Alphabeta MathHigher Homotopy Groups and Cofiber Sequences
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Modules, Exact Sequences, Projective and Injective Modules
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Tensor Products of Modules
Algebraic Topology
Category Theory
- Abelian Categories
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Homological Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Compactly generated conventions for based homotopy
Lemma
Kification, compact tests, and finite constructions
Lemma
Compact-test exponential law and products of quotient maps
Lemma
Weak Hausdorff diagonals and closed quotients
Lemma
Compact generation preserves the cylinder and closed pushouts
Lemma
Interval exponential law and quotient homotopies
Definition
Higher homotopy group by based cubes
Lemma
Cubical concatenation is well defined on higher homotopy classes
Theorem
Higher homotopy classes form groups and are abelian above degree one
Proposition
Cubical and spherical models of higher homotopy agree
Proposition
Higher homotopy groups are functorial and based homotopy invariant
Definition
Relative homotopy classes and groups
Lemma
Relative cubical disk model and compression
Lemma
Relative homotopy operations are well defined in their valid degrees
Theorem
Long exact sequence of relative homotopy groups
Definition
Cofibration and homotopy extension property
Proposition
Cofibrations are characterized by a retraction of the mapping cylinder strip
Definition
Mapping cylinder and mapping cone
Theorem
Mapping cylinder factorization
Lemma
Pushouts and products preserve the cofibrations used here
Definition
Reduced cone suspension and cofiber sequence
Lemma
Cofiber of a based cofibration is equivalent to the quotient
Lemma
Iterated cofibers rotate with suspension reflection
Lemma
Suspension homotopy classes have natural group structures
Theorem
Puppe sequence is exact after mapping into a based space
Proposition
Loop suspension adjunction on based homotopy classes
Corollary
Higher homotopy groups are iterated loop components
Proposition
Higher homotopy basepoint transport and moving homotopies
Definition
N connected space and n connected map
Lemma
Finite cw basepoints have explicit homotopy extension
Theorem
Lower-dimensional sphere maps are based nullhomotopic
Lemma
Based sphere maps have finite affine bubble normal forms
Lemma
Finite affine bubbles represent signed cubical sums
Theorem