The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory.
This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.
Readership: Graduate students and research mathematicians interested in algebraic topology and homotopy theory.
Author(s): Jeffrey Strom
Series: Graduate Studies in Mathematics 127
Publisher: American Mathematical Society
Year: 2011
Language: English
Pages: xxii+835
Preface
History
The Aim of This Book.
Omissions
Problems and Exercises
Audience
Teaching from This Book
Acknowledgements.
Part 1 The Language of Categories
Chapter 1 Categories and Functors
1.1. Diagrams
1.2. Categories
1.3. Functors
1.4. Natural Transformations
1.5. Duality
1.6. Products and Sums
1.7. Initial and Terminal Objects
1.8. Group and Cogroup Objects
1.9. Homomorphisms
1.10. Abelian Groups and Cogroups
1.11. Adjoint Functors
Chapter 2 Limits and Colimits
2.1. Diagrams and Their Shapes
2.2. Limits and Colimits
2.3. Naturality of Limits and Colimits
2.4. Special Kinds of Limits and Colimits
2.4.1. Pullback
2.4.2. Pushout.
2.4.3. Telescopes and Towers.
2.5. Formal Properties of Pushout and Pullback Squares
Part 2 Semi-Formal Homotopy Theory
Chapter 3 Categories of Spaces
3.1. Spheres and Disks
3.2. CW Complexes
3.2.1. CW Complexes and Cellular Maps
3.2.2. Some Topology of CW Complexes.
3.2.3. Products of CW Complexes
3.3. Example: Projective Spaces
3.3.1. Projective Spaces.
3.3.2. Cellular Decomposition of FP^n.
3.4. Topological Spaces
3.4.1. Mapping Spaces.
3.4.2. The Category of Unpointed Spaces
3.5. The Category of Pairs
3.6. Pointed Spaces
3.6.1. Pointed Mapping Spaces.
3.6.2. Products of Pointed Spaces
3.6.3. The Category of Pointed Spaces
3.7. Relating the Categories of Pointed and Unpointed Spaces
3.7.1. Various Pointed and Unpointed Products.
3.7.2. Some Mixed Adjunctions
3.8. Suspension and Loop
3.8.1. Suspension
3.8.2. Loop Spaces
3.9. Additional Problems and Projects
Chapter 4 Homotopy
4.1. Homotopy of Maps
4.1.1. The Deformation Approach.
4.1.2. Adjoint Definition of Homotopy
4.1.3. Homotopies of Paths.
4.1.4. Composing and Inverting Homotopies
4.2. Constructing Homotopies
4.2.1. Straight-Line Homotopy
4.2.2. Pushing a Map off of a Cell.
4.2.3. Pushing a Path off the Disk.
4.2.4. Cellular Approximation for 1-Dimensional Domains
4.2.5. Maps of Products.
4.3. Homotopy Theory
4.3.1. The Homotopy Category
4.3.2. Contractible Spaces and Nullhomotopic Maps
4.4. Groups and Cogroups in the Homotopy Category
4.5. Homotopy Groups
4.6. Homotopy and Duality
4.7. Homotopy in Mapping Categories
4.7.1. The Category of Maps
4.7.2. Weaker Notions of Homotopy Equivalence for Maps
4.7.3. Spaces under A or over B.
4.7.4. Pushouts and Pullbacks as Functors.
4.7.5. Maps into CW Pairs, Triples, etc.
4.8. Additional Problems
Chapter 5 Cofibrations and Fibrations
5.1. Cofibrations
5.1.1. The Homotopy Extension Property.
5.1.2. Point-Set Topology of Cofibrations
5.1.3. Two Reformulations.
5.1.4. Cofibrations and Pushouts.
5.2. Special Properties of Cofibrations of Spaces
5.2.1. The Power of a Parametrized Cylinder
