Alphabeta MathSimple Homotopy, Whitehead Groups, and Torsion
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 27 results · all verified · 7 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 20 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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 Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- 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
- The Group Algebra and Representations of Finite Groups
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
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Vector Spaces, Linear Subspaces, Span and Direct Sums
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
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- 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
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Elementary expansions and collapses of finite CW complexes
Definition
Simple homotopy equivalence
Definition
Stable general linear and elementary groups for right modules
Lemma
Stable elementary matrices equal the commutator subgroup
Definition
K₁ of a ring and the Whitehead group of a discrete group
Lemma
Integral group rings have invariant basis number
Lemma
A chain contraction makes the odd-to-even parity map invertible
Definition
Finite based free complexes and contraction torsion
Lemma
Contraction torsion does not depend on the contraction
Lemma
Basis-change, direct-sum and based exact-sequence formulas
Definition
Based cellular chains of a universal cover as finite free right group-ring modules
Lemma
Cellular basis ambiguities vanish in the Whitehead group
Lemma
Universal-cover boundaries, maps and homotopies respect the right group-ring action
Lemma
A lifted finite CW equivalence has a contractible group-ring mapping cone
Definition
Whitehead torsion of a finite CW homotopy equivalence
Theorem
Whitehead torsion is independent of all auxiliary choices
Theorem
Composition and based-pair sum formulas for Whitehead torsion
Lemma
An elementary CW expansion has zero Whitehead torsion
Theorem
Simple homotopy equivalences have zero torsion
Lemma
The target of a finite cellular mapping cylinder is a simple subcomplex
Lemma
Cell trading puts a finite relative equivalence in two high degrees
Lemma
Two high relative cell layers have free homotopy bases and their cellular boundary matrix
Lemma
Cell slides and stabilizations realize elementary group-ring matrices
Lemma
An identity relative homotopy matrix permits cell cancellation
Lemma
Zero relative torsion gives a finite relative elementary deformation
Lemma
Every Whitehead class is realized by a finite CW homotopy equivalence
Theorem
Whitehead torsion is the complete obstruction to finite CW simple homotopy
Sources
- Lück, §2.3, pp.34–35
- Cohen, §7, pp.24–26
- Casson, Simple Homotopy Theory, §4, p.30
- Lück, Definition 2.17, pp.34–35
- Lück, §2.1, pp.24–26
- Cohen, §8.3–8.4, pp.31–32
- Lück, Lemma 2.2, pp.25–26
- Lurie, Remark 12, p.4
- Cohen, §19, pp.62–65
- Lück, §2.2, contraction-torsion setup pp.27–28
- Lück, §2.2, equation (2.7), pp.27–28
- Lück, Lemma 2.9 and equation (2.11), pp.29–30
- Cohen, §§19–20, pp.62–69
- Lück, §2.2, pp.30–31
- Davis–Kirk, §11.4, p.343
- Davis–Kirk, Theorem 11.31(1), pp.343–344
- Cohen, §§19, 22, pp.62–65, 72–75
- Lück, A Basic Introduction to Surgery Theory, §2.2
- Lück, Definition 2.13, pp.30–31
- Cohen, §22, pp.72–75
- Lück, Definition 2.13 and Lemma 2.9, pp.29–31
- Cohen, §§19,22, pp.62–65,72–75
- Lück, Lemma 2.9, pp.29–30; Theorem 2.1, pp.23–24
- Cohen, §§20–23, pp.66–77
- Lurie, Lemma 2 and Proposition 4, pp.1–2
- Lück, Lemma 2.18(1), p.35
- Davis–Kirk, Theorem 11.31(2), p.344
- Cohen, §22, p.72
- Lück, Lemmas 2.19–2.20, p.36
- Cohen, §22.3, pp.72–73
- Cohen, §§7.3–7.4, pp.25–27
- Davis–Kirk, Theorem 11.31(3) sketch, pp.344–345
- Casson, Theorem 4.7, Chapter 4
- Cohen, §8.1 and beginning of §8.2, pp.28–30
- Lück, Theorem 2.21 proof sketch, pp.37–38
- Hatcher, Algebraic Topology, Proposition 4.21, and the relative Hurewicz and covering-space arguments of §4.2
- Hatcher, Algebraic Topology, §4.1 change-of-basepoint arguments
- Cohen, §§7.1, 7.4, 8.3–8.4, printed pp.23, 26–27, 31–32
- Casson, proof of Theorem 4.7, printed pp.32–34
- Cohen, §8.2, printed p.30
- Casson, proof of Theorem 4.7, printed pp.33–34
- Cohen, §§7.3–8.5, printed pp.25–33
- Casson, Theorem 4.7, printed pp.32–34
- Lück, Theorem 2.21, printed pp.37–38
- Lück, Lemma 2.18(2), printed pp.35–36
- Lurie, Remark 6, printed p.2
- Cohen, §§8.5 and 22, printed pp.32–33, 72–75