Alphabeta MathCup Cap Cross Products and Cohomology Rings
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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
- 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
Algebraic Topology
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
Homological Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- 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
- Topology of ℝ
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
2 · Logical flowchart
3 · Items
Definition
Alexander–Whitney map and diagonal approximation
Theorem
Alexander--Whitney and shuffle are natural chain-homotopy inverses
Definition
Singular cup product on cochains
Theorem
Cup product Leibniz identity
Definition
Singular cohomology ring
Proposition
Cup product is natural, unital and associative
Lemma
Factor reversal gives the commutativity chain homotopy
Theorem
Singular cohomology is graded commutative
Definition
Relative cup product for an excisive triad
Proposition
Relative cup products are natural and connector-compatible
Definition
Cap product with cohomology written first
Theorem
Cap product boundary identity
Proposition
Cap naturality and projection formula
Definition
Relative cap products with quotient domains displayed
Lemma
Relative singular product comparison for CW pairs
Lemma
Relative cohomological Kunneth under finite free homology hypotheses
Theorem
Cohomological Kunneth cross product is a ring isomorphism
Lemma
Local coordinate cup products generate top relative cohomology
Definition
Cup length over a coefficient ring
Proposition
Positive-degree cup products on a suspension vanish
Lemma
Integral surface cup pairing from the oriented polygon
Sources
- Hatcher, Algebraic Topology, section 3.2, pages 206–207
- Hatcher §3.2; Miller Lemma 33.2
- Hatcher §3.2, Cup Product; Miller Construction 28.1
- Hatcher Lemma 3.6; Miller Lecture 28
- Hatcher §3.2, The Cohomology Ring; Miller Definition 29.1
- Hatcher Proposition 3.10; Miller Proposition 28.3
- Hatcher proof of Theorem 3.11; Miller Lecture 29
- Hatcher Theorem 3.11; Miller Lecture 29
- Hatcher relative products after Proposition 3.10; Miller Lecture 34
- Hatcher relative cup products; Miller Lecture 34
- Hatcher §3.3, Cap Product; Miller Lecture 34
- Hatcher cap boundary formula p.239; Miller Lecture 34
- Hatcher cap formula (20); Miller Proposition 34.1
- Hatcher relative cap products pp.239--241; Miller Lecture 34
- Hatcher, Algebraic Topology, §3.B, printed p.276 relative quotient comparison and pp.277–278 shuffle; local proof gives the singular chain/cochain comparison
- Hatcher, §3.B p.276 relative quotient; §3.2 pp.216–219 cohomological comparison. Local algebra proves the exact finite-free homology version.
- Hatcher Theorems 3.15 and 3.18; Miller Proposition 29.2 and Theorem 33.3
- Hatcher, Algebraic Topology, proof of Theorem 3.19, printed p.221, local coordinate/cube product; detailed comparison supplied locally
- Hatcher §3.2 ring examples; Miller Lectures 28--29
- Hatcher relative cup method §3.2; Miller Lectures 28--29
- Hatcher, Example 3.7, pp207–208; complete signed singular fan calculation