5.2.2. Mapping Spaces into Cofibrations
5.2.3. Products and Cofibrations
5.3. Fibrations
5.3.1. Dualizing Cofibrations
5.3.2. Some Examples
5.3.3. Pullbacks of Fibrations
5.4. Factoring through Cofibrations and Fibrations
5.4.1. Mapping Cylinders.
5.4.2. Converting a Map to a Fibration.
5.5. More Homotopy Theory in Categories of Maps
5.5.1. Mapping Cylinders in Mapping Categories.
5.5.2. Homotopy Inverses for Pointwise Equivalences
5.6. The Fundamental Lifting Property
5.6.1. The Case i is a Homotopy Equivalence.
5.6.2. Relative Homotopy Lifting
5.6.3. The Case p is a Homotopy Equivalence
5.6.4. Mutual Characterization of Fibrations and Cofibrations
5.6.5. Some Consequences of the Mutual Characterization
5.7. Pointed Cofibrations and Fibrations
5.8. Well-Pointed Spaces
5.8.1. Well-Pointed Spaces
5.8.2. Cofibrations and Fibrations of Well-Pointed Spaces
5.8.3. Double Factorizations.
5.8.4. The Fundamental Lifting Property
5.9. Exact Sequences, Cofibers and Fibers
5.9.1. Exact Sequences in Homotopy Theory.
5.9.2. The Cofiber of a Map.
5.9.3. The Fiber of a Map
5.9.4. Cofibers of Maps out of Contractible Spaces
5.10. Mapping Spaces
5.10.1. Unpointed Mapping Spaces
5.10.2. Pointed Maps into Pointed Fibrations.
5.10.3. Applications
5.11. Additional Topics, Problems and Projects
5.11.1. Homotopy Equivalences in A | T | B
5.11.2. Comparing Pointed and Unpointed Homotopy Classes
5.11.3. Problems
Chapter 6 Homotopy Limits and Colimits
6.1. Homotopy Equivalence in Diagram Categories
6.2. Cofibrant Diagrams
6.2.1. Cofibrant Diagrams
6.2.2. An Instructive and Important Example
6.2.3. Cofibrant Replacements of Diagrams
6.3. Homotopy Colimits of Diagrams
6.3.1. The Homotopy Colimit of a Diagram.
6.3.2. Induced Maps of Homotopy Colimits
6.3.3. Example: Induced' Maps Between Suspensions
6.3.4. The Functorial Approach to Homotopy Colimits.
6.4. Constructing Cofibrant Replacements
6.4.1. Simple Categories
6.4.2. Recognizing Cofibrant Diagrams.
6.4.3. Colimits of Well-Pointed Spaces
6.4.4. Existence of Cofibrant Replacements
6.5. Examples: Pushouts, 3 x 3s and Telescopes
6.5.1. Homotopy Pushouts
6.5.2. Telescopes
6.5.3. 3 x 3 Diagrams
6.6. Homotopy Limits
6.6.1. Fibrant Diagrams of Unpointed Spaces
6.6.2. Homotopy Limits.
6.6.3. Existence of Fibrant Replacements
6.6.4. Homotopy Limits of Pointed Spaces.
6.6.5. Special Cases: Maps, Pullbacks, 3 x 3s and Towers
6.7. Functors Applied to Homotopy Limits and Colimits
6.7.1. The Unpointed Case.
6.7.2. The Pointed Case.
6.7.3. Contravariant Functors
6.8. Homotopy Colimits of More General Diagrams
6.9. Additional Topics, Problems and Projects
6.9.1. Rigidifying Homotopy Morphisms of Diagrams
6.9.2. Homotopy Colimits versus Categorical Colimits
6.9.3. Homotopy Equivalence in Mapping Categories
6.9.4. Problems and Projects
Chapter 7 Homotopy Pushout and Pullback Squares
7.1. Homotopy Pushout Squares
7.2. Recognition and Completion
7.2.1. Recognition.
7.2.2. Completion
7.3. Homotopy Pullback Squares
7.4. Manipulating Squares
7.4.1. Composition of Squares.
7.4.2. 3 x 3 Diagrams.
