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Simplicial Homology Tyler White MATH 493 Dr. Wanner.

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Presentation on theme: "Simplicial Homology Tyler White MATH 493 Dr. Wanner."— Presentation transcript:

1 Simplicial Homology Tyler White MATH 493 Dr. Wanner

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

3 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.

4 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

5  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.

6 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.

7  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

8 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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