Alphabeta MathGeneralized Cohomology and the Atiyah Hirzebruch Spectral Sequence
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Algebraic Topology
- Bocksteins Steenrod Squares and Cohomology Operations
- Complex Topological K Theory and Bott Periodicity
- 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
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- 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
- Spectra and Stable Homotopy Groups
- Topological Vector Bundles and Grassmannian Classification
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
Complex Analysis
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Simply Connected Plane Domains: the Grand Equivalence
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Projective and Injective Resolutions
- Spectral Sequences
- Universal Coefficients and Kunneth Theorems
Linear Algebra
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- 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
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- The Fundamental Theorems of Calculus
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- 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
- Applications of the Fundamental Group
- 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
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Reduced generalized cohomology theory
Definition
Reduced generalized homology theory
Definition
Coefficient groups of a generalized cohomology theory
Definition
Coefficient groups of a generalized homology theory
Proposition
Reduced and unreduced generalized cohomology theories correspond
Proposition
Degree-d sphere maps act by multiplication by d in any generalized theory
Definition
Skeletal filtration for generalized cohomology
Lemma
The AHSS E-one page is cellular cochains with theory coefficients
Lemma
The AHSS first differential is the cellular coboundary
Theorem
Cohomological Atiyah–Hirzebruch spectral sequence
Lemma
Homological AHSS exact couple from the skeletal filtration
Theorem
Homological Atiyah–Hirzebruch spectral sequence
Lemma
Edge maps of a bounded skeletal AHSS
Theorem
Naturality and edge maps of the AHSS
Lemma
Pairings of skeletal exact couples induce multiplicative AHSS
Theorem
Multiplicative AHSS for a multiplicative generalized theory
Proposition
AHSS collapse generally determines only the associated graded object
Corollary
Complex K-theory AHSS
Lemma
The first connective complex K-theory Postnikov invariant is integral Sq-three
Lemma
KU representability and the skeletal–Postnikov d-three comparison
Theorem
The first possible complex K-theory AHSS differential is integral Sq-three
Lemma
Reduction of the integral Bockstein is the first Steenrod square
Sources
- Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, §2, printed pp. 3–4
- James Davis and Paul Kirk, Lecture Notes in Algebraic Topology, §8.8, printed pp. 227–233
- James Davis and Paul Kirk, Lecture Notes in Algebraic Topology, Definition 8.27, printed pp. 227–229
- Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, §2, printed p. 3
- James Davis and Paul Kirk, Lecture Notes in Algebraic Topology, Definition 8.28, printed pp. 229–230
- Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, Lemma 2.3 and Remark 2.2, printed pp. 4–5
- Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, §3, printed pp. 4–7
- Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, §3, printed pp. 4–6
- Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, Theorem 3.2 and its diagram, printed pp. 5–6
- Yiannis Loizides, The Atiyah–Hirzebruch Spectral Sequence, §3, Theorems 3.2 and 3.4, printed pp. 4–8
- Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, §§3.1–3.2, printed pp. 10–12
- Davis–Kirk, Lecture Notes in Algebraic Topology, §8.8 and Theorem 9.6, printed pp. 227–246
- Davis–Kirk, Lecture Notes in Algebraic Topology, Theorem 9.6, printed pp. 242–243
- Haynes Miller, MIT 18.906 Algebraic Topology II, Lecture 26, printed pp. 89–92
- Davis–Kirk, Lecture Notes in Algebraic Topology, §9.1, printed pp. 237–246
- Daniel Dugger, Multiplicative structures on homotopy spectral sequences II, §3.1, §3.3 and Theorem 3.4, printed pp. 4–5
- Haynes Miller, MIT 18.906 Algebraic Topology II, Lecture 29, printed pp. 100–101
- Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, §1.3, printed p. 4
- Daniel Dugger, Multiplicative structures on homotopy spectral sequences II, §3.1 and Theorem 3.4, printed pp. 4–5
- Romyar Sharifi, Homological Algebra, §4.1, printed pp. 87–90
- Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, §3.1, printed pp. 10–11
- J. F. Adams, Stable Homotopy and Generalised Homology, Proposition 16.6 and its proof, printed pp. 391–393
- J. F. Adams, Stable Homotopy and Generalised Homology, Chapter 2, printed pp. 174–179; Chapter 6(v), printed pp. 245–246; Chapter 7, printed pp. 257–260; Proposition 16.6, printed pp. 391–393
- J. P. May, A Concise Course in Algebraic Topology, Chapter 22 §2, printed pp. 175–179; Chapter 24 §§1–2, printed pp. 204–208
- Caleb Ji, The Atiyah–Hirzebruch Spectral Sequence, Proposition 3.12, printed p. 12
- J. F. Adams, Stable Homotopy and Generalised Homology, Proposition 16.6, printed pp. 391–393
- Allen Hatcher, Algebraic Topology, §3.E, printed pp. 303–305