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Branched-cover correction terms and sliceness — Examples and Counterexamples
1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Absolute Values Completions and P Adic Numbers
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Analytic Majorants and the Cauchy–Kovalevskaya Theorem
- Analyticity of Holomorphic Functions; Liouville and Morera
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Arithmetic Functions and Dirichlet Convolution
- Average Orders Divisor Sums and Representation Counts
- Banach-Space Differential Calculus and Banach Manifolds
- Binary Operations, Monoids, Groups and Subgroups
- Bocksteins Steenrod Squares and Cohomology Operations
- Bounded Linear Operators and Quotient Spaces
- Bounded Variation and the Riemann–Stieltjes Integral
- Braided and Symmetric Monoidal Categories
- Branched-cover correction terms and sliceness
- Calderón–Zygmund Decomposition and Singular Integrals
- Cardinal Arithmetic, Cofinality and the Alephs
- Casson–Gordon invariants and sliceness obstructions
- 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
- Chains, Antichains, Sperner and Dilworth
- Characteristic Numbers and Cobordism Obstructions
- Characters and the Orthogonality Relations
- Chebyshev Bounds and Mertens Theorems
- Chern and Pontryagin Classes by Splitting and Complexification
- Chern–Weil Theory and Characteristic Forms
- Choice Strength in Baire, Urysohn, Stone, and Tychonoff
- Classification of Compact Connected Surfaces
- Classification of Covering Spaces
- Clifford Bundles, Dirac Operators, and Index Interfaces
- Codimension One Foliations, Secondary Classes and Characteristic Disk Foundations
- Compact Operators and Riesz Schauder Theory
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complex Power Series and Analytic Functions
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Conformal Mapping, Branches, and the Schwarz Lemma
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connected sums and prime decomposition
- Connectedness
- Connections Levi Civita and Parallel Transport
- 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
- Convergence: Nets and Filters
- Convexity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Cyclic branched covers and knot determinants
- Cyclic Groups and Direct Products
- Darboux, L'Hôpital, and Taylor's Theorem
- Dehn surgery on knots and homology
- Density Separability and Convolution in Lᵖ
- Derived Functors
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dirichlet Characters L Functions and Primes in Progressions
- Dirichlet Series and Euler Products
- Distributions Test Functions and Differentiation
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Divisors, Riemann--Roch, and Duality
- Double Complexes Exact Couples and Convergence
- Dual Spaces Adjoint Operators and Annihilators
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Duality and Rigidity in Monoidal Categories
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Equivalent Forms of Completeness
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Euclidean Surface Measure, Divergence, and Green Identities
- Exactness and the Member Calculus
- Exotic Smooth Structures and Milnor Spheres
- Ext and Balanced Resolutions
- Exterior Powers, Orientation and Hodge Duality
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Filters and Ultrafilters
- Finite Averaging and Character-Theory Prerequisites
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite Probability and the Probabilistic Method
- Formal Immersions and the Smale Hirsch Theorem
- Formal Power Series
- Foundations of the Real Numbers for Analysis
- Fourier Multipliers and Sobolev Characterisations
- Fourier Transform Convolution and Approximate Identities
- Fredholm Elliptic Problems and the Elliptic Spectrum
- 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 Solutions Newtonian Potentials and Green Functions
- Fundamental Trigonometric Identities
- Further Trigonometric Identities and Inverse Functions
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Generalized Cohomology and the Atiyah Hirzebruch Spectral Sequence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Gradient Like Vector Fields and Morse Trajectories
- Graphs, Walks and Connectivity
- Green Functions, Harmonic Measure, and Conformal Invariance
- Grid homology and combinatorial knot Floer theory
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hall–Mal’cev Coordinates and Bass–Guivarc’h Growth
- Handle Cancellation Slides and Elementary Moves
- Handle Decompositions Duality and Rearrangement
- Harmonic Functions and Mean Values in Rn
- Harmonic Functions and the Poisson Integral
- 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
- Hilbert Space Geometry and Riesz Representation
- Hilbert Symbols and the Quadratic Local Global Principle
- Holomorphic Functions of Several Complex Variables
- 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
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Incidence Algebras and Möbius Inversion
- Infinite Products and the Weierstrass Factorisation Theorem
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Integration of Forms and the General Stokes Theorem
- Interior and Boundary Sobolev Elliptic Regularity
- Intersection Pairings Self Intersection and Euler Classes
- Inverse Limits and Noetherian Completion
- Isolated Singularities and Laurent Series
- Isotopy Extension and Embedding Theory Beyond Whitney
- Knot exteriors and peripheral systems
- Koszul Complexes and Regular Sequences
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Lebesgue Measure on Euclidean Space
- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
- Lie Groups, Invariant Fields, and the Exponential Map
- Lie Subgroups, Actions, and Homogeneous Spaces
- Limits and Colimits
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Recurrences and Rational Generating Functions
