Alphabeta MathHomology Axioms Degree and Classical Applications — Examples
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1 · Prerequisites
- Abelian Categories
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- 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
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cw Complexes and Cellular Homology
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exactness and the Member Calculus
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Function Space Topologies and the Exponential Law
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Homology Axioms Degree and Classical Applications
- Homotopy and Homotopy Equivalence
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Limits and Colimits
- Limits of Real Functions
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Long Exact Sequences in Homology
- Metric Spaces
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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
- Polynomial Rings, the Division Algorithm and Roots
- Preadditive and Additive Categories and Biproducts
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Relative Homology Excision and Mayer Vietoris
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Splitting Fields
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Diagram Lemmas in an Abelian Category
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Universal Properties, Representables and the Yoneda Lemma
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Example
Degree of the circle power map
Example
Degree of a coordinate reflection on a sphere
Example
Degree of the antipodal map in low dimensions
Example
Local degrees of a polynomial map on the riemann sphere
Example
Two homology theories with different coefficient groups
Counterexample
Degree zero does not imply a sphere map is constant
Counterexample
Degree is not defined by top homology for self maps of s zero
Counterexample
Finite additivity alone does not prove infinite cw uniqueness
Sources
- Hatcher, Algebraic Topology, Example 2.32, p.137
- Hatcher, Algebraic Topology, Degree property (e), p.134
- Hatcher, Algebraic Topology, Degree property (f) and Theorem 2.28, pp.134–135
- Hatcher, Algebraic Topology, Proposition 2.30 and Example 2.32, pp.136–137
- Lebl, Guide to Cultivating Complex Analysis, Theorem 5.1.3 and Exercise 1.3.9
- Hatcher, Algebraic Topology, Axioms for Homology, coefficient discussion p.161
- Miller, Algebraic Topology I lecture notes, Definition 11.1, p.26
- Hatcher, Algebraic Topology, Example 2.31, p.136 (signed fold variation)
- Hatcher, Algebraic Topology, Degree opening paragraph, p.134, excludes n=0
- Hatcher, Algebraic Topology, §2.3 Exercise 2, p.165