Algebraic Topology - A Structural Introduction

Contents
Preface
Introduction
0.1 Investigating spaces with algebraic structures
0.2 Homology and cohomology theories
0.3 An outline
0.4 Translations and underlying affinities
0.5 An inductive approach on structural bases
0.6 Prerequisites and literature
0.7 Notation and conventions
0.8 Acknowledgements
1 Introducing Algebraic Topology
1.1 Classifying spaces and maps
1.1.1 The shape of a space
1.1.2 Classifying maps
1.1.3 Classifying the homotopy type
1.1.4 Hints at singular homology
1.1.5 The homology of the spheres
1.1.6 Studying retracts
1.1.7 Topological embeddings and projections
1.1.8 Cover Lemmas
1.2 Transforming problem
1.2.1 Hints at categories and functors
1.2.2 Exploring connectedness
1.2.3 Other exercises
1.2.4 Distinguishing shape and homotopy type
1.2.5 Free abelian groups
1.2.6 A structure theorem
1.3 The terminology of categories and functors
1.3.1 Definition
1.3.2 Small and large categories
1.3.3 Isomorphisms and retracts
1.3.4 Subcategories, quotients and products of categories
1.3.5 Functors
1.3.6 Isomorphic categories
1.3.7 Forgetful and structural functors
1.3.8 Hom functors
1.4 Paths and homotopy
1.4.1 Euclidean spaces
1.4.2 Paths and loops
1.4.3 Cylinder and homotopies
1.4.4 The homotopy category
1.4.5 Pointed homotopy
1.4.6 Basic homotopy functors
1.4.7 Homotopy of paths
1.4.8 Homotopy of loops
1.4.9 The Lebesgue Number Lemma
1.5 Natural transformations and equivalence of categories
1.5.1 Natural transformations
1.5.2 Two operations
1.5.3 The cylinder functor
1.5.4 Categories of functors
1.5.5 Equivalence of categories
1.5.6 *Presheaves
1.6 Products, sums and universal properties
1.6.1 Products
1.6.2 Sums
1.6.3 Comments
1.6.4 Finite direct sums
1.6.5 Biproducts and additive categories
1.6.6 Universal arrows
1.6.7 Exercises and complements
2 Singular homology
2.1 Chain complexes and their homology
2.1.1 Kernels and cokernels
2.1.2 Exercises and complements
2.1.3 Exact sequences of abelian groups
2.1.4 Induction Lemma
2.1.5 The category of chain complexes
2.1.6 Chain homology
2.1.7 Chain homotopy
2.1.8 Direct sum of chain complexes
2.1.9 *The cylinder functor
2.2 The singular homology groups
2.2.1 The geometry of the cubes
2.2.2 The singular cubical set of a space
2.2.3 The chain complex of a cubical set
2.2.4 Singular homology
2.2.5 Exercises and complements (Basic results)
2.2.6 Homotopy Invariance Theorem
2.2.7 Numerical invariants
2.2.8 Paths and homology in degree 1
2.3 Mayer–Vietoris and subdivision
2.3.1 Exact sequences of chain complexes
2.3.2 A category of short exact sequences
2.3.3 Theorem (The exact sequence of chain homology)
2.3.4 Small chains
2.3.5 Subdivision Theorem
2.3.6 Pasting two spaces
2.3.7 Theorem (The Mayer–Vietoris sequence)
2.3.8 Exercises (Completing the sequence of chain homology)
2.3.9 *Proof of the Subdivision Theorem
2.4 The homology of the spheres
2.4.1 The homology of the circle
2.4.2 Theorem (The homology of the spheres)
2.4.3 Applications, I
2.4.4 Exercises and complements (Applications, II)
2.4.5 The suspension
2.4.6 The winding number
2.4.7 Theorem (The antipodal map)
2.4.8 Theorem (Vector fields on spheres)
2.4.9 *Exercises (A homology generator of the sphere)
2.5 Computing homology
2.5.1 The figure-eight space
2.5.2 Exercises and complements
2.5.3 Bouquets of circles
2.5.4 Sums of pointed spaces
2.5.5 Proposition
2.5.6 Theorem
2.5.7 Bouquets of spheres
2.5.8 Spaces with non-trivial torsion
2.5.9 Exercises (Torsion coeficients)
2.6 Compact surfaces and projective spaces
