The first step towards the definition of simplicial homology is to find a good notion of ‘triangulation’ of a topological space.
The main idea is to construct a model of the topological space from little building blocks called ‘simplices’, like points, edges, triangles etc.
To give a mathematical defintion of simplices, we need to introduce the geometric concepts of affine and convex hull.

M.II.1 Affine- and convex hulls

Affine Combination：Consider $p+1$ points $u_0,u_1,..., u_p$ in $\mathbb{R^n}$ . A point $\sum^p_{i=0}\lambda_i u_i \text{ where }\sum^p_{i=0}\lambda_i=1$
with $\lambda_0,...,\lambda_p \in \mathbb{R^n}$ is called an affine combination of the points $u_0,..., u_p$ .

Affine Hull: The affine hull of the points $u_0,...,u_p$ is the subset of $\mathbb{R^n}$ consisting of all affine combinations of the given points $u_0,...,u_p$ .
$aff\{u_0,...,u_p\}:=\{x=\sum\lambda_i u_i | \lambda_i \in \mathbb{R}\ and\ \sum \lambda_i=1\}$

The points $u_0,...,u_p$ are called affinely independent if for any $v_0,...,v_p \in \mathbb{R}$ the following holds
$\left(\sum_{i=0}^{p} \nu_{i} u_{i}=0\ \&\ \sum_{i=0}^{p} \nu_{i}=0\right) \Longrightarrow \nu_{0}=\cdots=\nu_{p}=0$
We can have at most $n+1$ affinely independent points in $\mathbb{R^n}$ because there are at most $n$ linearly independent vectors in $\mathbb{R^n}$ .
The affine subspace $aff\{u_0,...,u_p\}$ has dimension $p$ (and is then called a $p-$ plane) if and only if: the $p$ vectors $u_1-u_0,...,u_p-u_0$ are linearly independent in $\mathbb{R^n}$ .
$\begin{array}{l} \Longleftrightarrow \sum_{i=1}^{p} \mu_{i}\left(u_{i}-u_{0}\right)=0 \text { iff } \mu_{1}=\cdots=\mu_{p}=0 \\ \Longleftrightarrow\left(\sum_{i=0}^{p} \nu_{i} u_{i}=0\ \&\ \sum_{i=0}^{p} \nu_{i}=0\right) \text { iff } \nu_{0}=\cdots=\nu_{p}=0 \end{array}$

To see what the affine hull means geometrically, we write
$\begin{aligned} x &=\sum_{i=0}^{p} \lambda_{i} u_{i} \\ &=\lambda_{0} u_{0}+\lambda_{1} u_{1}+\cdots+\lambda_{p} u_{p} \\ &=u_{0}+\lambda_{1}\left(u_{1}-u_{0}\right)+\cdots+\lambda_{p}\left(u_{p}-u_{0}\right) \end{aligned}$

(use that $\lambda_0=1-\lambda_1- ... - \lambda_p$ since $\sum \lambda_i=1$ . )

image.png

Convex Combination: An affine combination $x=\sum^p_{i=0}\lambda_i u_i$ with $\sum^p_{i=0} \lambda_i=1$ if $\lambda_i>0$ for all $0\leq i \leq p$ .

Convex Hull: The convex hull of the points $u_0,...,u_p$ is the subset of $\mathbb{R^n}$ consisting of all convex combinations of the given points.
$conv\{u_0,...,u_p\}:=\left\{x=\sum \lambda_{i} u_{i} \mid \lambda_{i} \in \mathbb{R} \text { and } \sum \lambda_{i}=1 \text { and } \lambda_{i} \geq 0\right\}$

Convex:A subset $A$ of $\mathbb{R^n}$ is said to be convex if for any two points $x$ and $y$ in $A$ the line segment $[x,y]$ joining them is contained in $A$ .

Exercise 7: Prove that the convex hull of any given $p+1$ points in $\mathbb{R^n}$ is a convex subset of $\mathbb{R^n}$ .
:
$[xy]=\left\{Q_{t}=x+t \cdot(y-x) \mid 0 \leqslant t \leqslant 1\right\}$
from $x,y\in X = \text{conv}\{u_0,...,u_p\}$ we know
$\begin{array}{ll}x=\sum \lambda_{i} u_{i} & \sum \lambda_{i}=1 \quad \lambda_{i} \geqslant 0 \\ y=\sum \mu_{i} u_{i} & \sum{\mu_{i}=1}\quad \mu_{i} \geqslant 0\end{array}$
Show that $Q_t\in X, \forall\ 0\leq t\leq 1$
$\begin{aligned} Q_{t} &=x+t \cdot(y-x) \\ &=\sum \lambda_{i} u_{i}+t \cdot\left(\sum \mu_{i} u_{i}-\sum{\lambda_i} u_{i}\right) \\ &=\sum\left(t \mu_{i}+(1-t) \lambda_{i}\right) u_{i} \end{aligned}$
since $t \mu_{i}+\underbrace{(1-t)}_{\geqslant 0} \lambda i \geqslant 0$
and
$\begin{array}{l} \sum(t \mu_i+(1-t) \lambda_i) &=\sum( t \mu_{i}+\lambda_{i}-t \lambda_{i}) \\ &=t \cdot \underbrace{\sum \mu_{i}}_{=1}+\underbrace{\sum \lambda_{i}}_{=1}-t \cdot \underbrace{\sum \lambda_i}_{=1}=\Sigma \lambda_{i}=1 \end{array}$
Conclusion:
$Q_{t} \in \operatorname{cov}\left\{u_{0}, \cdots, u_{p}\right\} \quad \forall t$