7.4.3. Application of Functors
7.5. Characterizing Homotopy Pushout and Pullback Squares
7.6. Additional Topics, Problems and Projects
7.6.1. Cartesian and Cocartesian Cubes.
7.6.2. Problems.
Chapter 8 Tools and Techniques
8.1. Long Cofiber and Fiber Sequences
8.1.1. The Long Cofiber Sequence of a Map.
8.1.2. The Long Fiber Sequence of a Map
8.2. The Action of Paths in Fibrations
8.2.1. Admissible Maps
8.3. Every Action Has an Equal and Opposite Coaction
8.3.1. Coactions in Cofiber Sequences
8.3.2. A Diagram Lemma.
8.3.3. Action of \OmegaY on F.
8.4. Mayer-Vietoris Sequences
8.5. The Operation of Paths
8.6. Fubini Theorems
8.7. Iterated Fibers and Cofibers
8.8. Group Actions
8.8.1. G-Spaces and G-Maps
8.8.2. Homotopy Theory of Group Actions.
8.8.3. Homotopy Colimits of Pointed G-Actions.
Chapter 9 Topics and Examples
9.1. Homotopy Type of Joins and Products
9.1.1. The Join of Two Spaces.
9.1.2. Splittings of Products.
9.1.3. Products of Mapping Cones
9.1.4. Whitehead Products
9.2. H-Spaces and co-H-Spaces
9.2.1. H-Spaces
9.2.2. Co-H-Space
9.2.3. Maps from Co-H-Spaces to H-Spaces
9.3. Unitary Groups and Their Quotients
9.3.1. Orthogonal, Unitary and Symplectic Groups
9.3.2. Topology of Unitary Groups and Their Quotients
9.3.3. Cellular Structure for Unitary Groups.
9.4. Cone Decompositions
9.4.1. Cone Decompositions
9.4.2. Cone Decompositions of Products.
9.4.3. Boundary Maps for Products
9.4.4. Generalized CW Complexes
9.5. Introduction to Phantom Maps
9.5.1. Maps out of Telescopes
9.5.2. Inverse Limits and lim' for Groups.
9.5.3. Mapping into a Limit.
9.6. G. W. Whitehead's Homotopy Pullback Square
9.7. Lusternik-Schnirelmann Category
9.7.1. Basics of Lusternik-Schnirelmann Category
9.7.2. Lusternik-Schnirelmann Category of CW Complexes
9.7.3. The Ganea Criterion for L-S Category
9.7.4. Category and Products
9.8. Additional Problems and Projects
Chapter 10 Model Categories
10.1. Model Categories
10.2. Left and Right Homotopy
10.3. The Homotopy Category of a Model Category
10.4. Derived Functors and Quillen Equivalence
10.4.1. Derived Functors
10.4.2. Quillen Equivalence of Model Categories
10.5. Homotopy Limits and Colimits
10.5.1. A Model Structure for Diagram Categories.
10.5.2. Homotopy Colimit.
Part 3 Four Topological Inputs
Chapter 11 The Concept of Dimension in Homotopy Theory
11.1. Induction Principles for CW Complexes
11.1.1. Attaching One More Cell.
11.1.2. Composing Infinitely Many Homotopies
11.2. n-Equivalences and Connectivity of Spaces
11.2.1. n-Equivalences
11.3. Reformulations of n-Equivalences
11.3.1. Equivalence of the (a) Parts
11.3.2. Equivalence of Parts (2) (a) and (2) (b).
11.3.3. Proof that Part (2) (b) Implies Part (3) (b).
11.3.4. Proof that Part (3) (b) Implies Part (1) (b).
11.4. The J. H. C. Whitehead Theorem
11.5. Additional Problems
Chapter 12 Subdivision of Disks
12.1. The Seifert-Van Kampen Theorem
12.2. Simplices and Subdivision
12.2.1. Simplices and Their Boundaries
12.2.2. Finite Simplicial Complexes
12.2.3. Barycentric Subdivision.
12.3. The Connectivity of Xn ---> X
12.4. Cellular Approximation of Maps
12.5. Homotopy Colimits and n-Equivalences
12.5.1. Homotopy Pushouts.
12.5.2. Telescope Diagrams
12.6. Additional Problems and Projects
Chapter 13 The Local Nature of Fibrations
13.1. Maps Homotopy Equivalent to Fibrations
13.1.1. Weak Fibrations.
13.1.2. Homotopy Pullbacks and Weak Fibrations
13.1.3. Weak Homotopy Lifting
13.2. Local Fibrations Are Fibrations
13.3. Gluing Weak Fibrations
13.3.1. Tabs and Glue.
13.3.2. Gluing Weak Fibrations with Tabs.
13.4. The First Cube Theorem
Chapter 14 Pullbacks of Cofibrations
14.1. Pullbacks of Cofibrations
14.2. Pullbacks of Well-Pointed Spaces
14.3. The Second Cube Theorem
Chapter 15 Related Topics
