Alphabeta MathComplex Topological K Theory and Bott Periodicity
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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 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
- Tensor Products of Modules
Algebraic Topology
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Higher Homotopy Groups and Cofiber Sequences
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
- Spectra and Stable Homotopy Groups
- Topological Vector Bundles and Grassmannian Classification
Category Theory
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
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- 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
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Monotone Functions, Discontinuities, and Continuity Sets
- 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
Topology
- Applications of the Fundamental Group
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Covering Spaces and Lifting
- Hausdorff via the Diagonal
- 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
Definition
The Whitney-sum monoid of complex vector bundles
Definition
Complex topological K⁰ by Grothendieck completion
Proposition
Equality in K⁰ is stable isomorphism over compact bases
Definition
Grothendieck ring structure and rank map
Definition
Reduced complex K-theory
Proposition
K⁰ is contravariantly functorial and homotopy invariant
Theorem
Reduced K-theory exact sequence of a cofibration
Definition
External product in complex K-theory
Lemma
Determinant classifies loops in complex general linear groups
Theorem
Hopf-line calculation of K⁰(S²)
Lemma
Normalized clutching data for bundles over X×S²
Lemma
Uniform Laurent approximation through bundle automorphisms
Lemma
Negative Laurent powers are cleared by Hopf-line stabilization
Lemma
Polynomial clutching families stabilize to linear clutching
Lemma
Linear clutching splits into spectral subbundles
Theorem
Fundamental product theorem for complex K-theory
Definition
Negative-degree complex K-groups
Theorem
Complex Bott periodicity
Corollary
Complex K-theory of spheres
Theorem
Complex K-theory is a two-periodic generalized cohomology theory
Sources
- Hatcher, Vector Bundles & K-Theory, §2.1
- May, A Concise Course in Algebraic Topology, Chapter 24 §1
- Hatcher, Vector Bundles & K-Theory, Proposition 2.1 and following construction
- Hatcher, Vector Bundles & K-Theory, Proposition 2.9
- May, A Concise Course in Algebraic Topology, Chapter 24 §§1–2
- May, A Concise Course in Algebraic Topology, Chapter 24 §2
- Hatcher, Vector Bundles & K-Theory, proof of Proposition 1.11
- MIT 18.906 notes, Lectures 18 and 21
- Hatcher, Vector Bundles & K-Theory, Corollary 2.3 and Example 1.13
- Hatcher, Vector Bundles & K-Theory, proof of Theorem 2.2
- Hatcher, Vector Bundles & K-Theory, Proposition 2.6
- Hatcher, Vector Bundles & K-Theory, Proposition 2.7 and Lemma 2.8
- Hatcher, Vector Bundles & K-Theory, Theorem 2.2
- Hatcher, Vector Bundles & K-Theory, §2.2
- Hatcher, Vector Bundles & K-Theory, Theorem 2.11
- Hatcher, Vector Bundles & K-Theory, Corollary 2.12
- Hatcher, Vector Bundles & K-Theory, §§2.1–2.2