Alphabeta MathCasson–Gordon invariants and sliceness obstructions
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1 · Prerequisites
- Abelian Categories
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2 · Logical flowchart
3 · Items
Definition
Bounding-chain expression for a torsion linking pairing
Definition
Finite nonsingular linking forms and metabolizers
Definition
Oriented bordism groups of maps to a space
Definition
Poly-torsion-free abelian groups
Lemma
Local-coefficient cap duality on a coordinate ball without choice
Lemma
Choice-free Wall chain finiteness from an explicit finite domination
Lemma
Compact manifold fundamental groups have finite generators without choice
Lemma
A compact manifold's twisted singular chains admit a one-vertex reduction without choice
Lemma
Compact-support local Mayer–Vietoris and its cap ladder without choice
Lemma
Finite-order smooth line bundles have torsion reduced bundle class
Lemma
Finite von Neumann modules, Hilbert closures, and dimension-zero quotients
Lemma
Finite torsion-free abelian freeness by explicit rational coordinates without choice
Lemma
Finite Folner sets give both Ore conditions and a constructed skew field of fractions
Theorem
Compact manifolds are retracts of finite polyhedra without choice
Theorem
Explicit mitoses give functorial integrally acyclic group envelopes
Definition
Exterior of a smooth proper disk
Lemma
Finite smooth surface presentations admit prescribed collar matches
Lemma
Finite cellular L2 duality and weak exactness
Lemma
The torsion linking pairing of a rational homology sphere is nonsingular
Lemma
Oriented bordism classes over a space form an abelian group
Lemma
PTFA groups satisfy the finite Folner condition without a choice hypothesis
Lemma
Nonsplit torsion-free abelian extensions preserve absence of group-ring zero divisors
Lemma
Von Neumann inertia, dense congruences, and coisotropic reduction
Theorem
Compact oriented manifold duality with local coefficients without choice
Theorem
Compact twisted signature operators give additive bundle indices and finite-character signatures
Theorem
Compact-support duality for finite-atlas manifolds without choice
Theorem
Finite geometric chains and compact manifold chain domination without choice
Definition
Zero-framed surgery on a knot
Lemma
Twisted right-chain duals realize local coefficient duality
Lemma
Integral and field PTFA group rings have compatible Ore fractions and involutions
Lemma
The boundary kernel of a rational homology ball is a linking metabolizer
Lemma
The exterior of a smooth proper disk has the homology of a circle
Theorem
The relevant third oriented bordism group is finite
Theorem
Local coefficient Lefschetz duality without choice
Theorem
PTFA group rings embed in skew fields of fractions
Definition
Infinite and finite cyclic covers of a slice-disk exterior
Lemma
Equivariant intersections under collared bordism gluing
Lemma
Disk-band Seifert surfaces give actual cyclic zero-surgery bounding manifolds
Lemma
Hermitian symmetry of the relative equivariant intersection pairing
Lemma
Prime-power cyclic covers of a slice-disk exterior over the defining field
Theorem
Topological finite-projective L2 duality and whole-component boundary gluing
Definition
Cyclic cover of zero surgery on a knot
Definition
Finite-order characters of a cyclic branched cover
Definition
Integral half-solutions, Strebel classes, and L²-signatures
Definition
Cyclic branched cover of a smooth proper disk
Theorem
Topological five-dimensional bordisms preserve ordinary and L2 signatures
Definition
Localized Blanchfield pairings and higher-order rho-invariants
Lemma
Finite cyclic von Neumann signatures are exact character averages
Lemma
A prime-power branched cover of a smooth proper disk is a rational homology ball
Lemma
Signature additivity under gluing
Lemma
Homology splitting for the cyclic cover of zero surgery
Theorem
Smooth group signature defects are independent and uniformly bounded
Definition
Torsion linking form and Casson–Gordon bordism data
Lemma
A slice disk induces a metabolizer on the branched-cover linking form
Theorem
Canonical finite cyclic knot signature defects equal Seifert signature averages
Lemma
Prime-power characters admit a multiple bounding bordism
Lemma
