Alphabeta MathLeray–Hirsch, the Thom Isomorphism, and Gysin Sequences
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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
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
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
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- Relative Homology Excision and Mayer Vietoris
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
- The Serre Spectral Sequence and Applications
- Topological Vector Bundles and Grassmannian Classification
Category Theory
- Abelian Categories
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- 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
Homological Algebra
Linear Algebra
Number Theory
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
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
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- Order, Zorn's Lemma, and the Axiom of Choice
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Topology
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- 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
Lemma
A global fiber basis trivializes Serre monodromy
Lemma
The Leray–Hirsch associated-graded isomorphism lifts without extension ambiguity
Theorem
Leray–Hirsch module isomorphism
Definition
Disk, sphere, and Thom spaces of a metric vector bundle
Proposition
Thom spaces of zero and trivial bundles
Lemma
Disk-pair cohomology over an arbitrary commutative ring
Definition
R-oriented vector bundle and orientation local system
Definition
Thom class by fiberwise normalization
Theorem
Thom isomorphism for a trivial oriented bundle
Lemma
Thom isomorphisms glue over two trivializing opens
Lemma
Thom isomorphism extends over a finite numerable trivializing cover
Lemma
General Thom isomorphism from the relative Serre spectral sequence
Theorem
Thom isomorphism for oriented vector bundles
Theorem
Naturality and uniqueness of Thom classes
Theorem
External-product and Whitney-sum formulas for Thom classes
Definition
Thom diagonal and zero-section collapse
Definition
Thom-defined Euler class of an oriented vector bundle
Definition
Gysin pushforward for an oriented zero section
Theorem
Gysin long exact sequence of an oriented sphere bundle
Proposition
Thom and Gysin constructions respect pullback and composition
Sources
- Miller, MIT 18.906 notes, proof of Theorem 33.5
- Miller, MIT 18.906 notes, Theorem 33.5
- Hatcher, Vector Bundles and K-Theory
- May, A Concise Course in Algebraic Topology, Chapter 23 §5
- Miller, MIT 18.906 notes, Lectures 34–35
- Miller, MIT 18.906 notes, Proposition 35.2
- Miller, MIT 18.906 notes, Lectures 26 and 35