Alphabeta MathHurewicz Whitehead Freudenthal and Cw Approximation
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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
- 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
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- 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
- 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
Lemma
A low-dimensional disk can be pushed off a higher cell
Theorem
Cellular approximation for maps of CW pairs
Corollary
Each homotopy representative is supported on a finite CW subcomplex
Lemma
Cellular attachments with finite boundary support form a CW complex
Theorem
CW approximation of an arbitrary space
Lemma
Cubical pinch is additive on relative homology
Definition
Absolute and relative Hurewicz homomorphisms
Proposition
The first Hurewicz map is abelianization
Definition
Weak homotopy equivalence
Lemma
A weak equivalence has vanishing mapping-cylinder relative groups
Lemma
Vanishing relative homotopy extends an inverse over cells
Lemma
Cellular mapping cylinders and relative cylinders are CW complexes
Theorem
Whitehead theorem
Corollary
A weakly contractible CW complex is contractible
Definition
Connectivity of a CW pair
Lemma
High relative cells do not change lower homotopy
Lemma
Weak homotopy equivalences induce integral homology isomorphisms without choice
Lemma
A connected CW pair has a model without low relative cells
Lemma
Finite relative homotopy lifting across a weak equivalence
Lemma
Weak equivalences glue along a common connected CW subcomplex
Lemma
Weak equivalences of pairs induce isomorphisms on relative homotopy
Lemma
Homotopy excision for a single relative cell layer
Lemma
Relative homotopy exact sequence of a triple in group degrees
Theorem
Homotopy excision
Lemma
CW quotients and collapse of a contractible subcomplex
Lemma
Relative homotopy compares with the CW quotient in the connectivity range
Lemma
The first potentially nonzero homotopy group of a wedge of higher spheres has its cell basis
Lemma
A CW quotient induces relative singular homology isomorphisms
Theorem
Relative cellular homology computes relative singular homology
Lemma
Integral homology of a wedge of higher spheres has its cell basis
Lemma
A relative single cell layer has compatible homotopy and homology bases
Lemma
Cellular reduction for a highly connected pair
Theorem
Relative Hurewicz theorem in the simple-connectivity range
Lemma
Relative Hurewicz comparison through a choice-free weak model
Theorem
Absolute Hurewicz theorem at the first nonzero degree
Definition
Double-mapping-cylinder homotopy pushout and path-space homotopy pullback
Theorem
Blakers--Massey connectivity for a homotopy-pushout square
Theorem
Freudenthal suspension theorem
Sources
- Hatcher, Algebraic Topology, Lemma 4.10 and proof of Theorem 4.8, pp.349–351
- Hatcher Theorem 4.8 and Lemma 4.10; May Chapter 10 §4
- Hatcher Appendix A and Theorem 4.8
- Hatcher, expanded Appendix A, Proposition A.2, pp.3–4; local choice-free separation construction
- Hatcher Proposition 4.13; May Chapter 10 §5
- Hatcher, Algebraic Topology, Proposition4.36, pp369–370, with the relative wedge splitting and cube reparametrizations supplied locally
- Hatcher Proposition 4.36 and preceding definition; May Chapter 15 §1
- Hatcher Hurewicz discussion §2.A/§4.2; May Chapter 15 §1
- Hatcher §4.1; May Chapter 10 §3
- Hatcher proof of Theorem 4.5
- Hatcher, Algebraic Topology, Lemma 4.6 and the subcomplex case of Theorem 4.5, printed pp.346–347
- Hatcher, Algebraic Topology, proof of Theorem4.5 p347 and Proposition4.15 pp353–354; ordinary CW topology and relative cylinder details supplied locally
- Hatcher Theorem 4.5; May Whitehead Theorem, Chapter 10 §3
- Hatcher Whitehead theorem consequence; May Chapter 10 §3
- Hatcher, Algebraic Topology, printed p346, equivalent definitions of relative connectivity
- Hatcher, Algebraic Topology, Corollary4.9, Lemma4.10 and the cell-attachment argument on pp351–353; relative compact-domain proof supplied locally
- Hatcher, Algebraic Topology, Lemma4.6 pp346–347 and Proposition4.21 pp356–357; complete proofs read, finite-choice and finite face-realization details supplied locally
- Hatcher, Algebraic Topology, Proposition4.15 pp353–354 and construction on pp352–353; all-data choice-free weak model and relative inverse details supplied locally
- May, A Concise Course in Algebraic Topology, Chapter10 §3 HELP, p75; Hatcher Lemma4.6 p347; finite-relative ordinary-space proof with explicit time schedule
- May, A Concise Course, Chapter11 §3 p87, weak CW-triad reduction; Chapter10 §3 p75 HELP; finite-data proof supplied locally
- May, A Concise Course, Chapter10 §3 pp75–76, HELP and its cylinder proof of injectivity; relative cubical version proved here
- Hatcher, Algebraic Topology, Lemma4.10 pp350–351 and Theorem4.23 proof Cases1–2 pp361–362; explicit rank, graph and pair comparisons supplied here
- May, A Concise Course, Chapter11 §3 p86, triple sequence; Hatcher Theorem4.23 Case3 p363. Complete group-degree proof supplied locally.
- Hatcher Theorem 4.23 and full proof; May Chapter 11 §§1--3
- Hatcher, Algebraic Topology, quotient CW construction Chapter0 pp8–10 and Proposition0.17 pp15–16; ordinary topology and basepoint details supplied here
- Hatcher, Algebraic Topology, Proposition4.28 p364; full CW and quotient comparisons supplied here
- Hatcher, Algebraic Topology, Example4.26 p363; finite sphere-product topology and finite-support proof supplied locally
- Hatcher, Algebraic Topology, Proposition2.22 and Proposition0.17 pp15–16; mapping-cylinder collar reduction supplied locally
- J. Peter May, A Concise Course in Algebraic Topology, Chapter 13
- Allen Hatcher, Algebraic Topology, Proposition A.5 and Section 2.1
- Hatcher, Algebraic Topology, Example 2.23 and wedge homology; finite splitting supplied explicitly from the pair sequence
- Hatcher, Algebraic Topology, Example4.26, Proposition4.28 and the proof of Theorem4.32; based characteristic details supplied locally
- Hatcher Proposition 4.15, proof of Theorem 4.32, and Lemma 4.38
- Hatcher Theorem 4.32 and proof; May Chapter 15 §1
- Hatcher, Algebraic Topology, Proposition4.21 pp356–357 and proof of Theorem4.32 pp366–374; local weak-model comparison makes choice accounting explicit
- Hatcher, Algebraic Topology, Theorem4.32 pp366–367 and Hurewicz definition pp369–370; all-basepoint and actual CW-type transfer supplied locally
- May, A Concise Course in Algebraic Topology, Chapter 10 §7 double mapping cylinder
- Rezk, Proof of the Blakers--Massey theorem, §3
- Hatcher Theorem4.23 and complete local homotopy-excision proof; relative-cube to fiber and total-space comparisons derived explicitly here
- Rezk §1.6 and Theorem3.1, connectivity convention and statement only; this item uses the classical excision proof, not Rezk's truncation proof
- Hatcher Corollary 4.24; May Chapter 11 §2