Alphabeta MathHomology Axioms Degree and Classical Applications
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1 · Prerequisites
- Abelian Categories
- 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
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cw Complexes and Cellular Homology
- 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
- Homotopy and Homotopy Equivalence
- Limits and Colimits
- 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
- Preadditive and Additive Categories and Biproducts
- 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
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Subspaces, Products, and Quotients
- Suprema and Infima
- Tensor Products of Modules
- The Diagram Lemmas in an Abelian Category
- 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
Definition
Unreduced homology theory on cw pairs
Theorem
Singular homology satisfies homotopy exactness and excision
Theorem
Singular homology satisfies dimension and arbitrary additivity
Definition
Reduced homology theory and augmentation
Proposition
Unreduced pair and reduced quotient axioms are equivalent on cw pairs
Definition
Coefficient normalized morphism of ordinary homology theories
Lemma
Any ordinary homology theory computes relative cell groups from its coefficient group
Lemma
Any ordinary homology theory has a cellular chain complex on a cw pair
Lemma
Finite dimensional skeletal exactness computes axiomatic homology
Lemma
Oriented simplex comparison for an ordinary homology theory
Lemma
Finite simplicial approximation for homology comparison
Lemma
Subdivision compatible continuous polyhedral homology comparison
Lemma
Cw homotopy equivalence inclusions are strong deformation retracts
Lemma
Finite cw pairs admit finite simplicial homotopy models
Lemma
Coefficient comparison on finite cw pairs
Lemma
Sphere endomorphisms act by the same integer in every ordinary theory
Lemma
Axiomatic cellular boundaries are integral incidence matrices with coefficients
Lemma
Finite dimensional axiomatic homology has finite subcomplex support
Lemma
A comparison isomorphism propagates over one skeleton stage
Theorem
Eilenberg steenrod uniqueness on finite dimensional cw pairs
Proposition
Ordinary homology theories have mayer vietoris for cw covers
Definition
Skeletal mapping telescope of a cw pair
Lemma
The skeletal telescope projects by a homotopy equivalence of pairs
Lemma
A sequential abelian colimit is the cokernel of one minus shift
Lemma
Additivity and compact cell support control the infinite cw colimit
Theorem
Eilenberg steenrod uniqueness on all cw pairs
Definition
Degree of a self map of an oriented sphere
Proposition
Degree is homotopy invariant and multiplicative under composition
Proposition
Suspension preserves sphere map degree
Proposition
Degree of identity constant reflection and antipodal sphere maps
Lemma
A map of nonzero degree between spheres is surjective
Definition
Local degree at an isolated preimage
Lemma
Local sphere orientations and finite puncture excision
Theorem
Global sphere degree is the sum of local degrees
Proposition
Every integer occurs as the degree of a sphere map
Theorem
No retraction from a disk onto its boundary
Lemma
A fixed point free ball map produces a boundary retraction
Theorem
Brouwer fixed point theorem
Theorem
No nowhere zero tangent vector field on an even sphere
Proposition
An odd sphere admits a nowhere zero tangent vector field
Theorem
A sphere has a nowhere zero tangent vector field iff its dimension is odd
Corollary
A fixed point free sphere map has antipodal degree
Corollary
A group acting freely on a positive even sphere has at most two elements
Theorem
Invariance of dimension for euclidean spaces
Sources
- Miller, Algebraic Topology I lecture notes, Definition 11.1, printed pp.25–26
- May, A Concise Course in Algebraic Topology, 14§4, pp.110–111, CW-pair formulation
- Miller, Algebraic Topology I lecture notes, Definition 11.1 and ensuing verification, pp.25–26
- Hatcher, Algebraic Topology, Axioms for Homology, pp.160–162
- Miller, Algebraic Topology I lecture notes, Definition 11.1, pp.25–26
- May, A Concise Course in Algebraic Topology, 14§4, pp.110–111