2.6.1 Quotients of the square
2.6.2 The torus
2.6.3 Exercises (The homology of the torus)
2.6.4 The Klein bottle
2.6.5 The MŁobius band
2.6.6 The double torus
2.6.7 The projective spaces
2.6.8 Theorem (The homology of the projective spaces)
2.7 Diagram lemmas in Homological Algebra
2.7.1 The Snake Lemma
2.7.2 The 3 × 3 Lemma
2.7.3 Exercises (The Five Lemma)
2.7.4 Split exact sequences
2.7.5 Biproducts revisited
2.7.6 *Exact squares
2.8 Complements
2.8.1 Reduced singular homology
2.8.2 Exercises and complements
2.8.3 The simplicial form of singular homology
2.8.4 The geometry of tetrahedra
2.8.5 The singular simplicial set
2.8.6 Homology
2.8.7 Lemma
2.8.8 *Homotopy invariance
3 Relative singular homology and homology theories
3.1 Main definitions
3.1.2 Relative chains
3.1.3 Relative singular homology
3.1.4 Theorem (Homotopy invariance)
3.1.5 Theorem (The homology sequence of a pair)
3.1.6 Exercises and complements
3.1.7 Exercises and complements (Relative and reduced homology)
3.1.8 Theorem (The homology sequence of a triple)
3.1.9 *Comments
3.2 Excision and compact pairs
3.2.1 Theorem (Excision in singular homology)
3.2.2 Strong compact pairs
3.2.3 Lemma
3.2.4 Theorem
3.2.5 Relative homeomorphisms
3.2.6 Relative homeomorphism Theorem
3.3 Local homology and orientable manifolds
3.3.1 Orienting the spheres
3.3.2 Local homology
3.3.3 Local orientations
3.3.4 Topological manifolds and surfaces
3.3.5 Orientable manifolds
3.3.7 *The connected sum of manifolds and surfaces
3.3.8 *The Classification Theorem of Compact Surfaces
3.4 Eilenberg–Steenrod axioms for homology and cohomology
3.4.1 The axioms
3.4.2 The singular theory
3.4.3 Exercises and complements
3.4.4 Cohomology theories
3.4.5 Comments
3.4.6 Simplicial complexes
3.4.7 Triangulated pairs
3.4.8 *Uniqueness Theorem [ES]
3.4.9 *Simplicial homology
3.5 Alexander–Spanier cohomology
3.5.1 Cochain complexes
3.5.2 Complements
3.5.3 A preliminary cochain complex
3.5.4 Alexander–Spanier cochains
3.5.5 Alexander–Spanier cohomology
3.5.6 Relative Alexander–Spanier cohomology
3.5.7 The exact cohomology sequence
3.5.8 Exercises and complements (Degree 0)
3.5.9 *The axioms
3.6 The product in Alexander–Spanier cohomology
3.6.1 Algebras on a commutative ring
3.6.2 Graded algebras
3.6.3 The cohomology algebra
3.6.4 The Alexander–Spanier graded algebra
3.7 Hints at de Rham cohomology
3.7.1 An introduction
3.7.2 Main definitions
3.7.3 The product of differential forms
3.7.4 The differential
3.7.5 Cohomology
3.7.6 Smooth maps
3.7.7 Functoriality
3.7.8 *Complements
4 Singular homology with coefficients
4.1 Tensor product of modules and Hom
4.1.1 Linear and multilinear mappings
4.1.2 Tensor product of modules
4.1.3 Remarks and complements
4.1.4 Tensor product of homomorphisms
4.1.5 Basic properties of the tensor product
4.1.6 The functor Hom
4.1.7 The exponential law for sets and modules
4.1.8 *Exercises and complements (Tensor product of vector spaces)
4.2 Additive and exact functors
4.2.1 Additive functors
4.2.2 Exercises and complements (Exactness properties)
4.2.3 Lemma (Extending additive functors)
4.2.4 Theorem (Exactness properties of the tensor product)
4.2.5 Theorem (Exactness properties of Hom)
4.2.6 Flat, projective and injective modules
4.2.7 Exercises and complements
4.2.8 Exercises and complements on abelian groups, I
4.2.9 Exercises and complements on abelian groups, II
4.3 Relative singular homology with coecients
4.3.1 Extending tensor products, I
4.3.2 Chains with coecients
4.3.3 Singular homology with coecients
