Alphabeta MathChern and Pontryagin Classes by Splitting and Complexification
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- 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
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Algebraic Topology
- Bocksteins Steenrod Squares and Cohomology Operations
- Complex Topological K Theory and Bott Periodicity
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Generalized Cohomology and the Atiyah Hirzebruch Spectral Sequence
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- 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
- Spectra and Stable Homotopy Groups
- Stiefel Whitney and Euler Classes by Universal Constructions
- The Serre Spectral Sequence and Applications
- Topological Vector Bundles and Grassmannian Classification
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
Complex Analysis
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Simply Connected Plane Domains: the Grand Equivalence
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Projective and Injective Resolutions
- Spectral Sequences
- Universal Coefficients and Kunneth Theorems
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
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Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
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- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
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- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- The Fundamental Theorems of Calculus
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
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Topology
- Applications of the Fundamental Group
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- 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
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Lemma
The complex orientation of the underlying real bundle
Definition
Complex projective bundle and tautological complex line
Lemma
Integral cohomology ring of complex projective space
Lemma
Cohomology ring of infinite complex projective space
Lemma
The complex tautological Euler class restricts to the projective-fiber generator
Theorem
Integral complex projective bundle theorem
Definition
Chern classes from the projective-bundle relation
Definition
Complex flag bundle and Chern roots
Theorem
Complex splitting principle with integral injective pullback
Theorem
Naturality, normalization, and Whitney sum for Chern classes
Theorem
Uniqueness of Chern classes from the splitting principle
Lemma
The universal complex flag bundle is BT-n
Theorem
Integral cohomology of BU(n)
Theorem
The first Chern class classifies complex line bundles
Proposition
First Chern class of tensor, dual, and conjugate lines
Theorem
Top Chern class equals Euler class of the underlying real bundle
Theorem
Mod-two reduction of Chern classes
Proposition
Complexification is conjugation invariant
Corollary
Odd Chern classes of a complexified real bundle are two-torsion
Definition
Pontryagin classes by complexification
Theorem
Naturality, stability, and mod-two reduction of Pontryagin classes
Theorem
Pontryagin Whitney product away from two
Theorem
Top Pontryagin class is the square of the Euler class
Lemma
The universal oriented sphere-bundle total space has the homotopy type of BSO(n-1)
Lemma
Rational transfer identifies a finite regular cover with deck invariants
Theorem
Rational cohomology of BO and BSO by Pontryagin and Euler classes
Lemma
Cohomology of a finite CW complex vanishes above its dimension
Definition
Chern character of a complex vector bundle
Theorem
Chern character is a natural ring homomorphism on K-zero
Definition
Graded Chern character by suspension and Bott periodicity
Lemma
The graded Chern character respects relative maps and skeletal filtrations
Lemma
Chern character induces the rational isomorphism on AHSS E-two
Theorem
Rational Chern character isomorphism for finite CW complexes
Sources
- Milnor and Stasheff, Characteristic Classes, Lemma 14.1 and section 15
- Hatcher, Vector Bundles & K-Theory, section 3.2
- Rolf Schon, Fibrations Over a CWh-Base, Theorem 2
- Hatcher, Vector Bundles & K-Theory, section 3.1
- Miller, MIT 18.906 Algebraic Topology II, Lectures 34-35
- Hatcher, Algebraic Topology, section 3.2 and Example 4.42
- Miller, MIT 18.906 Algebraic Topology II, Lecture 35
- May, A Concise Course in Algebraic Topology, Chapter 24 section 3
- May, A Concise Course in Algebraic Topology, Chapter 23 section 7
- May, A Concise Course in Algebraic Topology, Chapter 24 section 4
- Miller, MIT 18.906 Algebraic Topology II, Lecture 36
- Milnor and Stasheff, Characteristic Classes, section 14
- Milnor and Stasheff, Characteristic Classes, section 15
- Hatcher, Vector Bundles & K-Theory, Theorem 3.16
- Hatcher, Vector Bundles & K-Theory, Proposition 3.15(b)
- Miller, MIT 18.906 Algebraic Topology II, Lectures 35-36
- Hatcher, Vector Bundles & K-Theory, Theorem 3.16 proof
- Hatcher, Algebraic Topology, section 3.G
- Hatcher, Algebraic Topology, section 2.2
- Hatcher, Vector Bundles & K-Theory, section 4.1
- Milnor and Stasheff, Characteristic Classes, Problem 16-B
- Hatcher, Vector Bundles & K-Theory, section 5.1
- Caleb Ji, The Atiyah-Hirzebruch Spectral Sequence