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Simplicial Homology Tyler White MATH 493 Dr. Wanner

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Overview Introduction to Simplicial Homology Prerequisite Definitions Definition of a Simplex Boundary Maps on Simplicies

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Introduction to Simplicial Homology Homology theory was originally developed with simplicies Every cubical set can be represented in terms of simplicies, but there are sets that can be represented in terms of simplicies but are not cubical sets. Cubical Homology works well for a wide range of computational problems, however it is easier to work with simplicies when working in a more abstract setting.

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Prerequisite Definitions Def. A subset K of R d is called convex if, given any two points x,y in K, the line segment [x,y] := {λx + (1 – λ)y | 0 ≤ λ ≤ 1 } [x,y] := {λx + (1 – λ)y | 0 ≤ λ ≤ 1 } joining x to y is contained in K. joining x to y is contained in K. Def. The convex hull conv A of a subset A of R d is the intersection of all closed and convex sets containing A. Theorem: Let V = {v 0, v 1, …, v n } є R d be a finite set. Then conv(V) is the set of those x є R d that can be written as n n n n x = Σ λ i v i, 0 ≤ λ i ≤ 1, Σ λ i = 1 x = Σ λ i v i, 0 ≤ λ i ≤ 1, Σ λ i = 1 i=0 i=0 i=0 i=0

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Def A finite set V = {v 0, v 1, …, v n } in R d is geometrically independent if, for any x є conv(V), the coefficients λ i are unique. Proposition: Let V = {v 0, v 1, …, v n } є R d. Then V is geometrically independent if and only if the set of vectors {v 1 – v 0, v 2 – v 0, …, v n – v 0 } is linearly independent.

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Definition of a Simplex Def: Let V = {v 0, v 1, …, v n } be geometrically independent. The set S = conv(V) is called a simplex or, more specifically, an n-simplex spanned by the vertices v 0, v 1, …, v n. The number n is called the dimension of S. If V’ is a subset of V of k ≤ n vertices, the set S’ = conv(V’) is called a k-face of S Theorem: Any two n-simplices are homeomorphic. Definition: A simplicial complex S is a finite collection of simplices such that 1. every face of a simplex in S is in S, 1. every face of a simplex in S is in S, 2. the intersection of any two simplices in S is a face of each of them. 2. the intersection of any two simplices in S is a face of each of them.

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Def: Given a simplicial complex S in R d, the union of all simplices of S is called the polytope of S and is denoted by |S|. A subset P of R d is a polyhedron if P is the polytope of some simplicial complex S. In this case S is called a triangulation of P

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Boundary Maps Restricting ourselves to Z 2 we can define the boundary maps as: n δ n (S) = Σ conv(V/{v i }) δ n (S) = Σ conv(V/{v i }) i=0 i=0 Proposition: δ n-1 δ n = 0 for all n The simplicial boundary operator with integer coefficients is: n δ k [v 0, v 1, …, v n ] = Σ(-1) i [v 0, v 1, …, v i-1, v i+1, …, v n ] δ k [v 0, v 1, …, v n ] = Σ(-1) i [v 0, v 1, …, v i-1, v i+1, …, v n ] i=0 i=0

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