- Hatcher, Algebraic Topology, Axioms for Homology, pp.160–161
- May, A Concise Course in Algebraic Topology, 14§4, CW definition and theorem, pp.110–111
- May, A Concise Course in Algebraic Topology, 15§2, pp.119–120, natural comparison and boundary compatibility
- Hatcher, Algebraic Topology, Axioms for Homology, p.161, uniqueness with coefficients
- May, A Concise Course in Algebraic Topology, 14§3, suspension corollaries, p.109; 15§2 p.120
- May, A Concise Course in Algebraic Topology, 15§2, definition of C_n and d, p.119
- May, A Concise Course in Algebraic Topology, 15§2, second theorem and full exact-diagram proof, pp.119–120
- May, A Concise Course in Algebraic Topology, 15§2, both cellular comparison theorems, pp.119–120
- Hatcher, Algebraic Topology, §2C, Theorem 2C.1 setting, pp.177–179
- Hatcher, Algebraic Topology, §2C, Lemma 2C.2 and Theorem 2C.1, pp.177–179
- May, A Concise Course in Algebraic Topology, 15§2, exact-diagram naturality and boundary compatibility, pp.119–120
- Hatcher, Algebraic Topology, Theorem 2C.1 and proof, pp.177–179
- Hatcher, Algebraic Topology, Propositions 0.18–0.19 and Corollary 0.20, complete proofs pp.16–17
- Hatcher, Algebraic Topology, Theorem 2C.5, construction and proof pp.182–184
- May, A Concise Course in Algebraic Topology, 15§2, uniqueness theorem pp.119–120
- Hatcher, Algebraic Topology, Theorem 2C.5 pp.182–184; Axioms for Homology p.161
- May, A Concise Course in Algebraic Topology, 15§2, first theorem and coefficient paragraph p.119
- Hatcher, Algebraic Topology, Theorem 4.59 homology argument, printed pp.399–401 (in chapter 4); alternative simplicial route specified in notes
- May, A Concise Course in Algebraic Topology, 15§2, first theorem and arbitrary-coefficient paragraph p.119
- May, A Concise Course in Algebraic Topology, 15§2, cellular calculation pp.119–120
- Hatcher, Algebraic Topology, Lemma 2.34 p.138, finite support reasoning
- May, A Concise Course in Algebraic Topology, 15§2, skeletal exactness pp.119–120
- Miller, Algebraic Topology I lecture notes, Proposition 9.6, opening p.22
- May, A Concise Course in Algebraic Topology, 15§2, pp.119–120
- Hatcher, Algebraic Topology, Axioms for Homology, Mayer–Vietoris derivation p.162
- Miller, Algebraic Topology I lecture notes, Lemma 11.6, pp.27–28
- May, A Concise Course in Algebraic Topology, 14§5, first Mayer–Vietoris theorem pp.112–113
- Hatcher, Algebraic Topology, Lemma 2.34, telescope construction p.138
- May, A Concise Course in Algebraic Topology, 14§6, telescope construction pp.114–116
- Hatcher, Algebraic Topology, Lemma 2.34, complete telescope deformation pp.138–139
- Hatcher, Algebraic Topology, Proposition 0.16 p.15 and Theorem A.6 p.524
- May, A Concise Course in Algebraic Topology, 14§6, algebraic lemma p.114
- Hatcher, Algebraic Topology, Theorem 3F.8 proof pp.314–315
- May, A Concise Course in Algebraic Topology, 14§6, theorem and telescope proof pp.114–116
- Hatcher, Algebraic Topology, Theorem 3F.8 pp.314–315
- May, A Concise Course in Algebraic Topology, 14§6 pp.114–116 and 15§2 pp.119–120
- Hatcher, Algebraic Topology, Axioms for Homology p.161
- Hatcher, Algebraic Topology, Degree, definition p.134
- Hatcher, Algebraic Topology, Degree properties (c),(d), p.134
- Hatcher, Algebraic Topology, Proposition 2.33, p.137
- Hatcher, Algebraic Topology, Degree properties (a),(e),(f) and (b), pp.134–135
- Hatcher, Algebraic Topology, Degree property (b), p.134
- Hatcher, Algebraic Topology, Local degree construction, pp.135–136
- Hatcher, Algebraic Topology, Local-degree diagram and Proposition 2.30 proof, pp.135–136
- Hatcher, Algebraic Topology, Proposition 2.30, p.136
- Hatcher, Algebraic Topology, Example 2.31, p.136
- Miller, Algebraic Topology I lecture notes, Theorem 10.7 proof, p.24
- Miller, Algebraic Topology I lecture notes, Theorem 10.7 and proof, p.24
- Hatcher, Algebraic Topology, Theorem 2.28, p.135
- Hatcher, Algebraic Topology, Theorem 2.28, positive direction p.135
- Hatcher, Algebraic Topology, Degree property (g), p.134
- Hatcher, Algebraic Topology, Proposition 2.29, p.135
- Miller, Algebraic Topology I lecture notes, Corollaries 10.5 and 10.6, pp.23–24