15.1. Locally Trivial Bundles
15.1.1. Bundles and Fibrations.
15.1.2. Example: Projective Spaces
15.2. Covering Spaces
15.2.1. Unique Lifting
15.2.2. Coverings and the Fundamental Group
15.2.3. Lifting Criterion.
15.2.4. The Fundamental Group of S^1.
15.3. Bundles Built from Group Actions
15.3.1. Local Sections for Orbit Spaces.
15.3.2. Stiefel Manifolds and Grassmannians
15.4. Some Theory of Fiber Bundles
15.4.1. Transition Functions.
15.4.2. Structure Groups
15.4.3. Change of Fiber and Principal Bundles.
15.5. Serre Fibrations and Model Structures
15.5.1. Serre Fibrations.
15.5.2. The Serre-Quillen Model Structure.
15.6. The Simplicial Approach to Homotopy Theory
15.6.1. Simplicial Complexes.
15.6.2. The Functorial Viewpoint
15.7. Quasifibrations
15.8. Additional Problems and Projects
Part 4 Targets as Domains, Domains as Targets
Chapter 16 Constructions of Spaces and Maps
16.1. Skeleta of Spaces
16.1.1. Formal Properties of Skeleta.
16.1.2. Construction of n-Skeleta
16.2. Connectivity and CW Structure
16.2.1. Cells and n-Equivalences
16.2.2. Connectivity and Domain-Type Constructions
16.3. Basic Obstruction Theory
16.4. Postnikov Sections
16.5. Classifying Spaces and Universal Bundles
16.5.1. The Simple Construction.
16.5.2. Fixing the Topology
16.5.3. Using EG for EH.
16.5.4. Discrete Abelian Torsion Groups.
16.5.5. What do Classifying Spaces Classify?
16.6. Additional Problems and Projects
Chapter 17 Understanding Suspension
17.1. Moore Paths and Loops
17.1.1. Spaces of Measured Paths
17.1.2. Composing Infinite Collections of Homotopies
17.2. The Free Monoid on a Topological Space
17.2.1. The James Construction
17.2.2. The Algebraic Structure of the James Construction
17.3. Identifying the Suspension Map
17.4. The Freudenthal Suspension Theorem
17.5. Homotopy Groups of Spheres and Wedges of Spheres
17.6. Eilenberg-Mac Lane Spaces
17.6.1. Maps into Eilenberg-Mac Lane Spaces
17.6.2. Existence of Eilenberg-Mac Lane Spaces
17.7. Suspension in Dimension 1
17.8. Additional Topics and Problems
17.8.1. Stable Phenomena
17.8.2. The James Splitting
17.8.3. The Hilton-Milnor Theorem
Chapter 18 Comparing Pushouts and Pullbacks
18.1. Pullbacks and Pushouts
18.1.1. The Fiber of $\psi$: Q ---> D
18.1.2. Ganea's Fiber-Cofiber Construction.
18.2. Comparing the Fiber of f to Its Cofiber
18.3. The Blakers-Massey Theorem
18.4. The Delooping of Maps
18.4.1. The Connectivity of Looping
18.4.2. The Kernel and Cokernel of Looping
18.5. The n-Dimensional Blakers-Massey Theorem
18.5.1. Blakers-Massey Theorem for n-Cubes
18.5.2. Recovering X from \Sigma X.
18.6. Additional Topics, Problems and Projects
18.6.1. Blakers-Massey Exact Sequence of a Cofibration
18.6.2. Exact Sequences of Stable Homotopy Groups
18.6.3. Simultaneously Cofiber and Fiber Sequences
18.6.4. The Zabrodsky Lemma
18.6.5. Problems and Projects.
Chapter 19 Some Computations in Homotopy Theory
19.1. The Degree of a Map S^n ---> S^n
19.1.1. The Degree of a Reflection and the Antipodal Map
19.1.2. Computation of Degree
19.2. Some Applications of Degree
19.2.1. Fixed Points and Fixed Point Free Maps
19.2.2. Vector Fields on Spheres.
19.2.3. The Milnor Sign Convention
19.2.4. Fundamental Theorems of Algebra
19.3. Maps Between Wedges of Spheres
19.4. Moore Spaces
19.5. Homotopy Groups of a Smash Product
19.5.1. Algebraic Properties of the Smash Product.
19.5.2. Nondegeneracy.
19.6. Smash Products of Eilenberg-Mac Lane Spaces
19.7. An Additional Topic and Some Problems
19.7.1. Smashing Moore Spaces
19.7.2. Problems
Chapter 20 Further Topics
20.1. The Homotopy Category Is Not Complete
20.2. Cone Decompositions with Respect to Moore Spaces
20.3. First p-Torsion Is a Stable Invariant
20.3.1. Setting Up
