Alphabeta MathSingular Cohomology and Coefficient Theorems — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- 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 over a Principal Ideal Domain and the Canonical Forms
- 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
- Cw Complexes and Cellular Homology
- Homology Axioms Degree and Classical Applications
- 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
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
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Functors
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Universal Coefficients and Kunneth Theorems
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
Example
Integral cohomology of real projective space from UCT
Example
Cohomology of lens spaces from UCT
Example
Field cohomology of an infinite wedge of circles
Example
Homology of a product of spheres by Kunneth
Example
Tor term in the homology of a product of real projective spaces
Example
Kronecker pairing for a cellular circle generator
Counterexample
The integral Kronecker map need not be an isomorphism
Counterexample
The UCT splitting is not natural
Sources
- Miller, section17 and section27, printed pages43 and73–74
- Hatcher, Example 2.43, complete finite lens-space construction and boundaries, printed pages144–146
- Hatcher, singular cohomology and unrestricted cochains, printed pages197–202; explicit CW and dual calculation
- Miller, Theorem25.15 and product calculations, printed pages66–67
- Miller, section 25, product Kunneth calculations
- Hatcher, section 3.1, evaluation and universal coefficients
- Hatcher, section 3.1, universal coefficients and real projective space
- Hatcher, section 3.1, nonnaturality of universal coefficient splittings