Contents

Introduction

1. What is algebraic topology?

2. The fundamental group

4. Homotopy invariance

5. Calculations: π1(R) = 0 and π1(S1) = Z

7. The fundamental theorem of algebra

2. Functors

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

1. The definition of covering spaces

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

2. Edge paths and trees

3. The homotopy types of graphs

5. Applications to groups

1. The definition of compactly generated spaces

2. The category of compactly generated spaces

1. The definition of cofibrations

2. Mapping cylinders and cofibrations

4. A criterion for a map to be a cofibration

5. Cofiber homotopy equivalence

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

2. Cones, suspensions, paths, loops

3. Based cofibrations

4. Cofiber sequences

6. Fiber sequences

7. Connections between cofiber and fiber sequences

2. Long exact sequences associated to pairs

4. A few calculations

5. Change of basepoint

6. n-Equivalences, weak equivalences, and a technical lemma

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

1. Statement of the homotopy excision theorem

2. The Freudenthal suspension theorem

3. Proof of the homotopy excision theorem

2. Maps and homotopies of maps of chain complexes

3. Tensor products of chain complexes

4. Short and long exact sequences

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

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

1. The Hurewicz theorem

2. The uniqueness of the homology of CW complexes

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

1. Universal coefficients in homology

2. The Künneth theorem

3. Hom functors and universal coefficients in cohomology

5. Relations between (Circled Times) and Hom

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

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

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

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

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

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

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

1. The cobordism groups of smooth closed manifolds

2. Sketch proof that N* 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

2. Textbooks in algebraic topology and homotopy theory

4. Differential forms and Morse theory

9. The Eilenberg-Moore spectral sequence

12. Characteristic classes

14. Hopf algebras; the Steenrod algebra, Adams spectral sequence

17. Quillen model categories

19. Infinite loop space theory

21. Follow-ups to this book

Index