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More Concise Algebraic Topology

Localization, Completion, and Model Categories

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.


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

Zentralblatt MATH

“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

Table of Contents


Some conventions and notations

Part 1. Preliminaries: basic homotopy theory and nilpotent spaces  
Chapter 1. Cofibrations and fibrations  
 1.1. Relations between cofibrations and fibrations
       1.2. The fill-in and Verdier lemmas  
       1.3. Based and free cofibrations and fibrations  
       1.4. Actions of fundamental groups on homotopy classes of maps 
       1.5. Actions of fundamental groups in fibration sequences  

Chapter 2. Homotopy colimits and homotopy limits; lim1
       2.1. Some basic homotopy colimits
       2.2. Some basic homotopy limits
       2.3. Algebraic properties of lim1
       2.4. An example of nonvanishing lim1 terms
       2.5. The homology of colimits and limits
       2.6. A profinite universal coefficient theorem

Chapter 3. Nilpotent spaces and Postnikov towers
       3.1. A -nilpotent groups and spaces
       3.2. Nilpotent spaces and Postnikov towers
       3.3. Cocellular spaces and the dual Whitehead theorem
       3.4. Fibrations with fiber an Eilenberg–MacLane space
       3.5. Postnikov A -towers

Chapter 4. Detecting nilpotent groups and spaces
       4.1. Nilpotent Actions and Cohomology
       4.2. Universal covers of nilpotent spaces
       4.3. A -Maps of A -nilpotent groups and spaces
       4.4. Nilpotency and fibrations
       4.5. Nilpotent spaces and finite type conditions

Part 2. Localizations of spaces at sets of primes

Chapter 5. Localizations of nilpotent groups and spaces
       5.1. Localizations of abelian groups
       5.2. The definition of localizations of spaces
       5.3. Localizations of nilpotent spaces
       5.4. Localizations of nilpotent groups
       5.5. Algebraic properties of localizations of nilpotent groups
       5.6. Finitely generated T -local groups

Chapter 6. Characterizations and properties of localizations
       6.1. Characterizations of localizations of nilpotent spaces
       6.2. Localizations of limits and fiber sequences
       6.3. Localizations of function spaces
       6.4. Localizations of colimits and cofiber sequences
       6.5. A cellular construction of localizations
       6.6. Localizations of H-spaces and co-H-spaces
       6.7. Rationalization and the finiteness of homotopy groups
       6.8. The vanishing of rational phantom maps

Chapter 7. Fracture theorems for localization: groups
       7.1. Global to local pullback diagrams
       7.2. Global to local: abelian and nilpotent groups
       7.3. Local to global pullback diagrams
       7.4. Local to global: abelian and nilpotent groups
       7.5. The genus of abelian and nilpotent groups
       7.6. Exact sequences of groups and pullbacks

Chapter 8. Fracture theorems for localization: spaces
       8.1. Statements of the main fracture theorems
       8.2. Fracture theorems for maps into nilpotent spaces
       8.3. Global to local fracture theorems: spaces
       8.4. Local to global fracture theorems: spaces
       8.5. The genus of nilpotent spaces
       8.6. Alternative proofs of the fracture theorems

Chapter 9. Rational H-spaces and fracture theorems
       9.1. The structure of rational H-spaces
       9.2. The Samelson product and H?(X;Q)
       9.3. The Whitehead product
       9.4. Fracture theorems for H-spaces

Part 3. Completions of spaces at sets of primes

Chapter 10. Completions of nilpotent groups and spaces
       10.1. Completions of abelian groups
       10.2. The definition of completions of spaces at T
       10.3. Completions of nilpotent spaces
       10.4. Completions of nilpotent groups

Chapter 11. Characterizations and properties of completions
       11.1. Characterizations of completions of nilpotent spaces
       11.2. Completions of limits and fiber sequences
       11.3. Completions of function spaces
       11.4. Completions of colimits and cofiber sequences
       11.5. Completions of H-spaces
       11.6. The vanishing of p-adic phantom maps

