Alphabeta MathBocksteins Steenrod Squares and Cohomology Operations
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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
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- 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
- 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
Stable natural cohomology operation
Definition
Bockstein connecting operation
Lemma
The Bockstein is independent of lift and representative
Proposition
Bocksteins are natural and stable
Proposition
The mod-two Bockstein is a derivation
Lemma
Natural higher diagonal approximations
Definition
Higher cup-i products
Theorem
Cup-i coboundary identity
Definition
Steenrod squares from cup-i
Theorem
Steenrod squares are well-defined and natural
Proposition
Steenrod normalization, instability, suspension, and top square
Lemma
Cartan coherence for higher diagonals
Theorem
Cartan formula for Steenrod squares
Lemma
Free cyclic resolution, group cohomology, and cochain transfer
Lemma
Equivariant p-fold external power and diagonal decomposition
Lemma
Wreath double-power comparison and coefficient transposition
Lemma
Finite-cellular cyclic squares, Cartan formula, and cyclic-basis action
Lemma
Adem double-power comparison
Lemma
Finite-cellular cyclic squares agree with singular cup-i squares
Lemma
Natural singular-cohomology identities are detected on finite regular complexes
Theorem
Adem relations for Steenrod squares
Lemma
Bockstein parity recurrence for Steenrod squares
Proposition
Sq¹ is the mod-two Bockstein
Definition
Total Steenrod square
Definition
Wu classes of a closed manifold
Lemma
Cyclic p-fold power construction
Definition
Mod-p reduced power operations
Theorem
Reduced powers satisfy naturality, instability, Cartan, and Adem relations
Lemma
Mod-two cohomology ring of infinite real projective space
Lemma
Mod-two cohomology rings of complex projective spaces
Sources
- May, A Concise Course in Algebraic Topology
- Hatcher, Algebraic Topology
- Mosher and Tangora, Cohomology Operations and Applications
- Medina-Mardones, New formulas for cup-i products and fast computation of Steenrod squares
- N. E. Steenrod and D. B. A. Epstein, Cohomology Operations
- Allen Hatcher, Algebraic Topology
- Mosher and Tangora, Cohomology Operations and Applications in Homotopy Theory
- Husemöller, Joachim, Jurčo, and Schottenloher, Basic Bundle Theory and K-Cohomology Invariants