Lecture notes in algebraic topology / James F. Davis, Paul Kirk.

Format:

Book

Author/Creator:

Davis, James F. (James Frederic), 1955-

Contributor:

Kirk, P. (Paul)

Series:

Graduate studies in mathematics 1065-7339 ; v. 35.

Graduate studies in mathematics, 1065-7339 ; v. 35

Language:

English

Subjects (All):

Algebraic topology.

Physical Description:

xv, 367 pages : illustrations ; 27 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2001]

Contents:

Chapter 1. Chain Complexes, Homology, and Cohomology 1

1.1. Chain complexes associated to a space 1

1.2. Tensor products, adjoint functors, and Hom 8

1.3. Tensor and Hom functors on chain complexes 12

1.4. Singular cohomology 14

1.5. The Eilenberg-Steenrod axioms 19

Chapter 2. Homological Algebra 23

2.1. Axioms for Tor and Ext; projective resolutions 23

2.2. Projective and injective modules 29

2.3. Resolutions 33

2.4. Definition of Tor and Ext - existence 35

2.5. The fundamental lemma of homological algebra 36

2.6. Universal coefficient theorems 43

Chapter 3. Products 51

3.1. Tensor products of chain complexes and the algebraic Kunneth theorem 51

3.2. The Eilenberg-Zilber maps 54

3.3. Cross and cup products 56

3.4. The Alexander-Whitney diagonal approximation 64

3.5. Relative cup and cap products 67

Chapter 4. Fiber Bundles 77

4.1. Group actions 77

4.2. Fiber bundles 78

4.3. Examples of fiber bundles 81

4.4. Principal bundles and associated bundles 84

4.5. Reducing the structure group 89

4.6. Maps of bundles and pullbacks 90

Chapter 5. Homology with Local Coefficients 95

5.1. Definition of homology with twisted coefficients 96

5.2. Examples and basic properties 98

5.3. Definition of homology with a local coefficient system 103

5.4. Functoriality 105

Chapter 6. Fibrations, Cofibrations and Homotopy Groups 111

6.1. Compactly generated spaces 111

6.2. Fibrations 114

6.3. The fiber of a fibration 116

6.4. Path space fibrations 120

6.5. Fiber homotopy 123

6.6. Replacing a map by a fibration 123

6.7. Cofibrations 127

6.8. Replacing a map by a cofibration 131

6.9. Sets of homotopy classes of maps 134

6.10. Adjoint of loops and suspension; smash products 136

6.11. Fibration and cofibration sequences 138

6.12. Puppe sequences 141

6.13. Homotopy groups 143

6.14. Examples of fibrations 145

6.15. Relative homotopy groups 152

6.16. The action of the fundamental group on homotopy sets 155

6.17. The Hurewicz and Whitehead theorems 160

6.18. Projects for Chapter 6 163

Chapter 7. Obstruction Theory and Eilenberg-MacLane Spaces 165

7.1. Basic problems of obstruction theory 165

7.2. The obstruction cocycle 168

7.3. Construction of the obstruction cocycle 169

7.4. Proof of the extension theorem 172

7.5. Obstructions to finding a homotopy 175

7.6. Primary obstructions 176

7.7. Eilenberg-MacLane spaces 177

7.8. Aspherical spaces 183

7.9. CW-approximations and Whitehead's theorem 185

7.10. Obstruction theory in fibrations 189

7.11. Characteristic classes 191

Chapter 8. Bordism, Spectra, and Generalized Homology 195

8.1. Framed bordism and homotopy groups of spheres 196

8.2. Suspension and the Freudenthal theorem 202

8.3. Stable tangential framings 204

8.4. Spectra 210

8.5. More general bordism theories 213

8.6. Classifying spaces 217

8.7. Construction of the Thom spectra 219

8.8. Generalized homology theories 227

Chapter 9. Spectral Sequences 237

9.1. Definition of a spectral sequence 237

9.2. The Leray-Serre-Atiyah-Hirzebruch spectral sequence 241

9.3. The edge homomorphisms and the transgression 245

9.4. Applications of the homology spectral sequence 249

9.5. The cohomology spectral sequence 254

9.6. Homology of groups 261

9.7. Homology of covering spaces 264

9.8. Relative spectral sequences 266

Chapter 10. Further Applications of Spectral Sequences 267

10.1. Serre classes of abelian groups 267

10.2. Homotopy groups of spheres 276

10.3. Suspension, looping, and the transgression 279

10.4. Cohomology operations 283

10.5. The mod 2 Steenrod algebra 288

10.6. The Thom isomorphism theorem 295

10.7. Intersection theory 299

10.8. Stiefel-Whitney classes 306

10.9. Localization 312

10.10. Construction of bordism invariants 317

Chapter 11. Simple-Homotopy Theory 323

11.2. Invertible matrices and K[subscript 1](R) 326

11.3. Torsion for chain complexes 334

11.4. Whitehead torsion for CW-complexes 343

11.5. Reidemeister torsion 346

11.6. Torsion and lens spaces 348

11.7. The s-cobordism theorem 357.

Notes:

Includes bibliographical references and index.

ISBN:

0821821601

OCLC:

46565070