Alphabeta MathGroup Homology Transfer and Low-Degree Exact Sequences
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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 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
Lemma
Diagonal bar coinvariants compute group homology
Definition
Finite-index transfer on normalized bar chains
Lemma
Bar transfer is a chain map independent of the transversal
Lemma
Corestriction after transfer multiplies by the index
Theorem
Positive integral homology is annihilated by the group order
Lemma
First integral homology and conjugation coinvariants
Lemma
Generator differences form a basis of the free-group augmentation ideal
Lemma
The low-degree filtration sequence of a first-quadrant bicomplex
Definition
The free-presentation Lyndon bar bicomplex
Lemma
Free-presentation total homology and its degree-one edges
Theorem
The free-presentation homology five-term sequence
Definition
Crossed homomorphisms and first cohomology
Definition
Degree-one restriction, inflation and quotient action
Lemma
Degree-one maps and the quotient action are well-defined
Theorem
Degree-one inflation–restriction is exact
Lemma
Bar two-cocycles classify abelian-kernel extensions
Lemma
Pullback and coefficient pushout realize bar cohomology maps
Definition
Low-degree transgression for a group extension
Lemma
The kernel of transgression is the image of restriction
Lemma
The kernel of degree-two inflation is the transgression image
Theorem
The inflation–restriction–transgression five-term sequence
Sources
- 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
- Loh, Group Cohomology, Definitions 1.7.12–13 and Theorem 1.7.15 pp.63–64; Theorem 4.1.18 pp.129–132
- Weibel, An Introduction to Homological Algebra, Definition 6.1.2 and Proposition 6.2.6–Corollary 6.2.7, pp.161,169
- Sharifi, Homological Algebra, Lemma 3.5.8, Proposition 3.5.9 and Remark 3.5.11, pp.66–68
- 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