**Graduate Studies in Mathematics**

Volume: 35;
2001;
367 pp;
Hardcover

MSC: Primary 55; 57;

Print ISBN: 978-0-8218-2160-2

Product Code: GSM/35

List Price: $68.00

Individual Member Price: $54.40

**Electronic ISBN: 978-1-4704-2088-8
Product Code: GSM/35.E**

List Price: $68.00

Individual Member Price: $54.40

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# Lecture Notes in Algebraic Topology

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*James F. Davis; Paul Kirk*

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.

#### Table of Contents

# Table of Contents

## Lecture Notes in Algebraic Topology

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface xi12 free
- Projects xiv15 free
- Chapter 1. Chain Complexes, Homology, and Cohomology 118 free
- Chapter 2. Homological Algebra 2340
- §2.1. Axioms for Tor and Ext; projective resolutions 2340
- §2.2. Projective and injective modules 2946
- §2.3. Resolutions 3350
- §2.4. Definition of Tor and Ext - existence 3552
- §2.5. The fundamental lemma of homologieal algebra 3653
- §2.6. Universal coefficient theorems 4360
- §2.7. Projects for Chapter 2 4966

- Chapter 3. Products 5168
- Chapter 4. Fiber Bundles 7794
- Chapter 5. Homology with Local Coefficients 95112
- Chapter 6. Fibrations, Cofibrations and Homotopy Groups 111128
- §6.1. Compactly generated spaces 111128
- §6.2. Fibrations 114131
- §6.3. The fiber of a fibration 116133
- §6.4. Path space fibrations 120137
- §6.5. Fiber homotopy 123140
- §6.6. Replacing a map by a fibration 123140
- 56.7. Cofibrations 127144
- §6.8. Replacing a map by a cofibration 131148
- §6.9. Sets of homotopy classes of maps 134151
- §6.10. Adjoint of loops and suspension; smash products 136153
- §6.11. Fibration and cofibration sequences 138155
- §6.12. Puppe sequences 141158
- §6.13. Homotopy groups 143160
- §6.14. Examples of fibrations 145162
- §6.15. Relative homotopy groups 152169
- §6.16. The action of the fundamental group on homotopy sets 155172
- §6.17. The Hurewicz and Whitehead theorems 160177
- §6.18. Projects for Chapter 6 163180

- Chapter 7. Obstruction Theory and Eilenberg-MacLane Spaces 165182
- §7.1. Basic problems of obstruction theory 165182
- §7.2. The obstruction cocycle 168185
- §7.3. Construction of the obstruction cocycle 169186
- §7.4. Proof of the extension theorem 172189
- §7.5. Obstructions to finding a homotopy 175192
- §7.6. Primary obstructions 176193
- §7.7. Eilenberg- MacLane spaces 177194
- §7.8. Aspherical spaces 183200
- §7.9. CW-approximations and Whitehead's theorem 185202
- §7.10. Obstruction theory in fibrations 189206
- §7.11. Characteristic classes 191208
- §7.12. Projects for Chapter 7 192209

- Chapter 8. Bordism, Spectra, and Generalized Homology 195212
- §8.1. Framed bordism and homotopy groups of spheres 196213
- §8.2. Suspension and the Freudenthal theorem 202219
- §8.3. Stable tangential framings 204221
- §8.4. Spectra 210227
- §8.5. More general bordism theories 213230
- §8.6. Classifying spaces 217234
- §8.7. Construction of the Thorn spectra 219236
- §8.8. Generalized homology theories 227244
- §8.9. Projects for Chapter 8 234251

- Chapter 9. Spectral Sequences 237254
- §9.1. Definition of a spectral sequence 237254
- §9.2. The Leray-Serre-Atiyah-Hirzebruch spectral sequence 241258
- §9.3. The edge homomorphisms and the transgression 245262
- §9.4. Applications of the homology spectral sequence 249266
- §9.5. The cohomology spectral sequence 254271
- §9.6. Homology of groups 261278
- §9.7. Homology of covering spaces 264281
- §9.8. Relative spectral sequences 266283
- §9.9. Projects for Chapter 9 266283

- Chapter 10. Further Applications of Spectral Sequences 267284
- §10.1. Serre classes of abelian groups 267284
- §10.2. Homotopy groups of spheres 276293
- §10.3. Suspension, looping, and the transgression 279296
- §10.4. Cohomology operations 283300
- §10.5. The mod 2 Steenrod algebra 288305
- §10.6. The Thorn isomorphism theorem 295312
- §10.7. Intersection theory 299316
- §10.8. Stiefel-Whitney classes 306323
- §10.9. Localization 312329
- §10.10. Construction of bordism invariants 317334
- §10.11. Projects for Chapter 10 319336

- Chapter 11. Simple-Homotopy Theory 323340
- §11.1. Introduction 323340
- §11.2. Invertible matrices and K[sub(1)](R) 326343
- §11.3. Torsion for chain complexes 334351
- §11.4. Whitehead torsion for CW-complexes 343360
- §11.5. Reidemeister torsion 346363
- §11.6. Torsion and lens spaces 348365
- §11.7. The s-cobordism theorem 357374
- §11.8. Projects for Chapter 11 357374

- Bibliography 359376
- Index 363380
- Back Cover Back Cover1385

#### Readership

Graduate students and research mathematicians interested in geometric topology and homotopy theory.

#### Reviews

Many exercises and comments in the book, which complement the material, as well as suggestions for further study, presented in the form of projects … The book is a nice advanced textbook on algebraic topology and can be recommended to anybody interested in modern and advanced algebraic topology.

-- European Mathematical Society Newsletter

The book might well have been titled ‘What Every Young Topologist Should Know’ … presents, in a self-contained and clear manner, all classical constituents of algebraic topology … recommend this book as a valuable tool for everybody teaching graduate courses as well as a self-contained introduction … for independent reading.

-- Mathematica Bohemica