Alphabeta MathTopological Vector Bundles and Grassmannian Classification
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1 · Prerequisites
Abstract Algebra
Algebraic Topology
Linear Algebra
Real Analysis
- 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
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- 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
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Real and complex topological vector bundles
Lemma
AC supplies the dependent-choice instances used in vector-bundle constructions
Theorem
Vector bundles are glued from transition cocycles
Definition
Bundle maps, sections, subbundles, and isomorphisms
Definition
Pullback vector bundles and sections
Proposition
Vector-bundle pullback is canonically functorial
Definition
Whitney sum, tensor, dual, Hom, and exterior-power bundles
Theorem
Numerable vector bundles admit bundle metrics
Corollary
Short exact sequences of numerable vector bundles split
Theorem
Finite-rank complement theorem over compact Hausdorff bases
Theorem
Homotopy invariance of vector-bundle pullback
Definition
Frame bundles and associated vector bundles
Definition
Oriented real bundles and oriented frame bundles
Proposition
Orientation is equivalent to an SO(n)-reduction
Definition
Stiefel spaces, Grassmannians, and tautological bundles
Definition
Oriented Grassmannians and the tautological oriented bundle
Theorem
Stable Stiefel space is contractible
Lemma
A bundle embedding produces its Grassmannian classifying map
Lemma
Homotopic Grassmannian maps classify isomorphic bundles and conversely
Theorem
Real and complex vector bundles are classified by stable Grassmannians
Theorem
Oriented real vector bundles are classified by BSO
Definition
Schubert cells in real and complex Grassmannians
Theorem
Schubert cells give the stable Grassmannian CW structure
Definition
Clutching construction for bundles over a suspension
Theorem
Clutching classifies vector bundles over spheres in the stable range
Theorem
Oriented clutching classifies oriented bundles over spheres
Sources
- Hatcher, Vector Bundles & K-Theory, §1.1
- Milnor and Stasheff, Characteristic Classes, §2
- Thomas J. Jech, The Axiom of Choice, §2.4.1
- Milnor and Stasheff, Characteristic Classes, §§2–3
- Milnor and Stasheff, Characteristic Classes, §3
- Hatcher, Vector Bundles & K-Theory, Proposition 1.2
- MIT 18.906 notes, Lecture 16
- Hatcher, Vector Bundles & K-Theory, Proposition 1.4
- MIT 18.906 notes, Lecture 20
- Hatcher, Vector Bundles & K-Theory, Theorem 1.6 and Proposition 1.7
- MIT 18.906 notes, Lecture 17
- MIT 18.906 notes, Lectures 16 and 18
- Hatcher, Vector Bundles & K-Theory, §§1.1–1.2
- MIT 18.906 notes, Lecture 18
- Hatcher, Vector Bundles & K-Theory, §1.2
- Milnor and Stasheff, Characteristic Classes, §5
- MIT 18.906 notes, Lecture 21
- Franklin and Thomas, A Survey of k-omega Spaces
- Hatcher, Vector Bundles & K-Theory, proof of Theorem 1.16
- Hatcher, Vector Bundles & K-Theory, Theorem 1.16
- MIT 18.906 notes, Lectures 19–21
- Hatcher, Vector Bundles & K-Theory, Proposition 1.17
- Milnor and Stasheff, Characteristic Classes, §6
- Hatcher, Vector Bundles & K-Theory, Propositions 1.11 and 1.14
- MIT 18.906 notes, Lectures 17 and 21
- Hatcher, Vector Bundles & K-Theory, Proposition 1.14