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The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems.

To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.

Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book.

The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic K-theory and the s-cobordism theorem.

A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the “big picture”, teaches them how to give mathematical lectures, and prepares them for participating in research seminars.

The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.

Chapter 1. Chain Complexes, Homology, and Cohomology 18
§1.1. Chain complexes associated to a space 18
§1.2. Tensor products, adjoint functors, and Horn 25
§1.3. Tensor and Horn functors on chain complexes 29
§1.4. Singular cohomology 31
§1.5. The Eilenberg-Steenrod axioms 36
§1.6. Projects for Chapter 1 39
Chapter 2. Homological Algebra 40
§2.1. Axioms for Tor and Ext; projective resolutions 40
§2.2. Projective and injective modules 46
§2.3. Resolutions 50
§2.4. Definition of Tor and Ext - existence 52
§2.5. The fundamental lemma of homologieal algebra 53
§2.6. Universal coefficient theorems 60
§2.7. Projects for Chapter 2 66
Chapter 3. Products 68
§3.1. Tensor products of chain complexes and the algebraic Kiinneth theorem 68
§3.2. The Eilenberg-Zilber maps 71
§3.3. Cross and cup products 73
§3.4. The Alexander-Whitney diagonal approximation 81
§3.5. Relative cup and cap products 84
§3.6. Projects for Chapter 3 87
Chapter 4. Fiber Bundles 94
§4.1. Group actions 94
§4.2. Fiber bundles 95
§4.3. Examples of fiber bundles 98
§4.4. Principal bundles and associated bundles 101
§4.5. Reducing the structure group 106
§4.6. Maps of bundles and pullbacks 107
§4.7. Projects for Chapter 4 109
Chapter 5. Homology with Local Coefficients 112
§5.1. Definition of homology with twisted coefficients 113
§5.2. Examples and basic properties 115
§5.3. Definition of homology with a local coefficient system 120
§5.4. Functoriality 122
§5.5. Projects for Chapter 5 125
Chapter 6. Fibrations, Cofibrations and Homotopy Groups 128
§6.1. Compactly generated spaces 128
§6.2. Fibrations 131
§6.3. The fiber of a fibration 133
§6.4. Path space fibrations 137
§6.5. Fiber homotopy 140
§6.6. Replacing a map by a fibration 140
56.7. Cofibrations 144
§6.8. Replacing a map by a cofibration 148
§6.9. Sets of homotopy classes of maps 151
§6.10. Adjoint of loops and suspension; smash products 153
§6.11. Fibration and cofibration sequences 155
§6.12. Puppe sequences 158
§6.13. Homotopy groups 160
§6.14. Examples of fibrations 162
§6.15. Relative homotopy groups 169
§6.16. The action of the fundamental group on homotopy sets 172
§6.17. The Hurewicz and Whitehead theorems 177
§6.18. Projects for Chapter 6 180
Chapter 7. Obstruction Theory and Eilenberg-MacLane Spaces 182
§7.1. Basic problems of obstruction theory 182
§7.2. The obstruction cocycle 185
§7.3. Construction of the obstruction cocycle 186
§7.4. Proof of the extension theorem 189
§7.5. Obstructions to finding a homotopy 192
§7.6. Primary obstructions 193
§7.7. Eilenberg- MacLane spaces 194
§7.8. Aspherical spaces 200
§7.9. CW-approximations and Whitehead's theorem 202
§7.10. Obstruction theory in fibrations 206
§7.11. Characteristic classes 208
§7.12. Projects for Chapter 7 209
Chapter 8. Bordism, Spectra, and Generalized Homology 212
§8.1. Framed bordism and homotopy groups of spheres 213
§8.2. Suspension and the Freudenthal theorem 219
§8.3. Stable tangential framings 221
§8.4. Spectra 227
§8.5. More general bordism theories 230
§8.6. Classifying spaces 234
§8.7. Construction of the Thorn spectra 236
§8.8. Generalized homology theories 244
§8.9. Projects for Chapter 8 251
Chapter 9. Spectral Sequences 254
§9.1. Definition of a spectral sequence 254
§9.2. The Leray-Serre-Atiyah-Hirzebruch spectral sequence 258
§9.3. The edge homomorphisms and the transgression 262
§9.4. Applications of the homology spectral sequence 266
§9.5. The cohomology spectral sequence 271
§9.6. Homology of groups 278
§9.7. Homology of covering spaces 281
§9.8. Relative spectral sequences 283
§9.9. Projects for Chapter 9 283
Chapter 10. Further Applications of Spectral Sequences 284
§10.1. Serre classes of abelian groups 284
§10.2. Homotopy groups of spheres 293
§10.3. Suspension, looping, and the transgression 296
§10.4. Cohomology operations 300
§10.5. The mod 2 Steenrod algebra 305
§10.6. The Thorn isomorphism theorem 312
§10.7. Intersection theory 316
§10.8. Stiefel-Whitney classes 323
§10.9. Localization 329
§10.10. Construction of bordism invariants 334
§10.11. Projects for Chapter 10 336
Chapter 11. Simple-Homotopy Theory 340
§11.1. Introduction 340
§11.2. Invertible matrices and K[sub(1)](R) 343
§11.3. Torsion for chain complexes 351
§11.4. Whitehead torsion for CW-complexes 360
§11.5. Reidemeister torsion 363
§11.6. Torsion and lens spaces 365
§11.7. The s-cobordism theorem 374
§11.8. Projects for Chapter 11 374
Bibliography 376
Index 380
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