Alphabeta MathFox–Milnor factorization and slice obstructions
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1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Alexander polynomials from Seifert matrices
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Asymptotic Cones and the Sublinear Triangle Criterion
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Variation and the Riemann–Stieltjes Integral
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Cayley Graphs, Word Metrics and Quasi-Isometry
- Chain Complexes and Homology
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Chain Homotopy and the Homotopy Category
- Chern and Pontryagin Classes by Splitting and Complexification
- Classification of Compact Connected Surfaces
- Classification of Covering Spaces
- Codimension One Foliations, Secondary Classes and Characteristic Disk Foundations
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connected sums and prime decomposition
- Connectedness
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Contour Integration
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Crossing signs, writhe, and linking number
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Cyclic Groups and Direct Products
- Darboux, L'Hôpital, and Taylor's Theorem
- Density Separability and Convolution in Lᵖ
- Derived Functors
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exactness and the Member Calculus
- Ext and Balanced Resolutions
- Exterior Powers, Orientation and Hodge Duality
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Filters and Ultrafilters
- Finite basis projector energies and variational electronic structure
- Finite Counting, Factorials and Binomial Coefficients
- Formal Immersions and the Smale Hirsch Theorem
- Foundations of the Real Numbers for Analysis
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Fubini and Change of Variables
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Geometric Braids and Artin Generators
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Gradient Like Vector Fields and Morse Trajectories
- Graphs, Walks and Connectivity
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hall–Mal’cev Coordinates and Bass–Guivarc’h Growth
- Handle Decompositions Duality and Rearrangement
- Hausdorff via the Diagonal
- Hecke Markov Traces and Polynomial Link Invariants
- Hereditary and Productive Behaviour of the Separation Axioms
- Higher Homotopy Groups and Cofiber Sequences
- Hnn Extensions and Brittons Lemma
- Homology Axioms Degree and Classical Applications
- Homotopy and Homotopy Equivalence
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Isotopy Extension and Embedding Theory Beyond Whitney
- Knot exteriors and peripheral systems
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Lebesgue Measure on Euclidean Space
- Levine–Tristram signatures and the Arf invariant
- Limits and Colimits
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Local Coefficients, Twisted Homology, and Duality
- Localisation of Modules and Support
- Long Exact Sequences in Homology
- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mittag-Leffler and Runge's Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Morse Critical Points Hessians and Indices
- Morse Functions Critical Values and Genericity
- Noetherian Rings and Hilbert Basis
- Normal Subgroups and Quotient Groups
- Normal-surface unknot recognition
- Normalization Finiteness for Affine Domains
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Orientations Poincare Lefschetz and Alexander Duality
- Oriented and Mod Two Intersection Numbers
- Oriented Links, Braid Closures, and Markov Equivalence
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Plane Graphs, Euler's Formula and the Five Colour Theorem
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Preadditive and Additive Categories and Biproducts
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Projective and Injective Resolutions
- Properties of the Integral and the Working FTC
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Rank Theorems and Embedded Submanifolds
- Reeb Stability and Global Foliation Constructions
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Relative Homology Excision and Mayer Vietoris
- Relative smooth triangulation and bundle fitting
- Riemannian Metrics Length Distance and Volume
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sard Theorem and Transversality
- Seifert matrices and the Alexander module
- Seifert surfaces and knot genus
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Sine, Cosine, and the Definition of Pi
- Singular Chains and Singular Homology
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
- Singular Cohomology and Coefficient Theorems
- Smooth and topological knot concordance
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Splitting Fields
- Sublevel Deformation and the Handle Attachment Theorem
- Subobject Lattices Generators and the Grothendieck Axioms
- Subspaces, Products, and Quotients
- Subspaces, Products, and Quotients: Examples and Counterexamples
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- Tensor Products of Modules
- The Burau Representations
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The de Rham Theorem and Degree
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Diagram Lemmas in an Abelian Category
- The Exponential Function
- The Exterior Derivative and Cartan Calculus
- The Field of Fractions and Localisation
- The Fundamental Group
- The Fundamental Group of the Circle
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Fundamental Theorems of Calculus
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Seifert–van Kampen Theorem
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The Winding Number and the Global Cauchy Theorem
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topological Vector Bundles and Grassmannian Classification
- Topology of ℝ
- Trees, Forests and Spanning Trees
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Uniform Spaces: the Three Definitions
- Universal Coefficients and Kunneth Theorems
- Universal Properties, Representables and the Yoneda Lemma
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Whitney Embedding Tubular Neighbourhoods and Approximation
2 · Logical flowchart
3 · Items
Lemma
A metabolic Seifert matrix has Fox–Milnor factorization
Theorem
Locally flat sliceness implies Fox–Milnor factorization
Corollary
Failure of Fox–Milnor factorization obstructs local sliceness
Theorem