculations
5. Change of basepoint
6. n-Equivalences, weak equivalences, and a technical lemma
Chapter 10. CW complexes
1. The definition and some examples of CW complexes
2. Some constructions on CW complexes
3. HELP and the Whitehead theorem
4. The cellular approximation theorem
5. Approximation of spaces by CW complexes
6. Approximation of pairs by CW pairs
7. Approximation of excisive triads by CW triads
Chapter 11. The homotopy excision and suspension theorems
1. Statement of the homotopy excision theorem
2. The Freudenthal suspension theorem
3. Proof of the homotopy excision theorem
Chapter 12. A little homological algebra
1. Chain complexes
2. Maps and homotopies of maps of chain complexes
3. Tensor products of chain complexes
4. Short and long exact sequences
Chapter 13. Axiomatic and cellular homology theory
1. Axioms for homology
2. Cellular homology
3. Verification of the axioms
4. The cellular chains of products
5. Some examples: T , K, and ℝPⁿ
Chapter 14. Derivations of properties from the axioms
1. Reduced homology; based versus unbased spaces
2. Cofibrations and the homology of pairs
3. Suspension and the long exact sequence of pairs
4. Axioms for reduced homology
5. Mayer-Vietoris sequences
6. The homology of colimits
Chapter 15. The Hurewicz and uniqueness theorems
1. The Hurewicz theorem
2. The uniqueness of the homology of CW complexes
Chapter 16. Singular homology theory
1. The singular chain complex
2. Geometric realization
3. Proofs of the theorems
4. Simplicial objects in algebraic topology
5. Classifying spaces and K(π, n)s
Chapter 17. Some more homological algebra
1. Universal coefficients in homology
2. The Künneth theorem
3. Hom functors and universal coefficients in cohomology
4. Proof of the universal coefficient theorem
5. Relations between ⊗ and Hom
Chapter 18. Axiomatic and cellular cohomology theory
1. Axioms for cohomology
2. Cellular and singular cohomology
3. Cup products in cohomology
4. An example: ℝPⁿ and the Borsuk-Ulam theorem
5. Obstruction theory
Chapter 19. Derivations of properties from the axioms
1. Reduced cohomology groups and their properties
2. Axioms for reduced cohomology
3. Mayer-Vietoris sequences in cohomology
4. Lim¹ and the cohomology of colimits
5. The uniqueness of the cohomology of CW complexes
Chapter 20. The Poincaré duality theorem
1. Statement of the theorem
2. The definition of the cap product
3. Orientations and fundamental classes
4. The proof of the vanishing theorem
5. The proof of the Poincaré duality theorem
6. The orientation cover
Chapter 21. The index of manifolds; manifolds with boundary
1. The Euler characteristic of compact manifolds
2. The index of compact oriented manifolds
3. Manifolds with boundary
4. Poincaré duality for manifolds with boundary
5. The index of manifolds that are boundaries
Chapter 22. Homology, cohomology, and K(π, n)s
1. K(π, n)s and homology
2. K(π, n)s and cohomology
3. Cup and cap products
4. Postnikov systems
5. Cohomology operations
Chapter 23. Characteristic classes of vector bundles
1. The classification of vector bundles
2. Characteristic classes for vector bundles
3. Stiefel-Whitney classes of manifolds
4. Characteristic numbers of manifolds
5. Thom spaces and the Thom isomorphism theorem
6. The construction of the Stiefel-Whitney classes
7. Chern, Pontryagin, and Euler classes
8. A glimpse at the general theory
Chapter 24. An introduction to K-theory
1. The definition of K-theory
2. The Bott periodicity theorem
3. The splitting principle and the Thom isomorphism
4. The Chern character; almost complex structures on spheres
5. The Adams operations
6. The Hopf invariant one problem and its applications
Chapter 25. An introduction to cobordism
1. The cobordism groups of smooth closed manifolds
2. Sketch proof that ?_∗ is isomorphic to π_∗(TO)
3. Prespectra and the algebra H_∗(TO; ℤ₂)
4. The Steenrod algebra and its coaction on H_∗(TO)
5. The relationship to Stiefel-Whitney numbers
6. Spectra and the computation of π_∗(TO) = π_∗(MO)
7. An introduction to the stable category
Suggestions for further reading
1. A classic book and historical references
2. Textbooks in algebraic topology and homotopy theory
3. Books on CW complexes
4. Differential forms and Morse theory
5. Equivariant algebraic topology
6. Category theory and homological algebra
7. Simplicial sets in algebraic topology
8. The Serre spectral sequence and Serre class theory
9. The Eilenberg-Moore spectral sequence
10. Cohomology operations
11. Vector bundles
12. Characteristic classes
13. K-theory
14. Hopf algebras; the Steenrod algebra, Adams spectral sequence
15. Cobordism
16. Generalized homology theory and stable homotopy theory
17. Quillen model categories
18. Localization and completion; rational homotopy theory
19. Infinite loop space theory
20. Complex cobordism and stable homotopy theory
21. Follow-ups to this book