Alphabeta MathLocal Coefficients, Twisted Homology, and Duality
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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
- 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
- Orientations Poincare Lefschetz and Alexander Duality
- 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
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
- 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
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Fundamental groupoid of a space
Proposition
Vertex groups recover the fundamental group
Definition
Local systems and pullback
Theorem
Local systems correspond to group-ring modules
Definition
Right action on universal-cover chains
Definition
Singular and cellular local chain complexes
Lemma
Twisted boundaries square to zero and ignore lift bases
Definition
Homology and cohomology with local coefficients
Proposition
Functoriality with coefficient morphisms
Theorem
Cellular chains compute local homology
Theorem
Pair exact sequences with local coefficients
Theorem
Excision and Mayer–Vietoris with local coefficients
Theorem
Cellular cochains compute cohomology with local coefficients
Definition
Compactly supported cohomology with local coefficients
Proposition
The orientation system is a local system
Definition
Orientation local system on a manifold with boundary
Lemma
Canonical twisted fundamental classes over compact subsets
Definition
Cup and cap products with local coefficients
Theorem
Poincare duality with the orientation local system
Theorem
Poincare–Lefschetz duality with local coefficients
Lemma
Fiber transport gives the Serre local systems
Sources
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §§3–4, pp.103–109
- Hatcher, Algebraic Topology, §3.H, pp.327–334
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §3, pp.103–107
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §§1,3, pp.95–97, 103–107
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §1, pp.95–97
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §§1–3, pp.95–105
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §2.1, pp.98–100
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §4, pp.105–109
- Hatcher, Algebraic Topology, Proposition 2.21 and §3.H, pp.123–124 and 332–334
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §4, pp.107–109
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §§2.1 and 4, pp.98–100, 107–109
- Hatcher, Algebraic Topology, §3.H, pp.334–335
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §2.2, pp.100–103
- Hatcher, Algebraic Topology, Proposition 3.42 and Theorem 3.43, pp.253–254
- Hatcher, Algebraic Topology, Lemma 3.27 and Theorem 3.43, pp.236–238, 253–254
- Hatcher, Algebraic Topology, §3.H, pp.335–336
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §2.2, Theorem 5.7 and stronger form, pp.101–103
- Hatcher, Algebraic Topology, Theorem 3H.6, pp.335–336
- Hatcher, correction to the last two paragraphs of Algebraic Topology p.335
- Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §2.2, Poincare–Lefschetz extension, pp.101–103
- Hatcher, Algebraic Topology, Theorem 3.43, pp.253–254