1. Tyler White MATH 493 Dr. Wanner
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 Rd is called convex if, given any two points x,y in K, the line segment
[x,y] := {λx + (1 – λ)y | 0 ≤ λ ≤ 1 }
joining x to y is contained in K.
Def. The convex hull conv A of a subset A of Rd is the intersection of all closed and convex sets containing A.
Theorem: Let V = {v0, v1, …, vn} є Rd be a finite set. Then conv(V) is the set of those x є Rd that can be written as
n n
x = Σ λivi, 0 ≤ λi ≤ 1, Σ λi = 1
i= i=0

5. Def A finite set V = {v0, v1, …, vn } in Rd is geometrically independent if, for any x є conv(V), the coefficients λi are unique.
Proposition: Let V = {v0, v1, …, vn } є Rd. Then V is geometrically independent if and only if the set of vectors {v1 – v0, v2 – v0, …, vn – v0} is linearly independent.

6. Definition of a Simplex
Def: Let V = {v0, v1, …, vn} be geometrically independent. The set S = conv(V) is called a simplex or, more specifically, an n-simplex spanned by the vertices v0, v1, …, vn. 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,
2. the intersection of any two simplices in S is a face of each of them.

7. Def: Given a simplicial complex S in Rd, the union of all simplices of S is called the polytope of S and is denoted by |S|. A subset P of Rd 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 Z2 we can define the boundary maps as: n
δn(S) = Σ conv(V/{vi})
i=0
Proposition: δn-1δn = 0 for all n
The simplicial boundary operator with integer coefficients is: n
δk[v0, v1, …, vn] = Σ(-1)i[v0, v1, …, vi-1, vi+1, …, vn]