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Modern Classical Homotopy Theory
 
Jeffrey Strom Western Michigan University, Kalamazoo, MI
Modern Classical Homotopy Theory
Softcover ISBN:  978-1-4704-7163-7
Product Code:  GSM/127.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-1188-6
Product Code:  GSM/127.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7163-7
eBook: ISBN:  978-1-4704-1188-6
Product Code:  GSM/127.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
Modern Classical Homotopy Theory
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Modern Classical Homotopy Theory
Jeffrey Strom Western Michigan University, Kalamazoo, MI
Softcover ISBN:  978-1-4704-7163-7
Product Code:  GSM/127.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-1188-6
Product Code:  GSM/127.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7163-7
eBook ISBN:  978-1-4704-1188-6
Product Code:  GSM/127.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 1272011; 835 pp
    MSC: 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.

    Readership

    Graduate 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. Semi-formal 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 Leray-Serre spectral sequence
    • Chapter 32. Application: Bott periodicity
    • Chapter 33. Using the Leray-Serre 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1272011; 835 pp
MSC: 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.

Readership

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. Semi-formal 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 Leray-Serre spectral sequence
  • Chapter 32. Application: Bott periodicity
  • Chapter 33. Using the Leray-Serre 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 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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