4.3.4 Subdivision Theorem
4.3.5 Theorem
4.3.6 Theorem (The Mayer–Vietoris sequence)
4.3.7 Acyclic spaces and maps
4.3.8 Exercises and complements
4.4 Relative singular cohomology with coefficients
4.4.1 Extending the Hom functor, I
4.4.2 Singular cochains
4.4.3 Singular cohomology
4.4.4 Subdivision Theorem
4.4.5 Theorem
4.4.6 Theorem (The Mayer–Vietoris sequence)
4.5 *Changing the coefficient group
4.5.1 Extending tensor products, II
4.5.2 The Bockstein operator in homology
4.5.3 Extending the Hom functor, II
4.5.4 The Bockstein operator in cohomology
4.5.5 Remarks
5 Derived functors, universal coefficients and products
5.1 Derived functors
5.1.1 Introduction
5.1.2 Projective resolutions
5.1.3 Left derived functors
5.1.4 Theorem (Lifting short exact sequences)
5.1.5 The sequence of derived functors
5.1.6 Exercises and complements
5.1.7 The case of abelian groups
5.1.8 Right derived functors
5.1.9 *The case of abelian groups
5.2 Torsion product and universal coecients
5.2.1 The torsion product
5.2.2 Proposition
5.2.3 Corollary
5.2.4 A natural morphism
5.2.5 Universal Coefficient Theorem (for chain homology)
5.2.7 Comments and exercises
5.2.8 *Complements
5.3 Ext functor and universal coecients
5.3.1 The Ext functor
5.3.2 Proposition
5.3.3 Corollary
5.3.4 A natural morphism
5.3.5 Universal Coefficient Theorem (for chain cohomology)
5.3.6 Corollary (universal coefficients for singular cohomology)
5.3.7 Betti numbers in cohomology
5.3.8 *Manifolds and Poincare duality
5.4 Tensor product of chain complexes
5.4.1 Extending the tensor product, III
5.4.2 The tensor functor
5.4.3 A natural morphism
5.4.4 Künneth Theorem (for chain complexes)
5.4.5 Künneth Theorem (for cochain complexes)
5.4.6 Proof of Theorem 5.4.4
5.4.7 Lemma
5.5 Products in singular cohomology
5.5.1 Singular cochains with coefficients in a ring
5.5.2 Cup product
5.5.3 The cohomology algebra
5.5.4 Path components and cup product
5.5.5 Exercises (Cohomology algebras)
5.5.6 The Alexander–Whitney diagonal approximation
5.5.7 Cross product
5.5.8 Complements
5.5.9 *The dual affinity of spaces and rings
5.6 *Acyclic models and products of spaces
5.6.1 Acyclic models
5.6.2 A more complex version
5.6.3 Acyclic Models Theorem
5.6.4 Theorem (Equivalence of the simplicial and cubical form)
5.6.5 Eilenberg–Zilber Theorem
5.6.6 Corollary (Eilenberg–Zilber)
5.6.7 Exercises and complements
5.6.8 Complements
5.6.9 Proof of Theorem 5.6.3
6 An introduction to homotopy groups
6.1 The fundamental groupoid
6.1.1 Reviewing paths
6.1.2 The fundamental groupoid
6.1.3 The fundamental-groupoid functor
6.1.4 Theorem (Homotopy equivariance)
6.1.5 Exercises and complements (Path connectedness)
6.1.6 Proposition (Cartesian products)
6.1.7 Double homotopies and 2-homotopies
6.1.8 Lemma
6.1.9 Representative subsets
6.2 The fundamental group
6.2.1 Definition
6.2.2 Comments and complements
6.2.3 Exercises and complements (Homotopy invariance)
6.2.4 Proposition (Cartesian products)
6.2.5 Theorem (The fundamental group of the circle)
6.2.6 Exercises (Playing with π1)
6.2.7 Exercises and complements (The n-torus)
6.2.8 Covering maps
6.2.9 Theorem (Lifting properties)
6.3 Moving or dropping the basepoint
6.3.1 The transition isomorphism
6.3.2 Theorem (Non-pointed homotopies)
6.3.3 The Hurewicz homomorphism
6.3.4 *Hurewicz Theorem (in dimension one)
6.3.5 Remarks and complements
6.3.6 Lemma
6.3.7 Theorem (Classification by degree)
6.4 The van Kampen Theorem
6.4.1 Free products of groups
6.4.2 Pushouts of groups
6.4.3 The van Kampen Theorem
