Alphabeta MathThe de Rham Theorem and Degree
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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
- Free Modules, Exact Sequences, Projective and Injective Modules
- 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
- Tensor Products of Modules
Algebraic Topology
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Homology Axioms Degree and Classical Applications
- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
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
- Subobject Lattices Generators and the Grothendieck Axioms
- 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
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
- 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 De Rham Complex Homotopy and Mayer Vietoris
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
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
- 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
- Constant Rank, Submersions, Immersions and Regular Level Sets
- 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
- Improper and Parameter-Dependent Multiple Integrals
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- 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 Divergence Theorem and Classical Stokes
- 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
- Volumes of Elementary Solids and Solids of Revolution
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
- Function Space Topologies and the Exponential Law
- 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 Fundamental Group of the Circle
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Standard orientation of the affine simplex
Definition
Integral of a form over a smooth singular simplex
Lemma
Simplex integrals are independent of affine coordinate identification
Lemma
Stokes theorem for the standard simplex
Theorem
Stokes theorem for smooth singular chains
Lemma
The de Rham complex and pullback extend to manifolds with boundary
Definition
De Rham integration cochain
Theorem
De Rham integration is a cochain map
Theorem
The de Rham map on cohomology is well defined
Proposition
Naturality of the de Rham map
Lemma
The de Rham homotopy formula extends to boundary manifolds
Lemma
The de Rham map is an isomorphism on convex coordinate domains
Lemma
De Rham Mayer–Vietoris with boundary and an explicit partition lift
Lemma
The de Rham and smooth singular Mayer–Vietoris diagram commutes away from connectors
Lemma
The de Rham map commutes with Mayer–Vietoris connectors
Theorem
The de Rham map is an isomorphism on a two-open union
Theorem
De Rham theorem for smooth singular cohomology
Corollary
De Rham vector-space comparison with continuous singular cohomology
Lemma
An affine cone homotopy from the diagonal to the front-back shuffle
Lemma
Integration over the signed shuffle equals the product of simplex integrals
Lemma
De Rham integration respects wedge and cup in cohomology
Theorem
The de Rham theorem
Corollary
De Rham cohomology depends only on the underlying homotopy type
Remark
The ring form of de Rham’s theorem needs the singular cup product
Definition
Compactly supported de Rham cohomology
Proposition
Proper smooth maps pull back compactly supported forms
Proposition
Compactly supported de Rham cohomology is contravariant for proper smooth maps
Lemma
Finite chart localization gives choice-free integration and compact Stokes
Theorem
Integration descends to compactly supported top de Rham cohomology
Lemma
Zero-integral compactly supported top forms on Euclidean space have compactly supported primitives
Lemma
Compactly supported top cohomology propagates across overlapping oriented coordinate balls
Theorem
Integration is an isomorphism on top compactly supported de Rham cohomology
Corollary
Top de Rham cohomology of a closed connected oriented manifold is real
Definition
Degree of a proper smooth map by compact-support cohomology
Lemma
Degree is well defined and independent of the normalized top form
Definition
Local orientation sign of a regular preimage
Theorem
Regular-value formula for compact-support degree
Lemma
Smooth orientation sign is the local integral homology multiplier
Theorem
Regular-value formula for degree
Corollary
Degree is an integer and independent of the regular value
Proposition
Degree is multiplicative under composition
Proposition
Degree of an orientation-preserving or reversing diffeomorphism
Theorem
Degree is invariant under proper smooth homotopy
Corollary
A nonzero-degree map to a connected manifold is surjective
Proposition
Degree of the antipodal map on the sphere
Proposition
Degree of the power map on the circle
False statement
The de Rham map is a cochain map without Stokes on simplices
False statement
Naturality alone gives Mayer–Vietoris connector compatibility
False statement
The vector-space de Rham comparison is automatically a ring isomorphism
False statement
Compactly supported cohomology is contravariant for every smooth map
False statement
Degree is the unsigned number of points in a regular fibre
False statement
A homotopy between proper maps is automatically proper
Sources
- Peter S. Park, Proof of de Rham's Theorem
- Ioan Mărcuț, Manifolds (2017 lecture notes), boundary smoothness and forms
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Chapters 16–17
- Nigel Hitchin, Differentiable Manifolds (2014), de Rham homotopy operator
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Theorem 17.20
- Hatcher, Algebraic Topology, section 3.2; affine-cone specialization of the diagonal comparison
- Local simplex product integration; shuffle convention from Miller Lecture 7
- Joel W. Robbin, The de Rham Theorem
- Robbin–Salamon, Introduction to Differential Topology
- Robbin-Salamon, Introduction to Differential Topology, section5.2.2 Lemma5.2.10 and section5.2.3 Theorem5.2.12; finite-support specialization
- Alexander Kupers, Algebraic Topology, section8.1 pp60–62; local degree and regular-value computation
- Robbin–Salamon, Introduction to Differential Topology, Theorem 5.4.1 and the consequence following it
- Robbin–Salamon, Introduction to Differential Topology, degree examples following Theorem 5.4.1
- Peter S. Park, Proof of de Rham's Theorem, Theorem 3.1, PDF p.5
- Peter S. Park, Proof of de Rham's Theorem, Proposition 3.2, PDF p.6
- Joel W. Robbin, The de Rham Theorem, multiplicative comparison discussion
- Robbin–Salamon, Introduction to Differential Topology, compactly supported pullback discussion
- Robbin–Salamon, Introduction to Differential Topology, Theorem 5.4.1, pp.191–192
- Robbin–Salamon, Introduction to Differential Topology, proper homotopy clause following Theorem 5.4.1