Alphabeta MathHurewicz Whitehead Freudenthal and Cw Approximation — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Group Homomorphisms and the Isomorphism Theorems
- 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
- Cw Complexes and Cellular Homology
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Relative Homology Excision and Mayer Vietoris
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
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
- Applications of the Fundamental Group
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Covering Spaces and Lifting
- 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 Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Example
First nonzero homotopy group of a sphere
Example
Hurewicz calculation for a wedge of simply connected spheres
Example
A simply connected CW homology equivalence is a homotopy equivalence under the stated choice conditions
Example
Freudenthal stable range for spheres
Counterexample
Whitehead theorem fails without CW type
Counterexample
A homology equivalence need not be a homotopy equivalence without simple connectivity
Sources
- Hatcher Corollary 4.25 and Theorem 4.32
- May Hurewicz wedge lemma, Chapter 15 §1
- Hatcher Corollary 4.33
- Hatcher Corollary 4.24; May Chapter 11 §2
- May Whitehead theorem hypotheses, Chapter 10 §3; explicit rational-space witness checked locally
- Hatcher discussion after Corollary 4.33; explicit two-cell witness checked locally