Chapter 12. Fracture theorems for completion: Groups
       12.1. Preliminaries on pullbacks and isomorphisms
       12.2. Global to local: abelian and nilpotent groups
       12.3. Local to global: abelian and nilpotent groups
       12.4. Formal completions and the ad`elic genus

Chapter 13. Fracture theorems for completion: Spaces
       13.1. Statements of the main fracture theorems
       13.2. Global to local fracture theorems: spaces
       13.3. Local to global fracture theorems: spaces
       13.4. The tensor product of a space and a ring
       13.5. Sullivan’s formal completion
       13.6. Formal completions and the ad`elic genus

Part 4. An introduction to model category theory

Chapter 14. An introduction to model category theory
       14.1. Preliminary definitions and weak factorization systems
       14.2. The definition and first properties of model categories
       14.3. The notion of homotopy in a model category
       14.4. The homotopy category of a model category

Chapter 15. Cofibrantly generated and proper model categories
       15.1. The small object argument for the construction of WFS’s
       15.2. Compactly and cofibrantly generated model categories
       15.3. Over and under model structures
       15.4. Left and right proper model categories
"       15.5. Left properness, lifting properties, and the sets [X, Y ] "

Chapter 16. Categorical perspectives on model categories
       16.1. Derived functors and derived natural transformations
       16.2. Quillen adjunctions and Quillen equivalences
       16.3. Symmetric monoidal categories and enriched categories
       16.4. Symmetric monoidal and enriched model categories
       16.5. A glimpse at higher categorical structures

Chapter 17. Model structures on the category of spaces
       17.1. The Hurewicz or h-model structure on spaces
       17.2. The Quillen or q-model structure on spaces
       17.3. Mixed model structures in general
       17.4. The mixed model structure on spaces
       17.5. The model structure on simplicial sets
       17.6. The proof of the model axioms

Chapter 18. Model structures on categories of chain complexes
       18.1. The algebraic framework and the analogy with topology
       18.2. h-cofibrations and h-fibrations in ChR
       18.3. The h-model structure on ChR
       18.4. The q-model structure on ChR
       18.5. Proofs and the characterization of q-cofibrations
       18.6. The m-model structure on ChR

Chapter 19. Resolution and localization model structures
       19.1. Resolution and mixed model structures
       19.2. The general context of Bousfield localization
       19.3. Localizations with respect to homology theories
       19.4. Bousfield localization at sets and classes of maps
       19.5. Bousfield localization in enriched model categories

Part 5. Bialgebras and Hopf algebras

Chapter 20. Bialgebras and Hopf algebras
       20.1. Preliminaries 
       "20.2. Algebras, coalgebras, and bialgebras "
       20.3. Antipodes and Hopf algebras 
       "20.4. Modules, comodules, and related concepts "

Chapter 21. Connected and component Hopf algebras 
      "21.1. Connected algebras, coalgebras, and Hopf algebras "
       21.2. Splitting theorems
       21.3. Component coalgebras and the existence of antipodes
       21.4. Self-dual Hopf algebras
       21.5. The homotopy groups of MO and other Thom spectra
       21.6. A proof of the Bott periodicity theorem

Chapter 22. Lie algebras and Hopf algebras in characteristic zero
       22.1. Graded Lie algebras
       22.2. The Poincar´e-Birkhoff-Witt theorem
       22.3. Primitively generated Hopf algebras in characteristic zero
       22.4. Commutative Hopf algebras in characteristic zero

Chapter 23. Restricted Lie algebras and Hopf algebras in characteristic p
       23.1. Restricted Lie algebras
       23.2. The restricted Poincar´e-Birkhoff-Witt theorem
       23.3. Primitively generated Hopf algebras in characteristic p
       23.4. Commutative Hopf algebras in characteristic p

Chapter 24. A primer on spectral sequences
       24.1. Definitions
       24.2. Exact Couples
       24.3. Filtered Complexes
       24.4. Products
       24.5. The Serre spectral sequence
       24.6. Comparison theorems
       24.7. Convergence proofs


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