Alphabeta MathSimplicial Subdivision and Simplicial Approximation
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1 · Prerequisites
- Abelian Categories
- 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
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Homotopy and Homotopy Equivalence
- Limits and Colimits
- Metric Spaces
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinals, Cardinals, and Transfinite Recursion
- Preadditive and Additive Categories and Biproducts
- Relations, Functions, and Quotients
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Simplicial Complexes and Simplicial Homology
- Subspaces, Products, and Quotients
- Suprema and Infima
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Universal Properties, Representables and the Yoneda Lemma
2 · Logical flowchart
3 · Items
Definition
Face poset and order complex
Definition
Barycentric subdivision of an abstract simplicial complex
Definition
Canonical barycentric realization map
Lemma
Finite simplicial weak topology agrees with euclidean topology
Lemma
Barycentric face chains triangulate a geometric simplex
Theorem
Barycentric subdivision realizes homeomorphically
Definition
Open and closed stars in a subdivision
Lemma
Compact subsets of an arbitrary simplicial realization meet finitely many open simplices
Lemma
An augmented simplicial cone has an explicit chain contraction
Lemma
Simplicial chain maps carried by specified cones are chain homotopic
Definition
Oriented simplicial subdivision operator
Lemma
Oriented simplicial subdivision commutes with boundary
Lemma
Last vertex map is carried by original simplices
Theorem
Simplicial subdivision is a chain map and homology isomorphism
Lemma
Mesh of iterated simplicial barycentric subdivision tends to zero
Lemma
The open star criterion produces a simplicial map
Theorem
Finite simplicial approximation for maps of pairs
Definition
Relative derived subdivision of a finite simplicial pair
Lemma
Relative derived subdivision makes the fixed subcomplex full
Lemma
Relative subdivision neighbourhood adjustment
Theorem
Relative simplicial approximation after subdivision
Definition
Finite convex cell complex and linear subdivision
Lemma
Intersections of finite linear complexes form a convex cell complex
Lemma
Finite convex cell complexes admit compatible triangulations
Lemma