20.3.2. Connectivity with Respect to P.
20.3.3. P-Connectivity and Moore Spaces
20.3.4. The First P-Torsion of a Smash Product
20.3.5. P-Local Homotopy Theory
20.4. Hopf Invariants and Lusternik-Schnirelmann Category
20.4.1. Berstein-Hilton Hopf Invariants
20.4.2. Stanley's Theorems on Compatible Sections
20.5. Infinite Symmetric Products
20.5.1. The Free Abelian Monoid on a Space
20.5.2. Symmetric Products of Cofiber Sequences
20.5.3. Some Examples.
20.5.4. Symmetric Products and Eilenberg-Mac Lane Spaces.
20.6. Additional Topics, Problems and Projects
20.6.1. Self-Maps of Projective Spaces.
20.6.2. Fiber of Suspension and Suspension of Fiber
20.6.3. Complexes of Reduced Product Type.
20.6.4. Problems and Projects
Part 5 Cohomology and Homology
Chapter 21 Cohomology
21.1. Cohomology
21.1.1. Represented Ordinary Cohomology
21.1.2. Cohomology Theories.
21.1.3. Cohomology and Connectivity.
21.1.4. Cohomology of Homotopy Colimits.
21.1.5. Cohomology for Unpointed Spaces
21.2. Basic Computations
21.2.1. Cohomology and Dimension.
21.2.2. Suspension Invariance
21.2.3. Exact Sequences
21.2.4. Cohomology of Projective Spaces
21.3. The External Cohomology Product
21.4. Cohomology Rings
21.4.1. Graded R-Algebras
21.4.2. Internalizing the Exterior Product
21.4.3. R-Algebra Structure
21.5. Computing Algebra Structures
21.5.1. Products of Spheres.
21.5.2. Bootstrapping from Known Cohomology.
21.5.3. Cohomology Algebras for Projective Spaces
21.6. Variation of Coefficients
21.6.1. Universal Coefficients
21.7. A Simple Kunneth Theorem
21.8. The Brown Representability Theorem
21.8.1. Representing Homotopy Functors
21.8.2. Representation of Cohomology Theories
21.8.3. Representing a Functor on Finite Complexes
21.9. The Singular Extension of Cohomology
21.10. An Additional Topic and Some Problems and Projects
21.10.1. Cohomology of BZ/n.
21.10.2. Problems and Projects
Chapter 22 Homology
22.1. Homology Theories
22.1.1. Homology Theories
22.1.2. Homology and Homotopy Colimits.
22.1.3. The Hurewicz Theorem.
22.1.4. Computation
22.2. Examples of Homology Theories
22.2.1. Stabilization of Maps.
22.2.2. Ordinary Homology.
22.2.3. Infinite Loop Spaces and Homology
22.3. Exterior Products and the Kunneth Theorem for Homology
22.3.1. The Exterior Product in Homology
22.4. Coalgebra Structure for Homology
22.5. Relating Homology to Cohomology
22.5.1. Pairing Cohomology with Homology
22.5.2. Nondegeneracy
22.6. H-Spaces and Hopf Algebras
22.6.1. The Pontrjagin Algebra of an H-Space
22.6.2. Pontrjagin and Kiinneth.
22.6.3. The Homology and Cohomology of an H-Space
Chapter 23 Cohomology Operations
23.1. Cohomology Operations
23.2. Stable Cohomology Operations
23.2.1. The Same Operation in All Dimensions
23.2.2. Extending an Operation to a Stable Operation.
23.2.3. Cohomology of BZ/p.
23.3. Using the Diagonal Map to Construct Cohomology Operations
23.3.1. Overview
23.3.2. The Transformation $\lamba$.
23.4. The Steenrod Reduced Powers
23.4.1. Unstable Relations
23.4.2. Extending the pth Power to a Stable Operation
23.5. The Adem Relations
23.5.1. Steenrod Operations on Polynomial Rings
23.5.2. The Fundamental Symmetry Relation
23.6. The Algebra of the Steenrod Algebra
23.6.1. Fundamental Properties of Steenrod Operations
23.6.2. Modules and Algebras over A.
23.6.3. Indecomposables and Bases
23.7. Wrap-Up
23.7.1. Delooping the Squaring Operation.
23.7.2. Additional Problems and Projects
Chapter 24 Chain Complexes
24.1. The Cellular Complex
24.1.1. The Cellular Cochain Complex of a Space.
24.1.2. Chain Complexes and Algebraic Homology
24.1.3. Computing the Cohomology of Spaces via Chain Complexes.
24.1.4. Chain Complexes for Homology Theories
24.1.5. Uniqueness of Cohomology and Homology