Characters vanishing on the disk metabolizer extend over the branched cover
Theorem
Finite cyclic smooth bounding signature defects are independent
Lemma
Nonsingularity of the Casson–Gordon twisted intersection form
Lemma
Bordism independence of the Casson–Gordon Witt class
Definition
Casson–Gordon Witt class and signature defect
Lemma
Casson–Gordon signature defects vanish on the slice metabolizer
Lemma
Prime-power cyclic branched covers of knots have finite first homology
Theorem
Casson–Gordon signature-defect obstruction to sliceness
Sources
- Patrick Gilmer, Slice Knots in S3
- Charles Livingston and Swatee Naik, Introduction to Knot Concordance
- Tim D. Cochran, Kent E. Orr, and Peter Teichner, “Knot concordance, Whitney towers and L²-signatures”
- Hatcher, Algebraic Topology, local calculation in Poincare duality and local coefficients
- Andrew Ranicki, The algebraic theory of finiteness obstruction, Math. Scand. 57 (1985), 105–126
- Steve Ferry and Andrew Ranicki, A survey of Wall's finiteness obstruction
- Allen Hatcher, Algebraic Topology
- Hatcher, Algebraic Topology, fundamental groups and finite graph generators
- Hatcher, Algebraic Topology, singular chains and local coefficients, Chapters2 and3.H
- Hatcher, Algebraic Topology, proof of Poincare duality and local coefficients, Chapters 3.3 and 3.H
- Michael Atiyah, K-Theory
- M. F. Atiyah, K-Theory
- Jae Choon Cha and Kent E. Orr, L2-signatures, homology localization, and amenable groups
- C. Löh, Geometric Group Theory: An Introduction, finitely generated abelian groups
- Keith Conrad, Modules over a PID
- Tim D. Cochran, Kent E. Orr, and Peter Teichner, Knot concordance, Whitney towers and L²-signatures, §2.1
- C. Löh, Geometric Group Theory: An Introduction
- Hatcher, Algebraic Topology, Appendix
- Jae Choon Cha, A topological approach to Cheeger-Gromov universal bounds for von Neumann rho-invariants
- A. J. Casson and C. McA. Gordon, Cobordism of Classical Knots (1986), with appendix by P. M. Gilmer
- Gallier and Xu, A Guide to the Classification Theorem for Compact Surfaces
- Lück and Schick, Various L2-signatures and a topological L2-signature theorem
- Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, Decision problems for 3-manifolds and their fundamental groups
- Wolfgang Lück and Thomas Schick, Various L2-signatures and a topological L2-signature theorem
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter5, local-coefficient Poincare duality
- Hatcher, Algebraic Topology, Poincare duality and local coefficients
- Jean-Pierre Demailly, Complex Analytic and Differential Geometry
- Hatcher, Algebraic Topology
- Tim D. Cochran, Christopher W. Davis, and Arunima Ray, “Injectivity of satellite operators in knot concordance”
- Tim D. Cochran, Bridget D. Franklin, Matthew Hedden, and Peter D. Horn, “Knot concordance and homology cobordism”
- Hatcher, Algebraic Topology, local coefficients and duality, Section3.H
- Cochran, Orr and Teichner, Knot concordance, Whitney towers and L2-signatures, Theorem2.13
- Cochran, Orr and Teichner, Knot concordance, Whitney towers and L2-signatures, Section2.1
- Cha and Kim, Iterated satellite operators on the knot concordance group
- Cochran, Orr and Teichner, Structure in the classical knot concordance group
- Cochran, Orr and Teichner, Knot concordance, Whitney towers and L2-signatures
- Lueck and Schick, Various L2-signatures and a topological L2-signature theorem
- Anthony Conway, Algebraic Concordance and Casson–Gordon Invariants
- Jae Choon Cha, “Amenable L²-Theoretic Methods and Knot Concordance”
- Jae Choon Cha and Kent E. Orr, “L²-signatures, homology localization, and amenable groups”
- Patrick Aceto, Joshua Meier, Allison N. Miller, Matthew Miller, JungHwan Park, and Andras I. Stipsicz, Branched covers bounding rational homology balls
- Wolfgang Lück and Thomas Schick, "Various L²-signatures and a topological L²-signature theorem"
- Tim D. Cochran, Kent E. Orr, and Peter Teichner, "Knot concordance, Whitney towers and L²-signatures"
- Daniel S. Freed, Bordism Old and New
- Stefan Friedl, Higher and Quantum Invariants of Knots
- Patrick Gilmer and Charles Livingston, On Surgery Curves for Genus-One Slice Knots
- Charles Livingston, Seifert Forms and Concordance
- Dale Rolfsen, Knots and Links