Alphabeta MathGroup Homology Transfer and Low-Degree Exact Sequences: Examples
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1 · Prerequisites
- Abelian Categories
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Derived Functors
- Exactness and the Member Calculus
- Ext and Balanced Resolutions
- Finite Counting, Factorials and Binomial Coefficients
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Cohomology as a Derived Functor
- Group Extensions Complements and Schur Zassenhaus
- Group Homology Transfer and Low-Degree Exact Sequences
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Limits and Colimits
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Preadditive and Additive Categories and Biproducts
- Projective and Injective Resolutions
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Semidirect Products, Automorphism Groups and Split Extensions
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
- The ZFC Axioms and the Basic Set Constructions
- Universal Properties, Representables and the Yoneda Lemma
2 · Logical flowchart
3 · Items
Example
Integral homology of a finite cyclic group
Example
A cyclic presentation and its five-term sequence
Example
The Heisenberg transgression and its sign
Sources
- Weibel, An Introduction to Homological Algebra, Calculation 6.2.1, Theorem 6.2.2 and Example 6.2.3, pp.167–168
- Loh, Group Cohomology, Definitions 1.7.12–13 and Theorem 1.7.15 pp.63–64; Theorem 3.2.18 pp.129–132
- Weibel, An Introduction to Homological Algebra, Chapter 6, Sections 6.4–6.8
- Dekimpe–Hartl–Wauters, A seven-term exact sequence for the cohomology of a group extension, Sections 2–5 pp.2–11 and Section 10.2 p.21