24.2. Applying Algebraic Universal Coefficients Theorems
24.2.1. Constructing New Chain Complexes
24.2.2. Universal Coefficients Theorems
24.3. The General Kunneth Theorem
24.3.1. The Cellular Complexes of a Product.
24.3.2. Kunneth Theorems for Spaces.
24.4. Algebra Structures on C*(X) and C(X)
24.5. The Singular Chain Complex
Chapter 25 Topics, Problems and Projects
25.1. Algebra Structures on R^n and C^n
25.2. Relative Cup Products
25.2.1. A New Exterior Cup Product
25.2.2. Lusternik-Schnirelmann Category and Products.
25.3. Hopf Invariants and Hopf Maps
25.3.1. The Hopf Invariant Is a Homomorphism.
25.3.2. The Hopf Construction
25.3.3. Hopf Invariant One
25.3.4. Generalization.
25.4. Some Homotopy Groups of Spheres
25.4.1. The Group \pi_n+1(S^n)
25.4.2. Composition of Hopf Maps.
25.5. The Borsuk-Ulam Theorem
25.6. Moore Spaces and Homology Decomposi
25.6.1. Homology of Moore Spaces
25.6.2. Cohomology Operations in Moore Spaces
25.6.3. Maps Between Moore Spaces
25.6.4. Homology Decompositions.
25.7. Finite Generation of \pi_n(X), and Hn(X)
25.8. Surfaces
25.9. Euler Characteristic
25.9.1. Independence of the Field
25.9.2. Axiomatic Characterization of Euler Characteristic.
25.9.3. Poincare Series
25.9.4. More Examples.
25.10. The Kunneth Theorem via Symmetric Products
25.11. The Homology Algebra of \Omega \Sigma X
25.12. The Adjoint \lambda_X of id_\omega X
25.13. Some Algebraic Topology of Fibrations
25.14. A Glimpse of Spectra
25.15. A Variety of Topics
25.15.1. Contractible Smash Products
25.15.2. Phantom Maps
25.15.3. The Serre Exact Sequence
25.15.4. The G. W. Whitehead Exact Sequences
25.15.5. Hopf Algebra Structure on the Steenrod Algebra
25.16. Additional Problems and Projects
Part 6 Cohomology, Homology and Fibrations
Chapter 26 The Wang Sequence
26.1. Trivialization of Fibrations
26.2. Orientable Fibrations
26.3. The Wang Cofiber Sequence
26.3.1. Fibrations over a Suspension
26.3.2. The Wang Exact Sequence
26.3.3. Proof of Theorem 26.10(a).
26.3.4. Proof of Theorem 26.10(b).
26.4. Some Algebraic Topology of Unitary Groups
26.4.1. The Cohomology of the Unitary Groups.
26.4.2. The Homology Algebra of the Unitary Groups
26.4.3. Cohomology of the Special Unitary Group
26.4.4. Cohomology of the Stiefel Manifolds
26.5. The Serre Filtration
26.5.1. The Fundamental Cofiber Sequence
26.5.2. Pullbacks over a Cone Decomposition of the Base
26.6. Additional Topics, Problems and Projects
26.6.1. Clutching
26.6.2. Orthogonal and Symplectic Groups
26.6.3. The Homotopy Groups of S^3.
Chapter 27 Cohomology of Filtered Spaces
27.1. Filtered Spaces and Filtered Groups
27.1.1. Subquotients and Correspondence
27.1.2. Filtered Spaces.
27.1.3. Filtered Algebraic Gadgets.
27.1.4. Linking Topological and Algebraic Filtrations
27.1.5. The Functors Gr* and Gr*
27.1.6. Convergence
27.1.7. Indexing of Associated Graded Objects
27.2. Cohomology and Cone Filtrations
27.2.1. Studying Cohomology Using Filtrations
27.2.2. Approximating Z^n,m and B^s,n.
27.3. Approximations for General Filtered Spaces
27.3.1. Algebraic Repackaging
27.3.2. Algebraic Homology and Exact Couples
27.3.3. Topological Boundary Maps for a Filtration
27.4. Products in E1'* (X )
27.4.1. The Exterior Product for Z1'*.
27.4.2. Boundary Maps for a Smash of Filtered Spaces
27.4.3. Internalizing the External Product.
27.5. Pointed and Unpointed Filtered Spaces
27.6. The Homology of Filtered Spaces
27.7. Additional Projects
Chapter 28 The Serre Filtration of a Fibration
28.1. Identification of E2 for the Serre Filtration
28.1.1. Cohomology with Coefficients in Cohomology
28.2. Proof of Theorem 28.1
28.2.1. Setting Up
28.2.2. The Topological Boundary Map
28.2.3. Identifying the Differential.
28.2.4. Naturality of E2'*
28.3. External and Internal Products