Example M.II.1

(a) $conv\{u_0\}=\{u+0\}$
(b) line segment joining $u_{0}$ and $u_{1}$
$\begin{aligned} \operatorname{conv}\left\{u_{0}, u_{1}\right\} &=\left\{\lambda_{0} u_{0}+\lambda_{1} u_{1} \mid \lambda_{0}, \lambda_{1} \geq 0 \& \lambda_{0}+\lambda_{1}=1\right\} \\ &=\left\{u_{0}+\lambda_{1}\left(u_{1}-u_{0}\right) \mid 0 \leq \lambda_{1} \leq 1\right\} \\ &=\left[u_{0} u_{1}\right]\end{aligned}$
(c) Triangle with vertices $u_0,u_1$ and $u_2$ .
$\begin{aligned} \operatorname{conv}\left\{u_{0}, u_{1},u_2\right\} &=\left\{\lambda_{0} u_{0}+\lambda_{1} u_{1} +\lambda_2 u_{2} \mid \lambda_{0}, \lambda_{1}, \lambda_{2}\ \geq 0\ \&\ \lambda_{0}+\lambda_{1}+\lambda_{2}=1\right\} \\ &=\left\{u_{0}+\lambda_{1}(u_{1}-u_{0})+\lambda_{2}(u_2-u_0) \mid 0 \leq \lambda_{1},\lambda_{2} \leq 1\right\} \\ \end{aligned}$

M.II.2 Simplices

Definition M.II.2 Simplices

The $p$ -simplex $\sigma$ spanned by $p+1$ affinely independent points $u_{0}, \ldots, u_{p}$ i n $\mathbb{R}^{n}$ is defined to be the convex hull of the points, given by
$\sigma =\operatorname{conv}\left\{u_{0}, \ldots, u_{p}\right\} =\left\{x=\sum \lambda_{i} u_{i} \mid \lambda_{i} \in \mathbb{R} \text { and } \sum \lambda_{i}=1 \text { and } \lambda_{i} \geq 0\right\}$

The points $u_0,...,u_p$ are called the vertices of $\sigma$ .
The number $p$ is called the dimension of the simplex $\sigma$ and is denoted by $dim\ \sigma$ .

There are special names for simplices in small dimensions.

dimension	name
0	vertex
1	edge
2	triangle
3	tetrahedron

Barycentric Coordinates: The coefficients $\lambda_i$ in $x=\sum \lambda_i u_i$ are called the barycentric coordinates of the points $x$ in the simplex $\sigma$ with respect to the points $u_0,...,u_p$ .
They are uniquely determined by the point $x$ since the points $u_0,...,u_p$ are assumed to be affinely independent.

Standard p-simplex: The $p$ -simplex in $\mathbb{R^{p+1}}$ spanned by the $p+1$ unit vectors.

Using barycentric coordinates, the standard $p$ -simplex is mapped to any given $p$ -simplex that is spanned by the points $u_0,...,u_p$ in $\mathbb{R^n}$ by the following affine transformation:
$\mathbb{R}^{p+1} \ni\left(t_{0}, \ldots, t_{p}\right) \longmapsto \sum_{i=0}^{p} t_{i} u_{i} \in \mathbb{R}^{n}$
which defines a homeomorphism between the standard $p$ -simplex and the simplex spanned by the points $u_0,...,u_p$

Exercise 8: Prove that this map defines a homeomorphism between the standard $p-$ simplex and the simplex spanned by the points $u_0,...,u_p$
T is well defined:
T is continuous: (ex4) all affine transformation is continuous
T is bijective
T inverse is continuous?
:
T(U) open if U is open?
:

Face and Coface: A simplex $\tau$ spanned by a (proper) subset of the vertex set $\left\{u_{0}, \ldots, u_{p}\right\}$ is called a (proper) face of $\sigma$ , denoted by $\tau \leq \sigma(\tau<\sigma) .$
In this case, $\sigma$ is also called a (proper) coface of $\tau$ .

Boundary: The union of all proper faces of the simplex $\sigma$ is called the boundary of $\sigma$ , denoted by $bd\ \sigma$ .
Interior: Boundary's complement in $\sigma$ is the interior of $\sigma$ , denoted by $int\ \sigma$ .
The boundary and the interior of $\sigma$ are related by $int\ \sigma=\sigma \backslash bd\ \sigma$ .

Observation M.II.3 basic facts about simplices

(a) Every simplex is convex
convex hull is convex set: from ex7

(b) A $p$ -simplex has $2^{p+1}-1$ faces
p-simplex has p+1 vertices, means $2^{p+1}$ faces.

