Alphabeta MathSingular Cohomology and Coefficient Theorems
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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
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
Definition
Singular cochain complex with coefficients
Lemma
The singular coboundary squares to zero
Definition
Singular cohomology with coefficients
Proposition
Singular cohomology is contravariantly functorial
Theorem
Homotopic maps induce equal maps in singular cohomology
Definition
Relative singular cochain complex
Theorem
Long exact sequence of a pair in singular cohomology
Theorem
Naturality of the singular cohomology pair sequence
Theorem
Excision for singular cohomology
Theorem
Mayer vietoris sequence in singular cohomology
Definition
Kronecker evaluation pairing
Lemma
The kronecker pairing is independent of cocycle and cycle representatives
Lemma
Singular UCT extension from cycle projections
Theorem
Topological universal coefficient short exact sequence for cohomology
Proposition
The cohomology universal coefficient sequence splits nonnaturally
Corollary
Cohomology over a field is dual to homology over that field
Corollary
Integral cohomology detects adjacent homology torsion
Lemma
Singular product chain equivalence by simplex models
Theorem
Topological Kunneth short exact sequence for homology
Proposition
The homology Kunneth sequence splits nonnaturally
Corollary
Field Kunneth isomorphism for homology of products
Definition
Additive singular cohomology cross product
Lemma
The additive singular cohomology cross product is well-defined
Theorem
Cohomological Kunneth isomorphism under finite free hypotheses
Corollary
Singular cohomology satisfies the Eilenberg Steenrod cohomology axioms
Lemma
Real projective space cellular homology and the pinch map
Sources
- Hatcher, section 3.1, Cohomology of Spaces, printed pages 197–198
- Hatcher, section 3.1, printed pages 197–198
- Hatcher, section 3.1, printed page 198
- Hatcher, section 3.1, printed pages 198–200
- Hatcher, section 3.1, Homotopy Invariance, printed page 201
- Hatcher, section 3.1, Relative Groups, printed pages 199–200
- Hatcher, section 3.1, Relative Groups and the Long Exact Sequence of a Pair, printed pages 199–200
- Hatcher, section 3.1, Induced Homomorphisms, printed page 201
- Hatcher, section 3.1, Excision, printed pages 201–202
- Hatcher, section 3.1, Mayer–Vietoris Sequences, printed pages 203–204
- Hatcher, section 3.1, printed pages 198 and 200
- Hatcher, section 3.1, evaluation and its naturality, printed pages 191 and 198–201
- Miller, section 27, Theorem 27.1 and proof, printed pages 73–74
- Miller, Theorem 27.1 and proof, printed pages 73–74
- Miller, Theorem 27.1, printed pages 73–74
- Miller, section 27, Remark 27.3 and Theorem 27.1, printed pages 73–74
- Miller, section 27, Example 27.4 and Theorem 27.1, printed pages 73–74
- Miller, section 25, Lemma 25.10 through Theorem 25.13, printed pages 64–66
- Miller, section 25, Theorem 25.15, printed page 66
- Miller, Theorem 25.15, printed page 66
- Miller, Example 25.16, printed pages 66–67
- Miller, Construction 28.1, printed page 76; positive coboundary convention translated locally
- Miller, Construction 28.1, printed page 76; positive differential convention verified locally
- Hatcher, Theorem 3.15, printed pages 215–218; different CW hypothesis, local proof establishes the stated finite-free homology version
- Hatcher, Cohomology Axioms, printed page 202; Miller section27 printed page75
- Miller, section 17, printed pages 42–44; pinch map checked locally