Alphabeta MathSublevel Deformation and the Handle Attachment Theorem
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1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- 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
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exactness and the Member Calculus
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Fubini and Change of Variables
- Gradient Like Vector Fields and Morse Trajectories
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits and Colimits
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Long Exact Sequences in Homology
- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- 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
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Partitions of Unity and Paracompactness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Preadditive and Additive Categories and Biproducts
- Properties of the Integral and the Working FTC
- Rank Theorems and Embedded Submanifolds
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Relative Homology Excision and Mayer Vietoris
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tangent Cotangent and the Differential
- Tensor Products of Modules
- 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 Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Universal Properties, Representables and the Yoneda Lemma
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Closed sublevel and level set of a smooth function
Lemma
Normalized gradient crosses a compact regular band in controlled time
Theorem
Regular interval diffeomorphism
Corollary
Regular sublevels are diffeomorphic
Proposition
Deformation lemma for a critical point free slab
Definition
K handle core cocore attaching region and belt sphere
Definition
Attaching a smooth handle with corner rounding
Lemma
Smooth handle attachment is independent of corner rounding up to diffeomorphism
Lemma
Adapted descending field near a compact morse band
Lemma
Local critical-value lowering preserves the upper sublevel
Lemma
Local morse sublevel pair is a handle pair
Lemma
Descending flow identifies the local and global attaching regions
Theorem
One critical point handle attachment
Corollary
Unstable disk is the handle core
Corollary
One critical point cell attachment homotopy type
Lemma
Relative homology of the standard handle pair
Corollary
Relative homology of a single handle pair
Proposition
Simultaneous attachment at a morse critical value
Corollary
Index zero handles create components
Corollary
Index n handles cap boundary spheres
Remark