(c) For a $p$ -simplex $\sigma$ in $\mathbb{R^n}$ . If $n=p$ then the interior $int\ \sigma$ of $\sigma$ is an open subset of $\mathbb{R^n}$ .(No longer true if $n>p$ )
any point in $int\ \sigma$ , its distance to the boundary is larger than zero, then we can define a number smaller than the distance as the radius of the open ball

(d) The interior and the boundary in form of barycentric coordinates: A point $x=\sum \lambda_i u_i$ in $\sigma$ belongs to the unique face of $\sigma$ spanned by those vertices $u_i$ for which $\lambda_i>0$ .

In particular, we have
$x\in int\ \sigma$ if and only if $\lambda_i>0$ for all $0\leq i \leq p$
$x\in bd\ \sigma$ if and only if $\lambda_i=0$ for some $i\in \{0,...,p\}$

(e) For a $p$ -simplex, there is a homeomorphism $\sigma \cong B^{p}$ between $\sigma$ and the $p$ -ball $B^{p}$ that maps the boundary $\mathrm{bd}\ \sigma$ onto the $(p-1)$ -sphere $S^{p-1}$ .

image.png

M.II.3 Geometric Simplicial Complexes

Definition M.II.4: Geometric simplicial complex

A geometric simplcial comlpex $K$ in $\mathbb{R}^n$ is a finite collection of simplices in $\mathbb{R}^n$ with the following two properties.

(1) Every face of a simplex in $K$ is contained in $K$ . $(\sigma \in K\ \text{and}\ \tau \leq \sigma\ \Longrightarrow \tau \in K)$
(2) The intersection of any two simplices in $K$ is either empty or a face of each of them. $(\sigma,\sigma’ \in\ K\ \Longrightarrow\ either\ \sigma\ \cap\ \sigma'=\emptyset\$ or $\ \sigma\cap\ \sigma' \leq\ \sigma, \sigma')$

Dimension: $dim\ K$ is the maximum dimension of any its simplices. If $dim\ K=p$ , then $K$ is a geometric simplicial $p$ -complex.

(full) Subcomplex: A simplicial complex $L\subseteq K$ . It is said to be full when it contains all simplices in $K$ that are spanned by vertices in $L$ .

Skeleton: The subcomplex of $K$ consisting of all simplicies of dimension at most $p$ is called the $p$ -skeleton of $K$ and is denoted by $K^{(p)}=\{\sigma\in K\ |\ dim\ \sigma \leq p \}$ .
The 0-skelekton is also called vertex set of $K$ and is denoted by $Vert\ K:=K^{(0)}$

Underlying space: For a simplicial complex $K$ in $\mathbb{R^n}$ , the union of all its simplices is the underlying space of $K$ , denoted by $|K|:=\sigma_{1} \cup \sigma_{2} \cup \cdots \cup \sigma_{m} \subset \mathbb{R}^{n}$ .
By endowing a underlying space with the subspace topology induced from the standard topology on $\mathbb{R^n}$ , we can get a topological space, which is also called a polyhedron

Remark M.II.6

(a) A geometric simplicial complex $K$ is a subset of some ambient space $\mathbb{R^n}$ .
(b) Assume $K$ to be finite, because point cloud data sets are finite sets,
(c) often label the simplices in $K$

Example M.II.5 Geometric simplicial complexes

(a) Take a bunch of points $u_{0}, u_{1}, u_{2}, u_{3}, u_{4}$ and consider the collection $K=\left\{\left\{u_{0}\right\},\left\{u_{1}\right\},\left\{u_{2}\right\},\left\{u_{3}\right\},\left\{u_{4}\right\}\right\}$
(b) Take 2 points $u_0,u_1$ and consider the collection $K=\{\{u_0\},\{u_1\},conv\{u_0,u_1\}\}$
(c) Take 3 points $u_0,u_1,u_2$ and consider the collection $K=\{\{u_0\},\{u_1\},\{u_2\},conv\{u_0,u_1\},conv\{u_0,u_2\}\}$
(e) Not a simplicial complex $K=\left\{\left\{u_{0}\right\},\left\{u_{1}\right\},\left\{u_{2}\right\},\left\{u_{3}\right\}, \operatorname{conv}\left\{u_{0}, u_{1}\right\}, \operatorname{conv}\left\{u_{2}, u_{3}\right\}\right\}$

Definition M.II.7 Triangulation

A triangulation of a topological space $X$ is a geometric simplicial complex $K$ together with a homeomorphism $|K|\stackrel{\cong}{\longrightarrow}X$
A topological space is said to be triangulable if it admits a triangulaiton.

Example M.II.8 Triangulations

(a) The circle again

Exercise 10: Give 2 triangulations of the 2-sphere with different simplicial cimplexes. Write down explicity the simplices in each simplcial complex and draw it.

Remark M.II.9

Not every space admits a triangulation. But triangulations do exist as long as the topological space is 'sufficient nice'. Our example will always be of that sort.

If we admit infinite simplicial complexes, the following is true:

Every smooth manifold admits a triangulation
Topological manifolds always admit triangulations in dimensionat most 3

Topological Data Analyze M.II.1-3