Alphabeta MathThe De Rham Complex Homotopy and Mayer Vietoris
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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
- 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
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
Differential Geometry
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Homological Algebra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Exterior Powers, Orientation and Hodge Duality
- 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
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- 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 Total Derivative in ℝᵐ → ℝⁿ
- 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
- Countability Axioms and Cardinal Functions
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
De rham cochain complex
Definition
Closed and exact differential forms
Definition
De rham cohomology
Proposition
Zero and out of range de rham cohomology
Theorem
Zero th de rham cohomology is locally constant functions
Proposition
Pullback is a morphism of de rham complexes
Theorem
Pullback induces a well defined map on de rham cohomology
Proposition
De rham cohomology is a contravariant functor
Lemma
Wedge with a closed form preserves exactness classes
Theorem
Wedge product descends to de rham cohomology
Definition
De rham cohomology ring
Proposition
Pullback is a homomorphism of de rham cohomology algebras
Definition
Integration along the unit interval for a differential form
Lemma
The interval homotopy operator is coordinate independent
Theorem
De rham homotopy formula on a product
Theorem
De rham homotopy formula for a smooth homotopy
Corollary
Smoothly homotopic maps induce the same de rham map
Theorem
De rham cohomology is smooth homotopy invariant
Corollary
De rham cohomology is continuous homotopy invariant on smooth manifolds
Definition
Radial contraction of a star shaped domain
Theorem
Poincare lemma for differential forms on star shaped domains
Corollary
Closed differential forms are locally exact
Theorem
De rham cohomology of a contractible smooth manifold
Definition
Two open set de rham mayer vietoris cochain maps
Lemma
The de rham mayer vietoris sequence is exact at the first two terms
Lemma
The de rham mayer vietoris difference map is surjective
Theorem
Short exact mayer vietoris sequence of de rham complexes
Theorem
Mayer vietoris sequence in de rham cohomology
Proposition
Explicit de rham mayer vietoris connecting class
Theorem
Naturality of de rham mayer vietoris for maps of covered manifolds
Corollary
De rham cohomology of a finite disjoint union is the direct sum
Theorem
De rham cohomology of spheres
Corollary
De rham cohomology of punctured euclidean space
False statement
The de rham cohomology class of a form is defined without closedness
False statement
Every smooth form is either closed or exact
False statement
Continuously homotopic smooth maps can be inserted directly into the differential form homotopy operator
False statement
The poincare lemma says every closed form is globally exact
False statement
The mayer vietoris sequence is obtained by restricting forms without a partition of unity
False statement