6.4.4 Exercises and complements (Computations)
6.4.5 The van Kampen Theorem for groupoids
6.4.6 Corollary
6.4.7 The fundamental groupoid of the circle
6.4.8 Lemma (Pushouts and retracts)
6.4.9 Proof of Theorem 6.4.5
6.5 Hints at the higher homotopy groups
6.5.1 Homotopy groups
6.5.2 Exercises and complements
6.5.3 Homotopy functors
6.5.4 Complements
6.5.5 *Hurewicz Theorem (in higher dimension)
6.5.6 Higher connectedness
7 Complements on categories and topology
7.1 An overview of categorical limits and adjoint functors
7.1.1 Limits and colimits
7.1.2 Examples and complements
7.1.3 Limits and colimits by universal arrows
7.1.4 Introducing adjunctions
7.1.5 Main definitions
7.1.6 Examples and comments
7.1.7 Main properties of adjunctions
7.1.8 Equivalence and adjunctions
7.2 The compact-open topology
7.2.1 The exponential law, II
7.2.2 Exponentiable objects and spaces
7.2.3 The compact-open topology
7.2.4 Theorem
7.2.5 The path functor
7.2.6 Complements
8 Solution of the exercises
8.1 Exercises of Chapter 1
8.1.1 Solutions of 1.1.1
8.1.2 Solutions of 1.1.2
8.1.3 Solutions of 1.1.3
8.1.4 Solutions of 1.1.6
8.1.5 Solutions of 1.1.8
8.1.6 Solutions of 1.2.2
8.1.7 Solutions of 1.2.3
8.1.8 Solutions of 1.2.4
8.1.9 Solutions of 1.3.3
8.1.10 Solutions of 1.4.2
8.1.11 Solutions of 1.4.3
8.1.12 Solutions of 1.4.6
8.1.13 Solutions of 1.4.7
8.1.14 Solutions of 1.5.1
8.1.15 Solutions of 1.5.2
8.1.16 Solutions of 1.5.3
8.1.17 Solutions of 1.5.4
8.1.18 Solutions of 1.5.5
8.1.19 Solutions of 1.6.4
8.1.20 Solutions of 1.6.5
8.1.21 Solutions of 1.6.7
8.2 Exercises of Chapter 2
8.2.1 Solutions of 2.1.2
8.2.2 Solutions of 2.1.3
8.2.3 Solutions of 2.1.6
8.2.4 Solutions of 2.1.7
8.2.5 Solutions of 2.1.9
8.2.6 Solutions of 2.2.1
8.2.7 Solutions of 2.2.3
8.2.8 Solutions of 2.2.5
8.2.9 Solutions of 2.2.8
8.2.10 Solutions of 2.3.4
8.2.11 Solutions of 2.3.6
8.2.12 Solutions of 2.3.8
8.2.13 Solutions of 2.4.5
8.2.14 Solutions of 24.9
8.2.15 Solutions of 2.5.2
8.2.16 Solutions of 2.5.3
8.2.17 Solutions of 2.5.4
8.2.18 Solutions of 2.5.7
8.2.19 Solutions of 2.5.8
8.2.20 Solutions of 2.5.9
8.2.21 Solutions of 2.6.3
8.2.22 Solutions of 2.6.4
8.2.23 Solution of 2.6.6
8.2.24 Solutions of 2.7.3
8.2.25 Solutions of 2.7.4
8.2.26 Solutions of 2.7.5
8.2.27 Solutions of 2.8.2
8.2.28 Solutions of 2.8.6
8.3 Exercises of Chapter 3
8.3.1 Solutions of 3.1.2
8.3.2 Solutions of 3.1.6
8.3.3 Solutions of 3.1.7
8.3.4 Solutions of 3.4.3
8.3.5 Solutions of 3.5.3
8.3.6 Solutions of 3.5.5
8.3.7 Solutions of 3.5.8
8.3.8 Solutions of 3.6.3
8.3.9 Solutions of 3.6.4
8.3.10 Solutions of 3.7.3
8.3.11 Solutions of 3.7.4
8.3.12 Solutions of 3.7.5
8.3.13 Solutions of 3.7.6
8.4 Exercises of Chapter 4
8.4.1 *Solutions of 4.1.8
8.4.2 Solutions of 4.2.1
8.4.3 Solutions of 4.2.7
8.4.4 Solutions of 4.2.8
8.4.5 Solutions of 4.2.9
8.4.6 Solutions of 4.3.3
8.4.7 Solutions of 4.3.8
8.4.8 Solutions of 4.4.3
8.5 Exercises of Chapter 5
8.5.1 Solutions of 5.1.3
8.5.2 Solutions of 5.1.6
8.5.3 Solutions of 5.2.1
8.5.4 Solutions of 5.2.7
8.5.5 Solutions of 5.3.1
8.5.6 Solutions of 5.4.1
8.5.7 Solutions of 5.4.2
8.5.8 Solutions of 5.4.3
8.5.9 Solutions of 5.5.2
8.5.10 Solutions of 5.5.6
8.5.11 Solutions of 5.5.7
8.5.12 Solutions of 5.6.7
8.6 Exercises of Chapter 6
8.6.1 Solutions of 6.1.2
8.6.2 Solutions of 6.1.3
8.6.3 Solutions of 6.1.5
8.6.4 Solutions of 6.1.9
8.6.5 Solutions of 6.2.3
8.6.6 Solutions of 6.3.1
8.6.7 Solutions of 6.3.3
8.6.8 Solutions of 6.4.1
8.6.9 Solutions of 6.4.2
8.6.10 Solutions of 6.4.4
8.6.11 Solutions of 6.5.2
References
Index