# Topology/Simplicial complexes

 Topology ← Induced homomorphism Simplicial complexes Barycentric coordinates →

A simplicial complex is a union of spaces known as simplicies, that are convex hulls of points in general position. In Euclidean space they can be thought of as a generalisation of the triangle. For the first few dimensions they are: the point, the line segment, the triangle and the tetrahedron.

## Definition of a simplex

Given n+1 points ${\displaystyle \{a_{0},a_{1},\dots ,a_{n}\}}$, in a space with dimension of at least n, such that no three points are colinear, the set

${\displaystyle {\mathcal {S}}=\{t_{0}a_{0}+t_{1}a_{1}+\dots +t_{n}a_{n}:\sum _{i=1}^{n}t_{i}=1{\text{ with }}t_{i}\in [0,1]\}}$

is a simplex, and more specifically an n-simplex since it has n+1 verticies.

## The standard n-simplex

For ease we will sometimes need a standardised co-ordinate system for a simplex. The standard n-simplex, a subset of Euclidean (n+1)-space, ${\displaystyle \mathbb {R} ^{n+1}}$, is

${\displaystyle \Delta _{n}=\{(t_{0},t_{1},\dots ,t_{n})\in \mathbb {R} ^{n+1}:t_{i}\geq 0{\text{ and }}\sum _{i=0}^{n}t_{i}=1\}}$

Note that the set is expressed in a very similar manner, but the co-ordinates are fixed in ${\displaystyle \mathbb {R} ^{n+1}}$.

A diagram of the standard n-simplex in R3, with n=2

## Face of a Simplex

A face of an n-simplex ${\displaystyle {\mathcal {S}}}$ is any (n-1)-simplex formed by an n-1 element subset of the verticies of ${\displaystyle {\mathcal {S}}}$.

## Defintion of a simplicial complex

A simplicial complex is a union of simplicies such that the intersection of any two simplicies is a simplex. Alternatively if ${\displaystyle {\mathcal {C}}}$ is a simplicial complex then

1. Every face of a simplex in ${\displaystyle {\mathcal {C}}}$ is a simplex in ${\displaystyle {\mathcal {C}}}$.

2. The intersection of two simplicies ${\displaystyle A,B\in {\mathcal {C}}}$ is a face of both of A and B.

## Example

The triangulation of a polygon in the plane is a simplcial complex. In fact it hints at the existence of simplicial complexes existing for all polytopes, e.g. every polyhedron can be expressed as tetrahedrons meeting full face to full face.

## Exercises

(under construction)

 Topology ← Induced homomorphism Simplicial complexes Barycentric coordinates →