- 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
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Maximum Principles Harnack and Liouville in Rn
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Metrization: Urysohn, Nagata–Smirnov, Bing, Smirnov
- Mittag-Leffler and Runge's Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Modular Categories, Verlinde Formula, and TQFT
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monoidal Categories and Monoidal Functors
- 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
- Morse Inequalities and the Handle Chain Complex
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- 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
- Orthonormal Bases, Parseval and Fourier Series
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Periods, Jacobians, and Abel--Jacobi Theory
- pi: the Equivalent Characterizations
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Pin and Spin Groups and Their Representations
- Plane Graphs, Euler's Formula and the Five Colour Theorem
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Pontryagin Thom and Framed Cobordism
- Power Series and Real-Analytic Functions
- Preadditive and Additive Categories and Biproducts
- Prime Spectra and Radicals
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Primitive Roots and Unit Groups Modulo N
- Product Measures and the Fubini Tonelli Theorems
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Projective and Injective Resolutions
- Properties of the Integral and the Working FTC
- Quadratic Reciprocity and the Jacobi Symbol
- Quadratic Residues and the Legendre Symbol
- Radon Measures and the Riesz Markov Kakutani Theorem
- Rank Theorems and Embedded Submanifolds
- Real Clifford Modules, KO Theory, and ABS Orientations
- Reeb Stability and Global Foliation Constructions
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Reflective Subcategories and the Adjoint Functor Theorems
- Reflexivity and Eberlein Smulian
- Regular Surfaces and Surface Integrals
- Relations, Functions, and Quotients
- Relative Homology Excision and Mayer Vietoris
- Relative smooth triangulation and bundle fitting
- Riemann Curvature and Riemannian Submanifolds
- Riemann Surfaces, Branched Maps, and Differentials
- 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
- Schwartz Space and the Plancherel Theorem
- 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
- Signed and Complex Measures Hahn and Jordan
- 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 Cohomology and Coefficient Theorems
- Smooth Approximation and Sobolev Extension
- Smooth Cobordism Relations Groups and Rings
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Surgery Traces and Handle Trading
- Smooth Vector Bundles and Sections
- Solvability by Radicals and Kummer Theory
- Spectra and Stable Homotopy Groups
- Spectral Sequences
- Splitting Fields
- Stable Unstable Manifolds and Morse Smale Transversality
- Stiefel Whitney and Euler Classes by Universal Constructions
- Stone–Weierstrass in General
- Strictification and Mac Lanes Coherence Theorem
- Strongly Continuous Semigroups and Hille Yosida
- Subharmonic Functions and the Dirichlet Problem
- Sublevel Deformation and the Handle Attachment Theorem
- Subobject Lattices Generators and the Grothendieck Axioms
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
- Symplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory
- Tangent Cotangent and the Differential
- Tempered Distributions and the Fourier Transform
- Tensor and Fusion Categories
- Tensor Fields Exterior Algebra and Differential Forms
- Tensor Products of Modules
- The Analytic Hahn Banach Theorem
- The Ascoli–Arzelà Theorem
- The Burau Representations
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The De Rham Complex Homotopy and Mayer Vietoris
- 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 Divergence Theorem and Classical Stokes
- The Duality of Lᵖ and L^q
- The Exponential Function
- The Exterior Derivative and Cartan Calculus
- The Field of Fractions and Localisation
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- 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 Gauss Bonnet Theorem for Riemannian Surfaces
- The Group Algebra and Representations of Finite Groups
- The Hirzebruch Signature Theorem
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Hopf Degree Theorem
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Riemann Sphere and Möbius Transformations
- The Riemann Zeta Function
- The Seifert–van Kampen Theorem
- The Serre Spectral Sequence and Applications
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The Whitney Trick and Surgery Below the Middle Dimension
- The Winding Number and the Global Cauchy Theorem
- The ZFC Axioms and the Basic Set Constructions
- Thom Spaces Normal Data and Collapse Maps
- Thom Spectra and Unoriented Bordism Detection
- Three-manifold surgery presentations and Kirby moves
- Topological Spaces and Continuity
- Topological Vector Bundles and Grassmannian Classification
- Topology of ℝ
- Tor Flatness and Global Dimension
- Trees, Forests and Spanning Trees
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Trigonometric and Oscillatory Examples in One Variable
- Uniform Spaces: the Three Definitions
- Universal Clifford Algebras and Their PBW Basis
- Universal Coefficients and Kunneth Theorems
- Universal Properties, Representables and the Yoneda Lemma
- Urysohn's Lemma and the Tietze Extension Theorem
- Vanishing Cycles, Novikov and Taut Foliations
- Vector Field Index Euler Characteristic and Poincare Hopf
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Derivatives and Sobolev Spaces
- Whitney Embedding Tubular Neighbourhoods and Approximation
- Yang–Baxter Operators and Quantum Braid Representations
2 · Logical flowchart
3 · Definitions, theorems and proofs
None yet.
