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.
Author(s): K. Ponto, J. P. May
Series: Chicago Lectures in Mathematics
Publisher: University Of Chicago Press
Year: 2012
Language: English
Pages: 544
Tags: Математика;Топология;Алгебраическая топология;
Contents......Page 6
Introduction......Page 12
Some conventions and notations......Page 22
Acknowledgments......Page 28
Part 1: Preliminaries: Basic homotopytheory and nilpotent spaces......Page 30
1. Cofibrations and Fibrations
......Page 32
2. Homotopy Colimits and Homotopy Limits; lim1
......Page 53
3. Nilpotent Spaces and Postnikov Towers
......Page 75
4. Detecting Nilpotent Groups and Spaces
......Page 98
Part 2: Localizations of spaces at sets of primes......Page 114
5. Localizations of Nilpotent Groups and Spaces
......Page 116
6. Characterizations and Properties of Localizations
......Page 140
7. Fracture Theorems for Localization: Groups
......Page 161
8. Fracture Theorems for Localization: Spaces
......Page 183
9. Rational H-Spaces and Fracture Theorems
......Page 204
Part 3: Completions of spaces at sets of primes......Page 218
10. Completions of Nilpotent Groups and Spaces
......Page 220
11. Characterizations and Properties of Completions
......Page 243
12. Fracture Theorems for Completion: Groups
......Page 255
13. Fracture Theorems for Completion: Spaces
......Page 272
Part 4: An introduction to model category theory......Page 294
14. An Introduction to Model Category Theory......Page 296
15. Cofibrantly Generated and Proper Model Categories
......Page 321
16. Categorical Perspectives on Model Categories
......Page 343
17. Model Structures on the Category of Spaces
......Page 368
18. Model Structures on Categories of Chain Complexes
......Page 401
19. Resolution and Localization Model Structures......Page 424
Part 5: Bialgebras and Hopf algebras......Page 444
20. Bialgebras and Hopf Algebras
......Page 446
21. Connected and Component Hopf Algebras
......Page 461
22. Lie Algebras and Hopf Algebras in Characteristic Zero
......Page 482
23. Restricted Lie Algebras and Hopf Algebras in Characteristic p
......Page 494
24. A Primer on Spectral Sequences
......Page 506
Bibliography......Page 526
Index......Page 534
