# More Concise Algebraic Topology

## Localization, Completion, and Model Categories

*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.

**Zentralblatt MATH**

**Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne**

Contents

Introduction

Some conventions and notations

Acknowledgements

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

Bibliography

Index

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