28.3.1. External Products for E*'* (p).
28.3.2. Internalizing Using the Diagonal
28.4. Homology and the Serre Filtration
28.5. Additional Problems
Chapter 29 Application: Incompressibility
29.1. Homology of Eilenberg-Mac Lane Spaces
29.1.1. Exponents for H* (K(Z/p''); G).
29.1.2. The Homology Algebra H* (K(Z, 2n); Z
29.2. Reduction to Theorem 29.1
29.2.1. Compressible Maps.
29.2.2. The Reduction. I
29.2.3. Maps from QS2n+l to K(G, 2n)
29.3. Proof of Theorem 29.2
29.3.1. Reduction to the Case G = Z/p"'.
29.3.2. Compressibility and the Serre Filtration
29.3.3. Consequences of Membership in Fo.
29.3.4. Completing the Proof.
29.4. Consequences of Theorem 29.1
29.4.1. The Connectivity of a Finite H-Spaces
29.4.2. Sections of Fibrations over Spheres.
29.5. Additional Problems and Projects
Chapter 30 The Spectral Sequence of a Filtered Space
30.1. Approximating Grs Hn (X) by E; 'n (X )
30.1.1. Topological Description of dr.
30.1.2. The Algebraic Approach.
30.2. Some Algebra of Spectral Sequences
30.2.1. The Category of Spectral Sequences
30.2.2. Exact Couples and Filtered Modules
30.2.3. Multiplicative Structure
30.2.4. Convergence of Spectral Sequences
30.3. The Spectral Sequences of Filtered Spaces
30.3.1. Multiplicative Structures
30.3.2. Convergence
30.3.3. The Grand Conclusion.
Chapter 31 The Leray-Serre Spectral Sequence
31.1. The Leray-Serre Spectral Sequence
31.1.1. The Spectral Sequences Associated to the Serre Filtration.
31.1.2. Nondegeneracy of the Algebra Structure
31.1.3. Two Relative Variants
31.1.4. The Homology Leray-Serre Spectral Sequence
31.2. Edge Phenomena
31.2.1. Edge Filtration Quotients
31.2.2. One Step Back
31.2.3. Edge Homomorphisms
31.2.4. The Transgression
31.3. Simple Computations
31.3.1. Fibration Sequences of Spheres.
31.3.2. Cohomology of Projective Spaces.
31.3.4. Rational Exterior and Polynomial Algebras
31.3.5. Construction of Steenrod Squares.
31.4. Simplifying the Leray-Serre Spectral Sequence
31.4.1. Two Simplifying Propositions.
31.4.2. The Leray-Hirsch Theorem.
31.4.3. Exact Sequences for Fibrations Involving Spheres
31.4.4. The Thom Isomorphism Theorem
31.4.5. The Serre Exact Sequence.
31.5. Additional Problems and Projects
Chapter 3 Application: Bott Periodicity
32.1. The Cohomology Algebra of BU(n)
32.2. The Torus and the Symmetric Group
32.2.1. The Action of the Symmetric Group.
32.2.2. Identifying H*(BU(n)) with Symmetric Polynomials
32.2.3. The Main Theorem
32.3. The Homology Algebra of BU
32.3.1. H-Structure for BU.
32.3.2. The Diagonal of H* (BU; 7G)
32.3.3. The Pontrjagin Algebra H* (BU; Z).
32.4. The Homology Algebra of $\Omega$SU(n)
32.5. Generating Complexes for $\Omega$SU and BU
32.5.1. Generating Complex for BU.
32.5.2. Generating Complexes for \OmegaSU(n)
32.6. The Bott Periodicity Theorem
32.6.1. Shuffling Special Unitary Groups.
32.6.2. Properties of the Bott Map.
32.6.3. Bott Periodicity
32.7. K-Theory
32.7.1. K-Theory and Vector Bundles
32.7.2. Cohomology Operations in K-Theory
32.8. Additional Problems and Projects
Chapter 33 Using the Leray-Serre Spectral Sequence
33.1. The Zeeman Comparison Theorem
33.2. A Rational Borel-Type Theorem
33.3. Mod 2 Cohomology of K(G, n)
33.3.1. The Transgression
33.3.2. Simple Systems of Generators
33.3.3. Borel's Theorem.
33.3.4. Mod 2 Cohomology of Eilenberg-Mac Lane Space
33.4. Mod p Cohomology of K(G, n)
33.4.1. The mod p Path-Loop Transgression
33.4.2. Postnikov's Theorem
33.4.3. Mod p Cohomology of Eilenberg-Mac Lane Spaces
33.5. Steenrod Operations Generate .Ar
33.6. Homotopy Groups of Spheres
33.6.1. Finiteness for Homotopy Groups of Spheres
33.6.2. Low-Dimensional p-Torsion
33.7. Spaces Not Satisfying the Ganea Condition
33.8. Spectral Sequences and Serre Classes