4 · Examples, counterexamples and false statements
ExampleConstruction: Literature-sourcedVerification: AI-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →
L(3,1) correction terms in the stated orientation convention
Example
Use the orientation and cyclic labels for which . For , the correction terms are , , and . Reversing orientation negates all three values.
Example references
Facts & Assumptions
Given: The stated orientation convention and the three labels .
[F1]
For every and canonical label , the stated correction-term formula holds; reversing orientation negates the value at the conjugate label (Lens-space correction-term formula for L(p,1)).
Proof
technique · substitute each label into the formula and simplify
Given: The data above.
1.1F1givenalgebra
For , and , so [F1] gives .
2.1F1step 1.1algebra∎
For , the numerator is ; for , it is . Both denominators are , so the other two values are . Orientation reversal negates each by [F1].
CounterexampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →
A nonextendible Spin-c structure with nonzero d on a rational-ball boundary
Statement refuted
The false claim is: if a closed oriented three-manifold with bounds a smooth compact four-manifold with , then a nonzero correction term for any boundary Spin structure obstructs the existence of .
Statement references
Example
Let be the rational homology ball bounded by constructed as the complement of a tubular neighborhood of a conic in . The canonical Spin label has . The extension set has exactly two elements and every extending structure has correction term zero. Therefore label is nonextendible, even though this rational-ball filling exists.
Example references
Facts & Assumptions
Given: AC, the rational ball with boundary , and the lens-space label .
[A1]
AC is assumed by the construction and the extension-set supplier (The Axiom of Choice).
[F1]
The conic complement is a rational homology four-ball with boundary in the selected orientation (A rational homology ball bounded by L(4,1)).
[F2]
The surgery matrix for is nonsingular and its first homology has order (Inverse-matrix formula for a surgery linking form).
[F3]
Over a rational-ball boundary, the extending Spin structures form a coset of order , and every extending structure has zero correction term (Spin-c extensions over a rational ball form a metabolizer coset, Correction terms vanish on extending Spin-c structures over rational homology balls).
[F4]
For the selected orientation, (Lens-space correction-term formula for L(p,1)).
Proof
technique · compute the extension-set size and the correction term at a label outside that set
Given: The data above.
1.1A1F1F2F3givenalgebra
By [F1] and [F2], is a rational homology ball with . Therefore [F3] gives exactly extending Spin structures.
2.1A1F3F4step 1.1algebra∎
At label , [F4] gives . Every extending structure has by [F3], so label is not in the two-element extension set. Thus a nonextending structure has nonzero correction term while the rational-ball filling exists, refuting the stated claim.
Sources
- Peter S. Ozsvath and Zoltan Szabo, Absolutely Graded Floer Homologies and Intersection Forms for Four-Manifolds with Boundary
- Jennifer Hom, Lecture Notes on Heegaard Floer Homology
- Paolo Aceto and Marco Golla, Dehn Surgeries and Rational Homology Balls
- Hisaaki Endo and Yusuf Z. Gurtas, Lantern Relations and Rational Blowdowns