Alphabeta MathObstruction Theory, Postnikov Towers, and Classifying Spaces
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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
- 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
- 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
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- 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
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Lemma
Extending over one cell is equivalent to nullhomotoping the attaching sphere
Definition
Homotopy-group local system along a cellular map
Definition
Primary cellular obstruction cochain
Theorem
The primary obstruction cochain is a cocycle
Definition
Difference cochain between two cellular extensions
Theorem
The primary obstruction class is independent of cellular choices
Theorem
Vanishing of the primary obstruction is equivalent to extension over the next skeleton
Theorem
Difference cochains classify homotopies of extensions in the stable stage
Theorem
Obstruction theory for lifting through a fibration
Definition
Eilenberg--Mac Lane space
Theorem
Existence and homotopy uniqueness of Eilenberg--Mac Lane spaces
Theorem
Eilenberg--Mac Lane spaces represent singular cohomology
Corollary
Cohomology operations are universal classes on Eilenberg--Mac Lane spaces
Definition
Postnikov section and Postnikov tower
Theorem
Postnikov towers exist for connected CW complexes
Definition
Postnikov k-invariant
Theorem
Simple Postnikov stages are classified by k-invariants
Definition
Universal principal bundles and classifying spaces
Definition
Milnor's infinite-join model of EG
Lemma
Finite join models for the circle and the two-point group
Theorem
Milnor's join model is a contractible free G-space
Theorem
Numerable principal bundles are classified by maps to BG
Proposition
The based loop space of BG recovers G weakly
Corollary
The classifying space of a discrete group is a K(G,1)
Sources
- James Davis and Paul Kirk, Lecture Notes in Algebraic Topology
- Haynes Miller, MIT 18.906 Algebraic Topology II lecture notes
- J. P. May and Kate Ponto, More Concise Algebraic Topology
- Rolf Schon, Fibrations Over a CWh-Base
- Dale Husemoller, Fibre Bundles, Third Edition
- John Milnor, Construction of Universal Bundles II
- Tammo tom Dieck, Algebraic Topology
- Stanley P. Franklin and Barbara V. Smith Thomas, A Survey of k-omega Spaces