Simplicial Homology

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

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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 Rd is called convex if, given
  any two points x,y in K, the line segment
• x,y ?x (1 ?)y 0 ? 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 ? Rd be a
  finite set. Then conv(V) is the set of those x ?
  Rd that can be written as
• n
  n
• x S ?ivi, 0 ?i 1, S ?i 1
• i0
  i0

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• Def A finite set V v0, v1, , vn in Rd is
  geometrically independent if, for any x ?
  conv(V), the coefficients ?i are unique.
• Proposition Let V v0, v1, , vn ? Rd.
  Then V is geometrically independent if and only
  if the set of vectors v1 v0, v2 v0, , vn
  v0 is linearly independent.

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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.

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• 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
• dn(S) S conv(V/vi)
• i0
• Proposition dn-1dn 0 for all n
• The simplicial boundary operator with integer
  coefficients is n
• dkv0, v1, , vn S(-1)iv0, v1, ,
  vi-1, vi1, , vn
• i0