目錄
Introduction
Some conventions and notations
Acknowledgments
PART 1 Preliminaries: Basic Homotopy Theory and Nilpotent Spaces
1.Cofibrations and fibrations
1.1 Relations between cofibrations and fibrations
1.2 The fill-in and Verdier lemmas
1.3 Based and free cofbrations and fibrations
1.4 Actions of fundamental groups on homotopy classes of maps
1.5 Actions of fundamental groups in fibration sequences
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 Anexample of nonvanishing liml terms
2.5 The homology of colimits and limits
2.6 Aprofinite universal coefficient theorem
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-Mac Lane space
3.5 Postnikov A-towers
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
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
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 Acellular 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
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 pulbacks
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
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
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
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
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 adelic genus
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 adelic genus
PART 4 An Introduction to Model Category Theory
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
15.Cofbrantly generated and proper model categories
15.1 The small object argument for the construction of WFSs
15.2 Compactly and cofbrantly generated model categories
15.3 Over and under model structures
15.4 Left and right proper model categories
15.5 Left propernes, lifting properties, and the sets [X, Y]
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 Symmetricmonoidal and entiched model catesoies
16.5 A glimpse at higher categoricalstructures
17.Model structures on the category of spaces
17.1 The Hurewicz or h-model structure on spaces
17.2 The Quillen or gmodel 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
18.Model structures on categories of chain complexes
18.1 The algebraic framework and the analogy with topology
18.2 h-cofibrations and h-fbrations in ChR
18.3 The h-model structure on ChR
18.4 The -model structure on ChR
18.5 Profs and the characterization of qcofibrations
18.6 The m-model structure on ChR
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
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
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
22.Lie algebras and Hopf algebras in characteristic zero
22.1 Graded Lie algebras
22.2 The Poincare-Birkhoff-Witt theorem
22.3 Primitively generated Hopf algebras in characteristic zero
22.4 Commutative Hopf algebras in characteristic zero
23.Restricted Lie algebras and Hopf algebras in characteristic p
23.1 Restricted Lie algebras
23.2 The restricted Poincare-Birkhoff-Witt theorem
23.3 Primitively generated Hopf algebras in characteristic p
23.4 Commutative Hopf algebras in characteristic p
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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