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Book DetailsGraduate Studies in MathematicsVolume: 127; 2011; 835 ppMSC: Primary 55;
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.ReadershipGraduate students and research mathematicians interested in algebraic topology and homotopy theory.

Table of Contents

Part 1. The language of categories

Chapter 1. Categories and functors

Chapter 2. Limits and colimits

Part 2. Semiformal homotopy theory

Chapter 3. Categories of spaces

Chapter 4. Homotopy

Chapter 5. Cofibrations and fibrations

Chapter 6. Homotopy limits and colimits

Chapter 7. Homotopy pushout and pullback squares

Chapter 8. Tools and techniques

Chapter 9. Topics and examples

Chapter 10. Model categories

Part 3. Four topological inputs

Chapter 11. The concept of dimension in homotopy theory

Chapter 12. Subdivision of disks

Chapter 13. The local nature of fibrations

Chapter 14. Pullbacks of cofibrations

Chapter 15. Related topics

Part 4. Targets as domains, domains as targets

Chapter 16. Constructions of spaces and maps

Chapter 17. Understanding suspension

Chapter 18. Comparing pushouts and pullbacks

Chapter 19. Some computations in homotopy theory

Chapter 20. Further topics

Part 5. Cohomology and homology

Chapter 21. Cohomology

Chapter 22. Homology

Chapter 23. Cohomology operations

Chapter 24. Chain complexes

Chapter 25. Topics, problems and projects

Part 6. Cohomology, homology and fibrations

Chapter 26. The Wang sequence

Chapter 27. Cohomology of filtered spaces

Chapter 28. The Serre filtration of a fibration

Chapter 29. Application: Incompressibility

Chapter 30. The spectral sequence of a filtered space

Chapter 31. The LeraySerre spectral sequence

Chapter 32. Application: Bott periodicity

Chapter 33. Using the LeraySerre spectral sequence

Part 7. Vistas

Chapter 34. Localization and completion

Chapter 35. Exponents for homotopy groups

Chapter 36. Classes of spaces

Chapter 37. Miller’s theorem

Appendix A. Some algebra


Additional Material

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 wellwritten, as I have already indicated, and Strom's "problems firstandforemost" 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


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

Part 1. The language of categories

Chapter 1. Categories and functors

Chapter 2. Limits and colimits

Part 2. Semiformal homotopy theory

Chapter 3. Categories of spaces

Chapter 4. Homotopy

Chapter 5. Cofibrations and fibrations

Chapter 6. Homotopy limits and colimits

Chapter 7. Homotopy pushout and pullback squares

Chapter 8. Tools and techniques

Chapter 9. Topics and examples

Chapter 10. Model categories

Part 3. Four topological inputs

Chapter 11. The concept of dimension in homotopy theory

Chapter 12. Subdivision of disks

Chapter 13. The local nature of fibrations

Chapter 14. Pullbacks of cofibrations

Chapter 15. Related topics

Part 4. Targets as domains, domains as targets

Chapter 16. Constructions of spaces and maps

Chapter 17. Understanding suspension

Chapter 18. Comparing pushouts and pullbacks

Chapter 19. Some computations in homotopy theory

Chapter 20. Further topics

Part 5. Cohomology and homology

Chapter 21. Cohomology

Chapter 22. Homology

Chapter 23. Cohomology operations

Chapter 24. Chain complexes

Chapter 25. Topics, problems and projects

Part 6. Cohomology, homology and fibrations

Chapter 26. The Wang sequence

Chapter 27. Cohomology of filtered spaces

Chapter 28. The Serre filtration of a fibration

Chapter 29. Application: Incompressibility

Chapter 30. The spectral sequence of a filtered space

Chapter 31. The LeraySerre spectral sequence

Chapter 32. Application: Bott periodicity

Chapter 33. Using the LeraySerre spectral sequence

Part 7. Vistas

Chapter 34. Localization and completion

Chapter 35. Exponents for homotopy groups

Chapter 36. Classes of spaces

Chapter 37. Miller’s theorem

Appendix A. Some algebra

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 wellwritten, as I have already indicated, and Strom's "problems firstandforemost" 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