Alphabeta MathOrientations Poincare Lefschetz and Alexander Duality
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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 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 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
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
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- 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
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- Metric Spaces
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Topological manifolds with and without boundary
Theorem
Local homology detects manifold dimension, interior, and boundary
Lemma
Coordinate-ball classes identify local homology stalks
Definition
Orientation local system and orientation cover
Definition
R-orientation of a topological manifold
Proposition
Every manifold is F2-orientable and orientability is componentwise
Lemma
Relative homology Mayer–Vietoris for closed supports
Lemma
Compatible orientation classes over compact subsets
Definition
Fundamental class of a compact oriented manifold
Theorem
Top homology of a connected manifold
Definition
Compactly supported singular cohomology
Definition
The cap-duality map of an oriented manifold
Lemma
Cap duality on a Euclidean coordinate ball
Lemma
Cap product and the Mayer–Vietoris duality ladder
Lemma
Cap duality passes to increasing open unions
Lemma
Cap duality for open subsets of Euclidean space
Lemma
Duality extends to finite unions of coordinate balls
Lemma
A manifold exhaustion passes duality to the colimit
Theorem
Poincaré duality for oriented topological manifolds
Lemma
Finite generation from cap with a finite fundamental cycle
Corollary
Poincaré duality gives a nonsingular cup pairing
Definition
Degree of a map between oriented closed manifolds
Proposition
Manifold degree is functorial and detected in top cohomology
Theorem
Compact topological manifold boundaries admit collars
Lemma
A collar constructs the relative orientation class and its boundary class
Definition
Relative fundamental class and boundary orientation
Theorem
Poincaré–Lefschetz duality
Theorem
Fully relative Poincaré–Lefschetz duality
Theorem
Compact locally contractible Euclidean subsets are neighborhood retracts
Theorem
Alexander duality for compact locally contractible subsets of a sphere
Theorem
Jordan–Brouwer separation
Theorem
Invariance of domain
Lemma
A horn replacement block has an injective commutator meridian
Lemma
A controlled nested horn construction embeds a closed three-ball
Definition
Lefschetz number of a finite CW self-map
Lemma
Hopf trace formula
Theorem
Simplicial approximation after sufficient subdivision
Lemma
Finite CW complexes are Euclidean neighborhood retracts
Theorem
Lefschetz fixed-point theorem for finite complexes
Sources
- Hatcher, Algebraic Topology, §3.3
- Hatcher, Algebraic Topology, §3.3, pp.230–231
- Hatcher, Algebraic Topology, local orientation comparisons, pp.233–234
- Hatcher, Algebraic Topology, §3.3, pp.233–235
- Hatcher, Algebraic Topology, §3.3, pp.235–236
- Hatcher, Algebraic Topology, Lemma 3.27 proof, first exact sequence, p.237
- Hatcher, Algebraic Topology, Lemma 3.27, pp.236–238
- Miller, Lectures on Algebraic Topology, Lecture 32
- Hatcher, Algebraic Topology, Theorem 3.26 and Lemma 3.27
- Hatcher, Algebraic Topology, Theorem 3.26 and Proposition 3.29
- Miller, Lectures on Algebraic Topology, Lectures 31–32
- Hatcher, Algebraic Topology, §3.3, pp.242–243
- May, A Concise Course in Algebraic Topology, Chapter 20 §5
- Hatcher, Algebraic Topology, construction preceding Theorem 3.35, p.245
- Hatcher, Algebraic Topology, Example 3.34 p.244 and proof of Theorem 3.35 step (1), p.248
- May, A Concise Course in Algebraic Topology, Chapter 20 §5, Step 1
- Hatcher, Algebraic Topology, Lemma 3.36 and its complete proof, pp.246–247
- May, A Concise Course in Algebraic Topology, Chapter 20 §5, Step 2
- Hatcher, Algebraic Topology, proof of Theorem 3.35, increasing-union step (B), p.248
- Hatcher, Algebraic Topology, proof of Theorem 3.35 step (2), p.248; rational boxes replace bounded convex sets
- Hatcher, Algebraic Topology, proof of Theorem 3.35, step (A) p.247 and steps (2)–(3) p.248
- May, A Concise Course in Algebraic Topology, Chapter 20 §5, Steps 2 and 4
- Hatcher, Algebraic Topology, proof of Theorem 3.35, pp.245–248
- Miller, Lectures on Algebraic Topology, Lecture 36
- Hatcher, Algebraic Topology, Theorem 3.35, pp.241–248
- Miller, Lectures on Algebraic Topology, Lectures 35–36
- Arvind Nair, Topology II (2024), Theorem 1.11.1, printed p.11; statement restricted here to PID coefficients with a direct finite-cap proof
- Hatcher, Algebraic Topology, Poincare duality and finite generation discussion, section 3.3
- Hatcher, Algebraic Topology, pp.249–250
- Miller, Lectures on Algebraic Topology, Theorem 38.8
- Hatcher, Algebraic Topology, section 3.3 Exercise 7, printed p.258
- Hatcher, Algebraic Topology, §3.3, pp.258–259
- Hatcher, Algebraic Topology, Proposition 3.42 and complete proof, printed p.253
- May, A Concise Course in Algebraic Topology, Chapter 21 §4
- Hatcher, Algebraic Topology, Proposition 3.42 and Theorem 3.43 exercise, pp.252–254
- May, A Concise Course in Algebraic Topology, Chapter 21 §4, pp.169–170
- Hatcher, Algebraic Topology, Theorem 3.43, pp.253–254
- May, A Concise Course in Algebraic Topology, relative Poincaré duality, pp.170–171
- Hatcher, Algebraic Topology, Theorem 3.43, pp.254–255
- Miller, Lectures on Algebraic Topology, Theorem 37.1
- Hatcher, Algebraic Topology, Appendix Theorem A.7, pp.8–10
- Hatcher, Algebraic Topology, Theorem 3.44 and Corollary 3.45, pp.254–256
- Miller, Lectures on Algebraic Topology, Theorem 38.4
- Hatcher, Algebraic Topology, Proposition 2B.1, pp.169–171
- J. J. Walton, Algebraic Topology IV, Theorem 4.7.3, pp.95–98
- Hatcher, Algebraic Topology, Theorem 2B.3, pp.172–173
- J. J. Walton, Algebraic Topology IV, Theorem 4.4.7, p.98
- Hatcher, Algebraic Topology, Example 2B.2 pp170–172; explicit marked-coordinate realization and collared-cover proof supplied locally
- Daverman–Venema, Embeddings in Manifolds, §2.1 pp47–51; explicit relative cap transport and finite inverse-control construction supplied locally
- Hatcher, Algebraic Topology, Example 2B.2 pp170–172; marked recursive horn replacement
- Hatcher, Algebraic Topology, §2.C, pp.179–180
- Hatcher, Algebraic Topology, Hopf trace formula, p.180
- Hatcher, Algebraic Topology, Theorem 2C.1, pp.177–179
- E. C. Zeeman, Relative simplicial approximation, pp.39–42
- Hatcher, Algebraic Topology, Appendix Corollary A.10, pp.10–11
- Hatcher, Algebraic Topology, Theorem 2C.3, pp.179–181