**Graduate Studies in Mathematics**

Volume: 127;
2011;
835 pp;
Hardcover

MSC: Primary 55;

Print ISBN: 978-0-8218-5286-6

Product Code: GSM/127

List Price: $95.00

Individual Member Price: $76.00

**Electronic ISBN: 978-1-4704-1188-6
Product Code: GSM/127.E**

List Price: $95.00

Individual Member Price: $76.00

#### You may also like

#### Supplemental Materials

# Modern Classical Homotopy Theory

Share this page
*Jeffrey Strom*

The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory.

This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

#### Table of Contents

# Table of Contents

## Modern Classical Homotopy Theory

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xvii18 free
- Part I. The language of categories 124 free
- Categories and functors 326
- Limits and colimits 2952
- Part II. Semi-formal homotopy theory 4366
- Categories of spaces 4568
- Homotopy 6992
- Cofibrations and fibrations 99122
- Homotopy limits and colimits 143166
- Homotopy pushout and pullback squares 181204
- Tools and techniques 199222
- Topics and examples 221244
- Model categories 261284
- Part III. Four topological inputs 273296
- The concept of dimension in homotopy theory 275298
- Subdivision of disks 289312
- The local nature of fibrations 305328
- Pullbacks of cofibrations 317340
- Related topics 323346
- Part IV. Targets as domains, domains as targets 351374
- Constructions of spaces and maps 353376
- Understanding suspension 373396
- Comparing pushouts and pullbacks 393416
- Some computations in homotopy theory 413436
- Further topics 435458
- Part V. Cohomology and homology 457480
- Cohomology 459482
- Homology 499522
- Cohomology operations 515538
- Chain complexes 541564
- Topics, problems and projects 553576
- Part VI. Cohomology, homology and fibrations 589612
- The Wang sequence 591614
- Cohomology of filtered spaces 605628
- The Serre filtration of a fibration 623646
- Application: Incompressibility 635658
- The spectral sequence of a filtered space 645668
- The Leray-Serre spectral sequence 659682
- Application: Bott periodicity 681704
- Using the Leray-Serre spectral sequence 699722
- Part VII. Vistas 719742
- Localization and completion 721744
- Exponents for homotopy groups 745768
- Classes of spaces 759782
- Miller’s theorem 773796
- Some algebra 789812
- References 811834
- Index of notation 821844 free
- Index 823846 free
- Back Cover Back Cover1862

#### Readership

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

#### Reviews

Obviously the book was a labor of love for its author: this is visible on every page. The coverage of the material is, in a word, amazing, even to an outsider like me. The book is well-written, as I have already indicated, and Strom's "problems first-and-foremost" approach is bound to be a big pedagogical hit for those who can handle it, both in front of the class and in it. The book under review is a wonderful contribution indeed.

-- MAA Reviews