More Concise Algebraic Topology Localization, Completion, and Model Categories
by J. P. May and K. Ponto
University of Chicago Press, 2011
Cloth: 978-0-226-51178-8 | Electronic: 978-0-226-51179-5
DOI: 10.7208/chicago/9780226511795.001.0001


With firm foundations dating only from the 1950s, algebraic topology is a relatively young area of mathematics. There are very few textbooks that treat fundamental topics beyond a first course, and many topics now essential to the field are not treated in any textbook. J. Peter May’s A Concise Course in Algebraic Topology addresses the standard first course material, such as fundamental groups, covering spaces, the basics of homotopy theory, and homology and cohomology. In this sequel, May and his coauthor, Kathleen Ponto, cover topics that are essential for algebraic topologists and others interested in algebraic topology, but that are not treated in standard texts. They focus on the localization and completion of topological spaces, model categories, and Hopf algebras.
The first half of the book sets out the basic theory of localization and completion of nilpotent spaces, using the most elementary treatment the authors know of. It makes no use of simplicial techniques or model categories, and it provides full details of other necessary preliminaries. With these topics as motivation, most of the second half of the book sets out the theory of model categories, which is the central organizing framework for homotopical algebra in general. Examples from topology and homological algebra are treated in parallel. A short last part develops the basic theory of bialgebras and Hopf algebras.


J. P. May is professor of mathematics at the University of Chicago; he is the author or coauthor of many papers and books, including Simplicial Objects in Algebraic Topology and A Concise Course in Algebraic Topology, both also in this series. K. Ponto is assistant professor of mathematics at the University of Kentucky.     


“May and Ponto have done an excellent job of assembling important results scattered throughout the mathematical literature, primarily in research articles, into a coherent, compelling whole. All researchers in algebraic topology should have at least a passing acquaintance with the material treated in this book, much of which does not appear in any of the standard texts.”
— Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne

“This book fills thus a gap in the literature and will certainly serve as a reference in the field.”
— Zentralblatt MATH



Some conventions and notations


Part 1: Preliminaries: Basic homotopytheory and nilpotent spaces

1. Cofibrations and Fibrations

2. Homotopy Colimits and Homotopy Limits; lim1

3. Nilpotent Spaces and Postnikov Towers

4. Detecting Nilpotent Groups and Spaces

Part 2: Localizations of spaces at sets of primes

5. Localizations of Nilpotent Groups and Spaces

6. Characterizations and Properties of Localizations

7. Fracture Theorems for Localization: Groups

8. Fracture Theorems for Localization: Spaces

9. Rational H-Spaces and Fracture Theorems

Part 3: Completions of spaces at sets of primes

10. Completions of Nilpotent Groups and Spaces

11. Characterizations and Properties of Completions

12. Fracture Theorems for Completion: Groups

13. Fracture Theorems for Completion: Spaces

Part 4: An introduction to model category theory

14. An Introduction to Model Category Theory

15. Cofibrantly Generated and Proper Model Categories

16. Categorical Perspectives on Model Categories

17. Model Structures on the Category of Spaces

18. Model Structures on Categories of Chain Complexes

19. Resolution and Localization Model Structures

Part 5: Bialgebras and Hopf algebras

20. Bialgebras and Hopf Algebras

21. Connected and Component Hopf Algebras

22. Lie Algebras and Hopf Algebras in Characteristic Zero

23. Restricted Lie Algebras and Hopf Algebras in Characteristic p

24. A Primer on Spectral Sequences