33.8.1. Serre Classes
33.8.2. Some Algebra of Serre Classes
33.8.3. Serre Classes and Topology.
33.9. Additional Problems and Projects
Part 7 Vistas
Chapter 34 Localization and Completion
34.1. Localization and Idempotent Functors
34.1.1. Idempotent Functors
34.1.2. Homotopy Idempotent Functors
34.1.3. Simple Explorations
34.2. Proof of Theorem 34.5
34.2.1. The Shape of a Small Object Argument.
34.2.2. The Property to Be Tested.
34.2.3. The Construction
34.2.4. Connectivity of Lf (X)
34.3. Homotopy Theory of P-Local Spaces
34.3.1. P-Localization of Spaces
34.3.2. Hands-On Localization of Simply-Connected Spaces
34.3.3. Localization of Homotopy-Theoretic Constructions
34.3.4. Recovering a Space from Its Localizations
34.4. Localization with Respect to Homology
34.4.1. Construction of h*-Localization
34.4.2. Ordinary Cohomology Theories
34.4.3. Other Connective Homology Theories
34.5. Rational Homotopy Theory
34.5.1. Suspensions and Loop Spaces
34.5.2. Sullivan Model
34.5.3. The Lie Model.
34.5.4. Elliptic and Hyperbolic
34.5.5. Lusternik-Schnirelmann Category of Rational Spaces
34.6. Further Topics
34.6.1. The EHP Sequence
34.6.2. Spheres Localized at P.
34.6.3. Regular Primes
Chapter 35 Exponents for Homotopy Groups
35.1. Construction of a
35.1.1. Deviation
35.1.2. Deviation and Lusternik-Schnirelmann Category
35.1.3. Deviation and Ganea Fibrations.
35.1.4. Compositions of Order p.
35.1.5. Definition of a.
35.2. Spectral Sequence Computations
35.2.1. The Dual of the Bockstein
35.2.2. The Homology Algebra of f2(S^3(3)).
35.2.3. The Homology Algebra of f l2 (S3 (3) )
35.2.4. The Homology Algebra H*(\OmegaS2p+1{p}).
35.3. The Map \lambda
35.4. Proof of Theorem 35.3
35.4.1. The Map Induced by the Hopf Invariant
35.4.2. Finishing the Argument
35.5. Nearly Trivial Maps
Chapter 36 Classes of Spaces
36.1. A Galois Correspondence in Homotopy Theory
36.2. Strong Resolving Classes
36.2.1. Manipulating Classes of Spaces.
36.2.2. Closure under Finite-Type Wedges
36.2.3. Desuspension in Resolving Classes
36.2.4. Spherical Resolvability of Finite Complexes
36.3. Closed Classes and Fibrations
36.3.1. Cellular Inequalities
36.3.2. Closed Classes and Fibration Sequences.
36.3.3. E. Dror Farjoun's Theorem
36.4. The Calculus of Closed Classes
36.4.1. Fibers and Cofibers
36.4.2. Loops and Suspensions
36.4.3. Adjunctions
36.4.4. A Cellular Blakers-Massey Theorem
Chapter 37 Miller's Theorem
37.1. Reduction to Odd Spheres
37.1.1. From Odd Spheres to Wedges of Spheres.
37.1.2. Vanishing Phantoms
37.1.3. Non-Simply-Connected Targets
37.2. Modules over the Steenrod Algebra
37.2.1. Projective ,A-Modules
37.2.2. Homological Algebra.
37.2.3. The Functor T
37.3. Massey-Peterson Towers
37.3.1. Relating Algebras and Modules
37.3.2. Topologizing Modules and Resolution
37.3.3. The Groups E2X, Y).
37.3.4. A Condition for the Omniscience of Cohomology
37.4. Extensions and Consequences of Miller's Theorem
37.4.1. The Sullivan Conjecture.
37.4.2. BZ/p-Nullification
37.4.3. Neisendorfer Localization
37.4.4. Serre's Conjecture
Appendix A Some Algebra
A.1. Modules, Algebras and Tensor Products
A.1.1. Modules
A.1.2. Bilinear Maps and Tensor Products
A.1.3. Algebras
A.2. Exact Sequences
A.3. Graded Algebra
A.3.1. Decomposables and Indecomposable
A.4. Chain Complexes and Algebraic Homology
A.4.1. Homology of Chain Complexes.
A.5. Some Homological Algebra
A.5.1. Projective Resolutions and TorR
A.5.2. Injective Resolutions and ExtR(? , ?
A.5.3. Algebraic Kunneth and Universal Coefficients Theorems
A.6. Hopf Algebras
A.6.1. Coalgebras
A.6.2. Hopf Algebras.
A.6.3. Dualization of Hopf Algebras
A.7. Symmetric Polynomials
A.8. Sums, Products and Maps of Finite Type
A.9. Ordinal Numbers
Bibliography
Index of Notation
Index
